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Acta Mathematicae Applicatae
Sinica, Chinese Series
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Current Issue
2012 Vol.28 Issue.1,
Published 20120115
ARTICLES
ARTICLES
1
Deterministic Timeinconsistent Optimal Control Problems — an Essentially Cooperative Approach
Jiongmin YONG
A general deterministic timeinconsistent optimal control problem is formulated for ordinary differential equations. To find a timeconsistent equilibrium value function and the corresponding timeconsistent equilibrium control, a noncooperative
N
person differential game (but essentially cooperative in some sense) is introduced. Under certain conditions, it is proved that the openloop Nash equilibrium value function of the
N
person differential game converges to a timeconsistent equilibrium value function of the original problem, which is the value function of a timeconsistent optimal control problem. Moreover, it is proved that any optimal control of the timeconsistent limit problem is a timeconsistent equilibrium control of the original problem.
2012 Vol. 28 (1): 130 [
Abstract
] (
1643
) [
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1KB] [
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1KB] (
1079
)
31
Asymptotic Estimates of GerberShiu Functions in the Renewal Risk Model with Exponential Claims
Li WEI
This paper continues to study the asymptotic behavior of GerberShiu expected discounted penalty functions in the renewal risk model as the initial capital becomes large. Under the assumption that the claimsize distribution is exponential, we establish an explicit asymptotic formula. Some straightforward consequences of this formula match existing results in the field.
2012 Vol. 28 (1): 3138 [
Abstract
] (
1520
) [
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1KB] [
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1KB] (
738
)
39
Causal Analysis of Futures Sugar Prices in Zhengzhou
Fang WANG, Xingwei TONG, Jing XU, Jianping CHEN
In this paper, we are interested in investigating the causal relationships among futures sugar prices in the Zhengzhou futures exchange market (ZF), the spot sugar prices in Zhengzhou (ZS) and the futures sugar prices in New York futures exchange market (NF). A useful tool called Bayesian network is introduced to analyze the problem. Since there are only three variables in our Bayesian network, the algorithm is straightforward: we display all the 25 possible network structures and adopt certain scoring metrics to evaluate them. We applied five different scoring metrics in total. Firstly, for each metric, we obtained 24 scores, each calculated from one of the 24 possible structures i.e. a Directed Acyclic Graph (DAG). Then we eliminated the network structure which represents the independence of the three variables according to our prior knowledge concerning the futures sugar market. After that, the optimal network structure which implies the causal relationships was selected according to the corresponding scoring metric. Finally, after comparing the results from different scoring metrics, we obtained the relatively affirmative conclusion that ZS causes ZF from both the Bayesian Dirichlet (BD) metric, Bayesian DirichletAkaike Information Criterion (BDAIC) metric, Bayesian DirichletBayesian Information Criterion (BDBIC) metric and Bayesian Information Criterion (BIC) metric. The conclusions that NF causes ZF and ZF causes ZS from the Akaike Information Criterion (AIC) metric and ZF causes ZS from the BIC metric were useful and significant to our investigation.
2012 Vol. 28 (1): 3948 [
Abstract
] (
1649
) [
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1KB] (
813
)
49
Heteroscedasticity and/or Autocorrelation Checks in Longitudinal Nonlinear Models with Elliptical and AR(1) Errors
ChunZheng CAO, JinGuan LIN
The aim of this paper is to study the tests for variance heterogeneity and/or autocorrelation in nonlinear regression models with elliptical and AR(1) errors. The elliptical class includes several symmetric multivariate distributions such as normal, Studentt, power exponential, among others. Several diagnostic tests using score statistics and their adjustment are constructed. The asymptotic properties, including asymptotic chisquare and approximate powers under local alternatives of the score statistics, are studied. The properties of test statistics are investigated through Monte Carlo simulations. A data set previously analyzed under normal errors is reanalyzed under elliptical models to illustrate our test methods.
2012 Vol. 28 (1): 4962 [
Abstract
] (
1556
) [
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1KB] [
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1KB] (
916
)
63
Analysis of
SE
^{τ}
IR
^{ω}
S
Epidemic Disease Models with Vertical Transmission in Complex Networks
Xia LIU, Deju XU
When the role of network topology is taken into consideration, one of the objectives is to understand the possible implications of topological structure on epidemic models. As most real networks can be viewed as complex networks, we propose a new delayed
SE
^{τ}
IR
^{ω}
S
epidemic disease model with vertical transmission in complex networks. By using a delayed ODE system, in a smallworld (SW) network we prove that, under the condition
R
_{0}
≤ 1, the diseasefree equilibrium (DFE) is globally stable. When
R
_{0}
> 1, the endemic equilibrium is unique and the disease is uniformly persistent. We further obtain the condition of local stability of endemic equilibrium for
R
_{0}
> 1. In a scalefree (SF) network we obtain the condition
R
_{1}
> 1 under which the system will be of nonzero stationary prevalence.
2012 Vol. 28 (1): 6374 [
Abstract
] (
1914
) [
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1KB] [
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1KB] (
922
)
75
Local Influence Analysis for Semiparametric Reproductive Dispersion Nonlinear Models
Xuedong CHEN, Niansheng TANG, XuerenWANG
The present paper proposes a semiparametric reproductive dispersion nonlinear model (SRDNM) which is an extension of the nonlinear reproductive dispersion models and the semiparameter regression models. Maximum penalized likelihood estimates (MPLEs) of unknown parameters and nonparametric functions in SRDNM are presented. Assessment of local influence for various perturbation schemes are investigated. Some local influence diagnostics are given. A simulation study and a real example are used to illustrate the proposed methodologies.
2012 Vol. 28 (1): 7590 [
Abstract
] (
1824
) [
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1KB] [
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1KB] (
911
)
91
On Exact Solutions to Partial Differential Equations by the Modified Homotopy Perturbation Method
Gang YANG, Ruyun CHEN, Luogen YAO
Based on the modified homotopy perturbation method (MHPM), exact solutions of certain partial differential equations are constructed by separation of variables and choosing the finite terms of a series in
p
as exact solutions. Under suitable initial conditions, the PDE is transformed into an ODE. Some illustrative examples reveal the efficiency of the proposed method.
2012 Vol. 28 (1): 9198 [
Abstract
] (
1919
) [
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1KB] [
PDF
1KB] (
978
)
99
Model Checking for a General Linear Model with Nonignorable Missing Covariates
Zhihua SUN, WaiCheung IP, Heung WONG
In this paper, we investigate the model checking problem for a general linear model with nonignorable missing covariates. We show that, without any parametric model assumption for the response probability, the least squares method yields consistent estimators for the linear model even if only the complete data are applied. This makes it feasible to propose two testing procedures for the corresponding model checking problem: a score type lackoffit test and a test based on the empirical process. The asymptotic properties of the test statistics are investigated. Both tests are shown to have asymptotic power 1 for local alternatives converging to the null at the rate
n
^{r}
, 0 ≤
r
< (1/2). Simulation results show that both tests perform satisfactorily.
2012 Vol. 28 (1): 99110 [
Abstract
] (
1638
) [
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1KB] [
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1KB] (
864
)
111
A Better Semionline Algorithm for Q3/s
_{1}
= s
_{2}
≤ s
_{3}
/C
_{min}
with the Known Largest Size
Shengyi CAI, Qifan YANG
This paper investigates the semionline machine covering problem on three special uniform machines with the known largest size. Denote by
s
_{j}
the speed of each machine,
j
= 1, 2, 3. Assume 0 <
s
_{1}
=
s
_{2}
=
r
≤
t
=
s
_{3}
, and let
s
=
t/r
be the speed ratio. An algorithm with competitive ratio max{2, (3
s
+6)/(
s
+6)} is presented. We also show the lower bound is at least max{2, (3
s
)/(
s
+6)}. For
s
≤ 6, the algorithm is an optimal algorithm with the competitive ratio 2. Besides, its overall competitive ratio is 3 which matches the overall lower bound. The algorithm and the lower bound in this paper improve the results of Luo and Sun.
2012 Vol. 28 (1): 111116 [
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1519
) [
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1KB] (
935
)
117
Some Numerical Quadrature Schemes of a Nonconforming Quadrilateral Finite Element
Xiaofei GUAN, Mingxia LI, Shaochun CHEN
Numerical quadrature schemes of a nonconforming finite element method for general second order elliptic problems in two dimensional (2D) and three dimensional (3D) space are discussed in this paper. We present and analyze some optimal numerical quadrature schemes. One of the schemes contains only three sampling points, which greatly improves the efficiency of numerical computations. The optimal error estimates are derived by using some traditional approaches and techniques. Lastly, some numerical results are provided to verify our theoretical analysis.
2012 Vol. 28 (1): 117126 [
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] (
1793
) [
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1KB] (
1054
)
127
ErdosKoRado Theorems of Labeled Sets
Xingbo GENG, Yushuang LI
For
k
= (
k
_{1}
, … ,
k
_{n}
) ∈
N
^{n}
, 1 ≤
k
_{1}
≤ … ≤
k
_{n}
, let
L
_{k}
^{r}
be the family of
labeled rsets
on
k
given by
L
_{k}
^{r}
:= {{(
a
_{1}
,
l
_{a1}
), … , (
a
_{r}
,
l
_{ar}
)} : {
a
_{1}
, … ,
a
_{r}
} ⊆ [
n
],
l
_{ai}
∈ [
k
_{ai}
], i = 1, … ,
r
}. A family
A
of labeled rsets is
intersecting
if any two sets in
A
intersect. In this paper we give the sizes and structures of intersecting families of labeled
r
sets.
2012 Vol. 28 (1): 127130 [
Abstract
] (
1451
) [
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1KB] [
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1KB] (
923
)
131
Stabilization of Nonuniform EulerBernoulli Beam with Locally Distributed Feedbacks
Xianbing CAO, Qingxu YAN
In this article, we study the stabilization problem of a nonuniform EulerBernoulli beam with locally distributed feedbacks. Firstly, using the semigroup theory, we establish the wellposedness of the associated closed loop system. Then by proving the uniqueness of the solution of a related ordinary differential equations, we derive the asymptotic stability of the closed loop system. Finally, by means of the piecewise frequency domain multiplier method, we prove that the corresponding closed loop system can be exponentially stabilized by only one of the two distributed feedback controls proposed in this paper.
2012 Vol. 28 (1): 131138 [
Abstract
] (
1759
) [
HTML
1KB] [
PDF
1KB] (
1007
)
139
The Additive Hazard Mixing Models
Ping LI, Xiaoliang LING
This paper is concerned with the aging and dependence properties in the additive hazard mixing models including some stochastic comparisons. Further, some useful bounds of reliability functions in additive hazard mixing models are obtained.
2012 Vol. 28 (1): 139148 [
Abstract
] (
1378
) [
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1KB] [
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1KB] (
879
)
149
Symmetry of the Point Spectrum of Infinite Dimensional Hamiltonian Operators and Its Applications
Hua WANG, Alatancang, Junjie HUANG
This paper studies the symmetry, with respect to the real axis, of the point spectrum of the upper triangular infinite dimensional Hamiltonian operator
H
. Note that the point spectrum of
H
can be described as
σ
_{p}
(
H
) =
σ
_{p}
(
A
)∪
σ
_{p}
^{1}
(
A
*). Using the characteristic of the set
σ
_{p}
^{1}
(
A
*), we divide the point spectrum
σ
_{p}
(
A
) of A into three disjoint parts. Then, a necessary and sufficient condition is obtained under which
σ
_{p}
^{1}
(
A
*) and one part of
σ
_{p}
(
A
) are symmetric with respect to the real axis each other. Based on this result, the symmetry of
σ
_{p}
(
H
) is completely given. Moreover, the above result is applied to thin plates on elastic foundation, plane elasticity problems and harmonic equations.
2012 Vol. 28 (1): 149156 [
Abstract
] (
1620
) [
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1KB] [
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1KB] (
983
)
157
Singular Quasilinear Elliptic Problems with Indefinite Weights and Critical Potential
Gao JIA, Qing ZHAO, Chunyan DAI
The Dirichlet problem for a quasilinear subcritical inhomogeneous elliptic equation with critical potential and singular coefficients, which has indefinite weights in
R
^{N}
, is studied in this paper. We discuss the corresponding eigenvalue problems by the variational techniques and Picone’s identity, and obtain the existence of nontrivial solutions for the inhomogeneous Dirichlet problem by using Hardy inequality, Mountain Pass Lemma in conjunction with the property of eigenvalues.
2012 Vol. 28 (1): 157164 [
Abstract
] (
1392
) [
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1KB] [
PDF
0KB] (
24
)
165
Random Attractors of Stochastic NonNewtonian Fluids
Chunxiao GUO, Boling GUO, Yongqian HAN
The present paper investigates the asymptotic behavior of solutions for stochastic nonNewtonian fluids in a twodimensional domain. Firstly, we prove the existence of random attractors Λ
_{H}
(
ω
) in H; Secondly, we prove the existence of random attractors Λ
_{V}
(
ω
) in
V
. Then we verify regularity of the random attractors by showing that Λ
_{H}
(
ω
) = Λ
_{V}
(
ω
), which implies the smoothing effect of the fluids in the sense that solution becomes eventually more regular than the initial data.
2012 Vol. 28 (1): 165180 [
Abstract
] (
1572
) [
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1KB] [
PDF
1KB] (
851
)
181
Exponential Stability Analysis of CohenGrossberg Neural Networks with Timevarying Delays
Yimin MENG, Lihong HUANG, Zhaohui YUAN
In this paper, we study CohenGrossberg neural networks (CGNN) with timevarying delay. Based on Halanay inequality and continuation theorem of the coincidence degree, we obtain some sufficient conditions ensuring the existence, uniqueness, and global exponential stability of periodic solution. Our results complement previously known results.
2012 Vol. 28 (1): 181192 [
Abstract
] (
1495
) [
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1KB] [
PDF
0KB] (
21
)
193
Ordering Graphs with Cut Edges by Their Spectral Radii
Kunfu FANG
Let
G
_{n}
^{k}
denote a set of graphs with
n
vertices and
k
cut edges. In this paper, we obtain an order of the first four graphs in
G
_{n}
^{k}
in terms of their spectral radii for 6 ≤
k
≤ (
n
2)/3 .
2012 Vol. 28 (1): 193200 [
Abstract
] (
1471
) [
HTML
1KB] [
PDF
1KB] (
840
)
201
Componentwise Complementary Cycles in Multipartite Tournaments
Zhihong HE, Guojun LI, Xueqin ZHOU
The problem of complementary cycles in tournaments and bipartite tournaments was completely solved. However, the problem of complementary cycles in semicomplete
n
partite digraphs with
n
≥ 3 is still open. Based on the definition of componentwise complementary cycles, we get the following result. Let
D
be a 2strong
n
partite (
n
≥ 6) tournament that is not a tournament. Let
C
be a 3cycle of
D
and
D
\
V
(
C
) be nonstrong. For the unique acyclic sequence
D
_{1}
,
D
_{2}
, … ,
D
_{α}
of
D
\
V
(
C
), where
α
≥ 2, let
D
_{c}
= {
D
_{i}

D
_{i}
contains cycles, i = 1, 2, … ,
α
},
D
_{c}
= {
D
_{1}
,
D
_{2}
, … ,
D
_{α}
} \
D
_{c}
. If
D
_{c}
≠ Ø, then
D
contains a pair of componentwise complementary cycles.
2012 Vol. 28 (1): 201208 [
Abstract
] (
1694
) [
HTML
1KB] [
PDF
1KB] (
980
)
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