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Acta Mathematicae Applicatae
Sinica, Chinese Series
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Current Issue
2018 Vol.34 Issue.1,
Published 20180215
ARTICLES
ARTICLES
1
Adjoining Batch Markov Arrival Processes of a Markov Chain
Xiaoyun MO, Xuyan XIANG, Xiangqun YANG
A batch Markov arrival process (BMAP)
X
^{*}
=(
N, J
) is a 2dimensional Markov process with two components, one is the counting process
N
and the other one is the phase process
J
. It is proved that the phase process is a timehomogeneous Markov chain with a finite statespace, or for short, Markov chain. In this paper, a new and inverse problem is proposed firstly:given a Markov chain
J
, can we deploy a process
N
such that the 2dimensional process
X
^{*}
=(
N, J
) is a BMAP? The process
X
^{*}
=(
N, J
) is said to be an adjoining BMAP for the Markov chain
J
. For a given Markov chain the adjoining processes exist and they are not unique. Two kinds of adjoining BMAPs have been constructed. One is the BMAPs with fixed constant batches, the other one is the BMAPs with independent and identically distributed (i.i.d) random batches. The method we used in this paper is not the usual matrixanalytic method of studying BMAP, it is a pathanalytic method. We constructed directly sample paths of adjoining BMAPs. The expressions of characteristic (
D
_{k}
,
k
=0, 1, 2…) and transition probabilities of the adjoining BMAP are obtained by the density matrix
Q
of the given Markov chain
J
. Moreover, we obtained two frontal Theorems. We present these expressions in the first time.
2018 Vol. 34 (1): 110 [
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11
Optimality Conditions of the Setvalued Optimization Problem with Generalized Cone Convex Setvalued Maps Characterized by Contingent Epiderivative
Zhiang ZHOU, Xinmin YANG, Qiusheng QIU
In this paper, firstly, a new notion of generalized cone convex setvalued map is introduced in real normed spaces. Secondly, a property of the generalized cone convex setvalued map involving the contingent epiderivative is obtained. Finally, as the applications of this property, we use the contingent epiderivative to establish optimality conditions of the setvalued optimization problem with generalized cone convex setvalued maps in the sense of Henig proper efficiency. The results obtained in this paper generalize and improve some known results in the literature.
2018 Vol. 34 (1): 1118 [
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] (
41
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19
Disjoint Cliques in Clawfree Graphs
Suyun JIANG, Jin YAN
A graph is said to be clawfree if it does not contain an induced subgraph isomorphic to
K
_{1,3}
. Let
s
and
k
be two integers with 0 ≤
s
≤
k
and let
G
be a clawfree graph of order
n
. In this paper, we investigate clique partition problems in clawfree graphs. It is proved that if
n
≥ 3
s
+4(
ks
) and
d
(
x
)+
d
(
y
) ≥
n
2
s
+2
k
+1 for any pair of nonadjacent vertices
x, y
of
G
, then
G
contains
s
disjoint
K
_{3}
s and
ks
disjoint
K
_{4}
s such that all of them are disjoint. Moreover, the degree condition is sharp in some cases.
2018 Vol. 34 (1): 1934 [
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] (
36
) [
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35
An Efficient Risk Estimator with External Information Under Additive Hazards Model
Xin WANG, Xiaoming XUE, Jie ZHOU, Liuquan SUN
Rare event data is encountered when the events of interest occur with low frequency, and the estimators based on the cohort data only may be inefficient. However, when external information is available for the estimation, the estimators utilizing external information can be more efficient. In this paper, we propose a method to incorporate external information into the estimation of the baseline hazard function and improve efficiency for estimating the absolute risk under the additive hazards model. The resulting estimators are shown to be uniformly consistent and converge weakly to Gaussian processes. Simulation studies demonstrate that the proposed method is much more efficient. An application to a bone marrow transplant data set is provided.
2018 Vol. 34 (1): 3550 [
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] (
23
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51
Bayesian Planning of Optimal Stepstress Accelerated Life Test for Loglocationscale Distributions
Qiang GUAN, Yincai TANG
This paper introduces some Bayesian optimal design methods for stepstress accelerated life test planning with one accelerating variable, when the acceleration model is linear in the accelerated variable or its function, based on censored data from a loglocationscale distributions. In order to find the optimal plan, we propose different Monte Carlo simulation algorithms for different Bayesian optimal criteria. We present an example using the lognormal life distribution with TypeI censoring to illustrate the different Bayesian methods and to examine the effects of the prior distribution and sample size. By comparing the different Bayesian methods we suggest that when the data have large(small) sample size
B
_{1}
(
τ
) (
B
_{2}
(
τ
)) method is adopted. Finally, the Bayesian optimal plans are compared with the plan obtained by maximum likelihood method.
2018 Vol. 34 (1): 5164 [
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] (
24
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65
A Geometric Flow Approach for Regionbased Image Segmentationtheoretical Analysis
Zhucui JING, Juntao YE, Guoliang XU
In this paper, we analyze the wellposedness of an image segmentation model. The main idea of that segmentation model is to minimize one energy functional by evolving a given piecewise constant image towards the image to be segmented. The evolution is controlled by a serial of mappings, which can be represented by Bspline basis functions. The evolution terminates when the energy is below a given threshold. We prove that the correspondence between two images in the segmentation model is an injective and surjective mapping under appropriate conditions. We further prove that the solution of the segmentation model exists using the direct method in the calculus of variations. These results provide the theoretical support for that segmentation model.
2018 Vol. 34 (1): 6576 [
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77
Asymptotic Properties of Wavelet Estimators in Partially Linear Errorsinvariables Models with Longmemory Errors
Hongchang HU, Hengjian CUI, Kaican LI
While the random errors are a function of Gaussian random variables that are stationary and long dependent, we investigate a partially linear errorsinvariables (EV) model by the wavelet method. Under general conditions, we obtain asymptotic representation of the parametric estimator, and asymptotic distributions and weak convergence rates of the parametric and nonparametric estimators. At last, the validity of the wavelet method is illuminated by a simulation example and a real example.
2018 Vol. 34 (1): 7796 [
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25
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97
Ergodicity of the 2D NavierStokes Equations with Degenerate Multiplicative Noise
Zhao DONG, Xuhui PENG
Consider the twodimensional, incompressible NavierStokes equations on torus
T
^{2}
=[
π, π
]
^{2}
driven by a degenerate multiplicative noise in the vorticity formulation (abbreviated as SNS):
dw
_{t}
=
ν
△
w
_{t}
dt
+
B
(
Kw
_{t}
,
w
_{t}
)
dt
+
Q
(
w
_{t}
)
dW
_{t}
. We prove that the solution to SNS is continuous differentiable in initial value. We use the Malliavin calculus to prove that the semigroup {
P
_{t}
}
_{t}
>0 generated by the SNS is asymptotically strong Feller. Moreover, we use the coupling method to prove that the solution to SNS has a weak form of irreducibility. Under almost the same Hypotheses as that given by Odasso,
Prob. Theory Related Fields
, 140:4182 (2005) with a different method, we get an exponential ergodicity under a stronger norm.
2018 Vol. 34 (1): 97118 [
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] (
24
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119
A Primaldual Largeupdate Interiorpoint Algorithm for
P
_{*}
(
κ
)LCP Based on a New Class of Kernel Functions
Ping JI, Mingwang ZHANG, Xin LI
In this paper, we propose a largeupdate primaldual interior point algorithm for
P
_{*}
(
κ
)linear complementarity problem. The method is based on a new class of kernel functions which is neither classical logarithmic function nor selfregular functions. It is determines both search directions and the proximity measure between the iterate and the center path. We show that if a strictly feasible starting point is available, then the new algorithm has
O
(1+2
κ
)
p
√
n
(1/
p
log
n
+1)
^{2}
log
n/ε
iteration complexity which becomes
O
((1+2
κ
)√
n
log
n
log
n/ε
) with special choice of the parameter
p
. It is matches the currently best known iteration bound for
P
_{*}
(
κ
)linear complementarity problem. Some computational results have been provided.
2018 Vol. 34 (1): 119134 [
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] (
25
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135
Neighbor Sum Distinguishing Chromatic Index of Sparse Graphs via the Combinatorial Nullstellensatz
Xiaowei YU, Yuping GAO, Laihao DING
Let
φ
:
E
(
G
) → {1, 2, …,
k
} be an edge coloring of a graph
G
. A proper edge
k
coloring of
G
is called neighbor sum distinguishing if ∑
_{e∋u}
φ
(
e
)≠∑
_{e∋v}
φ
(
e
) for each edge
uv
∈
E
(
G
). The smallest value
k
for which
G
has such a coloring is denoted by
χ
'
_{∑}
(
G
), which makes sense for graphs containing no isolated edge (we call such graphs normal). It was conjectured by Flandrin et al. that
χ
'
_{∑}
(
G
) ≤ △(
G
) + 2 for all normal graphs, except for
C
_{5}
. Let mad(
G
)=max { (2
E
(
H
))/(
V
(
H
))
H
⊆
G
} be the maximum average degree of
G
. In this paper, we prove that if
G
is a normal graph with △(
G
) ≥ 5 and mad(
G
) < 3 2/△(
G
), then
χ
'
_{∑}
(
G
) ≤ △(
G
) + 1. This improves the previous results and the bound △(
G
) + 1 is sharp.
2018 Vol. 34 (1): 135144 [
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27
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145
Meansquare Stability of Stochastic Agedependent Delay Population Systems with Jumps
Qiang LI, Qimin ZHANG, Boqiang CAO
In this paper, we present the compensated stochastic
θ
method for stochastic agedependent delay population systems (SADDPSs) with Poisson jumps. The definition of meansquare stability of the numerical solution is given and a sufficient condition for meansquare stability of the numerical solution is derived. It is shown that the compensated stochastic
θ
method inherits stability property of the numerical solutions. Finally, the theoretical results are also confirmed by a numerical experiment.
2018 Vol. 34 (1): 145154 [
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30
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155
Em Algorithm of the Truncated Multinormal Distribution with Linear Restriction on the Variables
Baisuo JIN, Jingjing HAN, Shu DING, Baiqi MIAO
A new expectationmaximization (EM) algorithm is proposed to estimate the parameters of the truncated multinormal distribution with linear restriction on the variables. Compared with the generalized method of moments (GMM) estimation and the maximum likelihood estimation (MLE) for the truncated multivariate normal distribution, the EM algorithm features in fast calculation and high accuracy which are shown in the simulation results. For the real data of the national college entrance exams (NCEE), we estimate the distribution of the NCEE examinees' scores in Anhui, 2003, who were admitted to the university of science and technology of China (USTC). Based on our analysis, we have also given the ratio truncated by the NCEE admission line of USTC in Anhui, 2003.
2018 Vol. 34 (1): 155162 [
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26
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163
Hosoya and MerrifieldSimmons Indices in Random Polyphenyl Chains
Weiling YANG
The Hosoya index of a graph is the total number of matchings in it. And the MerrifieldSimmons index is the total number of independent sets in it. They are typical examples of graph invariants used in mathematical chemistry for quantifying relevant details of molecular structure. In this paper, we obtain explicit analytical expressions for the expectations of the Hosoya index and the MerrifieldSimmons index of a random polyphenyl chain.
2018 Vol. 34 (1): 163172 [
Abstract
] (
23
) [
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173
On Local Spectral Properties of Hamilton Type Operators
Junli SHEN, Alatancang
In this paper, we introduce the class of Hamilton type operators and study various properties of this class. We show that every Hamilton type operator with property (
β
) or (
δ
) is decomposable. In addition, we prove that a Hamilton type operator
T
satisfies property (
β
), Dunford's property (
C
) and Weyl's theorem if and only if its adjoint does.
2018 Vol. 34 (1): 173182 [
Abstract
] (
28
) [
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183
Secondorder Optimality Conditions for Conesubarcwise Connected Setvalued Optimization Problems
Zhenhua PENG, Yihong XU
The concept of a cone subarcwise connected setvalued map is introduced. Several examples are given to illustrate that the cone subarcwise connected setvalued map is a proper generalization of the cone arcwise connected setvalued map, as well as the arcwise connected set is a proper generalization of the convex set, respectively. Then, by virtue of the generalized secondorder contingent epiderivative, secondorder necessary optimality conditions are established for a point pair to be a local global proper efficient element of setvalued optimization problems. When objective function is cone subarcwise connected, a secondorder sufficient optimality condition is also obtained for a point pair to be a global proper efficient element of setvalued optimization problems.
2018 Vol. 34 (1): 183196 [
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28
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197
A Distance Function for Computing on Finite Subsets of Euclidean Spaces
Hajar GhahremaniGol, Farzad Didehvar, Asadollah Razavi
In practical purposes for some geometrical problems, specially the fields in common with computer science, we deal with information of some finite number of points. The problem often arises here is:"How are we able to define a plausible distance function on a finite three dimensional space?" In this paper, we define such a distance function in order to apply it to further purposes, e.g. in the field settings of transportation theory and geometry. More precisely, we present a new model for traveling salesman problem and vehicle routing problem for two dimensional manifolds in three dimensional Euclidean space, the second problem on which we focus on this line is, three dimensional triangulation.
2018 Vol. 34 (1): 197208 [
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33
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