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Regression Analysis for the Additive Hazards Model with General Biased Survival Data
Xiaolin CHEN
Acta Mathematicae Applicatae Sinica(English Series). 2020, 36(3): 545556.
DOI:
10.1007/s1025502009499
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In survival analysis, data are frequently collected by some complex sampling schemes, e.g., length biased sampling, casecohort sampling and so on. In this paper, we consider the additive hazards model for the general biased survival data. A simple and unified estimating equation method is developed to estimate the regression parameters and baseline hazard function. The asymptotic properties of the resulting estimators are also derived. Furthermore, to check the adequacy of the fitted model with general biased survival data, we present a test statistic based on the cumulative sum of the martingaletype residuals. Simulation studies are conducted to evaluate the performance of proposed methods, and applications to the shrub and Welsh Nickel Refiners datasets are given to illustrate the methodology.
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Partial Regularity of Suitable Weak Solutions to the System of the Incompressible Shearthinning Flow
Yazhou CHEN, Hailiang LI, Xiaoding SHI
Acta Mathematicae Applicatae Sinica(English Series). 2021, 37(2): 348363.
DOI:
10.1007/s1025502110112
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This paper is devoted to the partial regularity of suitable weak solutions to the system of the incompressible shearthinning flow in a bounded domain Ω ⊂ R
^{n}
,
n
≥ 2. It is proved that there exists a suitable weak solution of the shearthinning fluid in the
n
D smooth bounded domain (for
n
≥ 2). For 3D model, it is proved that the singular points are concentrated on a closed set whose 1 dimensional Hausdorff measure is zero.
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A New Nonmonotone Trust Region BarzilaiBorwein Method for Unconstrained Optimization Problems
Xing LI, Wenli DONG, Zheng PENG
Acta Mathematicae Applicatae Sinica(English Series). 2021, 37(1): 166175.
DOI:
10.1007/s1025502109979
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147
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In this paper, we propose a new nonmonotone trust region BarzilaiBorwein (BB for short) method for solving unconstrained optimization problems. The proposed method is given by a novel combination of a modified Metropolis criterion, BBstepsize and trust region method. The new method uses the reciprocal of BBstepsize to approximate the Hessian matrix of the objective function in the trust region subproblems, and accepts some bad solutions according to the modified Metropolis criterion based on simulated annealing idea. Under some suitable assumptions, the global convergence of the new method is established. Some preliminary numerical results indicate that, the new method is more efficient compared with the existing trust region BB method.
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Convergence of Stochastic Gradient Descent in Deep Neural Network
Baicun ZHOU, Congying HAN, Tiande GUO
Acta Mathematicae Applicatae Sinica(English Series). 2021, 37(1): 126136.
DOI:
10.1007/s1025502109912
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198
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Stochastic gradient descent (SGD) is one of the most common optimization algorithms used in pattern recognition and machine learning. This algorithm and its variants are the preferred algorithm while optimizing parameters of deep neural network for their advantages of low storage space requirement and fast computation speed. Previous studies on convergence of these algorithms were based on some traditional assumptions in optimization problems. However, the deep neural network has its unique properties. Some assumptions are inappropriate in the actual optimization process of this kind of model. In this paper, we modify the assumptions to make them more consistent with the actual optimization process of deep neural network. Based on new assumptions, we studied the convergence and convergence rate of SGD and its two common variant algorithms. In addition, we carried out numerical experiments with LeNet5, a common network framework, on the data set MNIST to verify the rationality of our assumptions.
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Blow up of Solutions for a Nonlinear Petrovsky Type Equation with Timedependent Coefficients
Xiaoxiao ZHENG, Yadong SHANG, Xiaoming PENG
Acta Mathematicae Applicatae Sinica(English Series). 2020, 36(4): 836846.
DOI:
10.1007/s1025502009846
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In this paper, we study a nonlinear Petrovsky type equation with nonlinear weak damping, a superlinear source and timedependent coefficients
utt
+ △
^{2}
u
+
k
_{1}
(
t
)
u
_{t}

^{m2}
u
_{t}
=
k
_{2}
(
t
)
u

^{p2}
u
,
x
∈ Ω,
t
> 0,
where Ω is a bounded domain in
R
^{n}
. Under certain conditions on
k
_{1}
(
t
),
k
_{2}
(
t
) and the initialboundary data, the upper bound for blowup time of the solution with negative initial energy function is given by means of an auxiliary functional and an energy estimate method if
p
>
m
. Also, a lower bound of blowup time are obtained by using a Sobolevtype inequality and a first order differential inequality technique for
n
= 2, 3 and
n
> 4.
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An Efficient Parameterized Logarithmic Kernel Function for Semidefinite Optimization
Louiza DERBAL, Zakia KEBBICHE
Acta Mathematicae Applicatae Sinica(English Series). 2020, 36(3): 753770.
DOI:
10.1007/s102550200955y
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In this paper, we present a primaldual interior point algorithm for semidefinite optimization problems based on a new class of kernel functions. These functions constitute a combination of the classic kernel function and a barrier term.
We derive the complexity bounds for large and smallupdate methods respectively. We show that the best result of iteration bounds for large and smallupdate methods can be achieved, namely
O
(
q
√
n
(log √
n
)
^{q+1/q}
log
n/ε
) for largeupdate methods and
O
(
q
^{3/2}
(log √
q
)
^{q+1/q}
√
n
log
n/ε
) for smallupdate methods.
We test the efficiency and the validity of our algorithm by running some computational tests, then we compare our numerical results with results obtained by algorithms based on different kernel functions.
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Mallows Model Averaging Estimation for Linear Regression Model with Right Censored Data
Zhongqi LIANG, Xiaolin CHEN, Yanqiu ZHOU
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(1): 523.
DOI:
10.1007/s1025502210530
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47
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This paper is concerned with an optimal model averaging estimation for linear regression model with right censored data. The weights for model averaging are picked up via minimizing the Mallows criterion. Under some mild conditions, it is shown that the identified weights possess the property of asymptotic optimality, that is, the model averaging estimator corresponding to these weights achieves the lowest squared error asymptotically. Some numerical studies are conducted to evaluate the finitesample performance of our method and make comparisons with its intuitive competitors, while an application to the PBC dataset is provided to serve as an illustration.
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The Analysis of Impact of Brexit on the PostBrexit EU Using Intervented Multivariate Time Series
Yu TIAN, Shaopei MA, Rongxiang RUI, Zhen YU, Maozai TIAN
Acta Mathematicae Applicatae Sinica(English Series). 2021, 37(3): 441458.
DOI:
10.1007/s102550211022z
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The UK is the most important partner of the EU in terms of economic and other fields due to the geographical proximity. It was one of the largest economies in the EU and its per capita income is higher than the EU average, so it is a net contributor to the EU. With UKs membership of the EU ended on 31 January 2019, there are concerns that the Brexit may have a significant impact on the EU, resulting in social, economic, political, and institutional changes, etc. in EU. While the impact of Brexit on the UK has always been the subject of considerable scholarly interest in recent years, there is relatively little literature on the impact of Brexit on the EU. This paper focuses on the evaluation of the impact of Brexit on the EU economy and other relevant aspects along three dimensions: GDP, PPP, Quarterly GDP growth. Employing powerful quantitative analysis technology that includes vector autoregression model, multivariate time series model with intervention variables, and autoregression integrated moving average, this paper obtains the important and novel evidence about the potential impact of Brexit on the EU economy, pointing out that Brexit is of farreaching significance to the EU. This analysis uses several statistical models to screen out several key influencing factors, which can be used to predict the total GDP of EU in the next five years. The results show that EU economy will react negatively to "nodeal" Brexit, and its growth rate of economy will slow down significantly in next 5 years. Finally, we put forward relevant policy suggestions on how to deal with the negative impact of Brexit on EU.
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Existence and Properties of Solutions for a Class of Fractional Differential Equations
Yongqiang XU, Shuhong CHEN, Zhong TAN
Acta Mathematicae Applicatae Sinica(English Series). 2021, 37(3): 477484.
DOI:
10.1007/s1025502110259
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205
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In this paper, we consider the initial value problem of a class of fractional differential equations. Firstly, we obtain the existence and uniqueness of the solutions by using Picard’s method of successive approximation. Then we discuss the dependence of the solutions on the initial value.
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Antimagicness of Lexicographic Product Graph
G
[
P
_{n}
]
Yingyu LU, Guanghua DONG, Ning WANG
Acta Mathematicae Applicatae Sinica(English Series). 2020, 36(3): 603619.
DOI:
10.1007/s1025502009530
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110
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Hartsfield and Ringel conjectured that every connected graph other than
K
_{2}
is antimagic. Since then, many classes of graphs have been proved to be antimagic. But few is known about the antimagicness of lexicographic product graphs. In this paper, via the construction of a directed Eulerian circuit, the Siamese method, and some modification on graph labeling, the antimagicness of lexicographic product graph
G
[
P
_{n}
] is obtained.
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A Generalization of Implicit Orecondition for Hamiltonicity of
k
connected Graphs
Junqing CAI, Lin WANG
Acta Mathematicae Applicatae Sinica(English Series). 2020, 36(3): 620626.
DOI:
10.1007/s102550200956x
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In 2005, Flandrin et al. proved that if
G
is a
k
connected graph of order
n
and
V
(
G
)=
X
_{1}
∪
X
_{2}
∪ … ∪
X
_{k}
such that
d
(
x
)+
d
(
y
) ≥
n
for each pair of nonadjacent vertices
x, y
∈
X
_{i}
and each
i
with
i
=1, 2, …,
k
, then
G
is hamiltonian. In order to get more sufficient conditions for hamiltonicity of graphs, Zhu, Li and Deng proposed the definitions of two kinds of implicit degree of a vertex
v
, denoted by
id
_{1}
(
v
) and
id
_{2}
(
v
), respectively. In this paper, we are going to prove that if
G
is a
k
connected graph of order
n
and
V
(
G
)=
X
_{1}
∪
X
_{2}
∪ … ∪
X
_{k}
such that
id
_{2}
(
x
) +
id
_{2}
(
y
) ≥
n
for each pair of nonadjacent vertices
x, y
∈
X
_{i}
and each
i
with
i
=1, 2, …,
k
, then
G
is hamiltonian.
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Harnack Inequality and Applications for SDEs Driven by
G
Brownian Motion
Fenfen YANG
Acta Mathematicae Applicatae Sinica(English Series). 2020, 36(3): 627635.
DOI:
10.1007/s1025502009579
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86
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In this paper, Wang's Harnack and shift Harnack inequality for a class of stochastic differential equations driven by
G
Brownian motion are established. The results generalize the ones in the linear expectation setting. Moreover, some applications are also given.
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Acyclic Edge Coloring of ICplanar Graphs
Wenyao SONG, Yuanyuan DUAN, Juan WANG, Lianying MIAO
Acta Mathematicae Applicatae Sinica(English Series). 2020, 36(3): 581589.
DOI:
10.1007/s1025502009503
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108
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A proper edge coloring of a graph
G
is acyclic if there is no 2colored cycle in
G
. The
acyclic chromatic index
of
G
is the least number of colors such that
G
has an acyclic edge coloring and denoted by
χ
'
_{a}
(
G
). An ICplane graph is a topological graph where every edge is crossed at most once and no two crossed edges share a vertex. In this paper, it is proved that
χ
'
_{a}
(
G
) ≤ △(
G
) + 10, if
G
is an ICplanar graph without adjacent triangles and
χ
'
_{a}
(
G
) ≤ △(
G
) + 8, if
G
is a trianglefree ICplanar graph.
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Nullfree False Discovery Rate Control Using Decoy Permutations
Kun HE, Mengjie LI, Yan FU, Fuzhou GONG, Xiaoming SUN
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(2): 235253.
DOI:
10.1007/s1025502210775
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26
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The traditional approaches to false discovery rate (FDR) control in multiple hypothesis testing are usually based on the null distribution of a test statistic. However, all types of null distributions, including the theoretical, permutationbased and empirical ones, have some inherent drawbacks. For example, the theoretical null might fail because of improper assumptions on the sample distribution. Here, we propose a null distributionfree approach to FDR control for multiple hypothesis testing in the casecontrol study. This approach, named
targetdecoy procedure
, simply builds on the ordering of tests by some statistic or score, the null distribution of which is not required to be known. Competitive decoy tests are constructed from permutations of original samples and are used to estimate the false target discoveries. We prove that this approach controls the FDR when the score function is symmetric and the scores are independent between different tests. Simulation demonstrates that it is more stable and powerful than two popular traditional approaches, even in the existence of dependency. Evaluation is also made on two real datasets, including an arabidopsis genomics dataset and a COVID19 proteomics dataset.
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The Proportional Mean Residual Life Regression Model with Cure Fraction and Auxiliary Covariate
Shaojia JIN, Yanyan LIU, Guangcai MAO, Mingyu SHAN
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(2): 312323.
DOI:
10.1007/s1025502210784
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15
)
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As biological studies become more expensive to conduct, it is a frequently encountered question that how to take advantage of the available auxiliary covariate information when the exposure variable is not measured. In this paper, we propose an induced cure rate mean residual life time regression model to accommodate the survival data with cure fraction and auxiliary covariate, in which the exposure variable is only assessed in a validation set, but a corresponding continuous auxiliary covariate is ascertained for all subjects in the study cohort. Simulation studies elucidate the practical performance of the proposed method under finite samples. As an illustration, we apply the proposed method to a heart disease data from the Study of Left Ventricular Dysfunction.
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Least Squares Model Averaging Based on Generalized Cross Validation
Xinmin LI, Guohua ZOU, Xinyu ZHANG, Shangwei ZHAO
Acta Mathematicae Applicatae Sinica(English Series). 2021, 37(3): 495509.
DOI:
10.1007/s102550211024x
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161
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Frequentist model averaging has received much attention from econometricians and statisticians in recent years. A key problem with frequentist model average estimators is the choice of weights. This paper develops a new approach of choosing weights based on an approximation of generalized cross validation. The resultant least squares model average estimators are proved to be asymptotically optimal in the sense of achieving the lowest possible squared errors. Especially, the optimality is built under both discrete and continuous weigh sets. Compared with the existing approach based on Mallows criterion, the conditions required for the asymptotic optimality of the proposed method are more reasonable. Simulation studies and real data application show good performance of the proposed estimators.
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Numerical Approximation of Stochastic Theta Method for Random Periodic Solution of Stochastic Differential Equations
Rong WEI, Chuanzhong CHEN
Acta Mathematicae Applicatae Sinica(English Series). 2020, 36(3): 689701.
DOI:
10.1007/s1025502009597
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101
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In this paper, we make use of stochastic theta method to study the existence of the numerical approximation of random periodic solution. We prove that the error between the exact random periodic solution and the approximated one is at the 1/4 order time step in mean sense when the initial time tends to ∞.
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Construction of Normal Bimagic Squares of Order 2
u
Wen LI, Fengchu PAN, Guangzhou CHEN
Acta Mathematicae Applicatae Sinica(English Series). 2020, 36(4): 771789.
DOI:
10.1007/s1025502009775
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107
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An
n
×
n
matrix A consisting of nonnegative integers is
a general magic square of order
n
if the sum of elements in each row, column, and main diagonal is the same. A general magic square
A
of order
n
is called
a magic square
, denoted by MS(
n
), if the entries of A are distinct. A magic square A of order
n
is
normal
if the entries of
A
are
n
^{2}
consecutive integers. Let
A
^{*d}
denote the matrix obtained by raising each element of
A
to the
d
th power. The matrix
A
is a
d
multimagic square, denoted by MS(
n, d
), if
A
^{*e}
is an MS(
n
) for 1 ≤
e
≤
d
. In this paper we investigate the existence of normal bimagic squares of order 2
u
and prove that there exists a normal bimagic square of order 2
u
, where
u
and 6 are coprime and
u
≥ 5.
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An Explicit Lower Bound for Blow Up Time in a Class of Nonlinear Wave Equations with Nonlinear Damping and Source Terms
Xiaoming PENG, Yadong SHANG, Xueqin WANG
Acta Mathematicae Applicatae Sinica(English Series). 2021, 37(1): 148154.
DOI:
10.1007/s102550210995y
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109
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This paper deals with an initial boundary value problem for a class of nonlinear wave equation with nonlinear damping and source terms whose solution may blow up in finite time. An explicit lower bound for blow up time is determined by means of a differential inequality argument if blow up occurs.
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A Necessary and Sufficient Condition for the Solvability of the Nonlinear Schrödinger Equation on a Finite Interval
Ruomeng LI, Xianguo GENG
Acta Mathematicae Applicatae Sinica(English Series). 2021, 37(1): 75100.
DOI:
10.1007/s102550210994z
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195
)
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The admissibility of the initialboundary data, which characterizes the existence of solution for the initialboundary value problem, is important. Based on the Fokas method and the inverse scattering transformation, an approach is developed to solve the initialboundary value problem of the nonlinear Schrödinger equation on a finite interval. A necessary and sufficient condition for the admissibility of the initialboundary data is given, and the reconstruction of the potential is obtained.
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Maximum Likelihood Estimator of the Location Parameter under Moving Extremes Ranked Set Sampling Design
Wangxue CHEN, Chunxian LONG, Rui YANG, Dongsen YAO
Acta Mathematicae Applicatae Sinica(English Series). 2021, 37(1): 101108.
DOI:
10.1007/s1025502109988
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222
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Cost effective sampling design is a problem of major concern in some experiments especially when the measurement of the characteristic of interest is costly or painful or time consuming. In the current paper, a modification of ranked set sampling (RSS) called moving extremes RSS (MERSS) is considered for the estimation of the location parameter for location family. A maximum likelihood estimator (MLE) of the location parameter for this family is studied and its properties are obtained. We prove that the MLE is an equivariant estimator under location transformation. In order to give more insight into the performance of MERSS with respect to (w.r.t.) simple random sampling (SRS), the asymptotic efficiency of the MLE using MERSS w.r.t. that using SRS is computed for some usual location distributions. The relative results show that the MLE using MERSS can be real competitors to the MLE using SRS.
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Continuity of HardyLittlewood Maximal Function
Di WU, Dunyan YAN
Acta Mathematicae Applicatae Sinica(English Series). 2020, 36(4): 982990.
DOI:
10.1007/s102350200983z
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91
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In the paper, we characterize a necessary and sufficient condition which ensures the continuities of the noncentered HardyLittlewood maximal function
Mf
and the centered HardyLittlewood maximal function
M
_{c}
f
on R
^{n}
. As two applications, we can easily deduce that
M
_{c}
f
and
Mf
are continuous if
f
is continuous, and
Mf
is continuous if
f
is of local bounded variation on R.
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Number of Induced Matchings of Graphs
Yan CHEN, Yan LIU
Acta Mathematicae Applicatae Sinica(English Series). 2021, 37(1): 3547.
DOI:
10.1007/s102550210996x
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135
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A matching
M
of a graph
G
is an induced matching if no two edges in
M
are joined by an edge of
G
. Let
iz
(
G
) denote the total number of induced matchings of
G
, named izindex. It is well known that the Hosoya index of a graph is the total number of matchings and the Hosoya index of a path can be calculated by the Fibonacci sequence. In this paper, we investigate the
iz
index of graphs by using the FibonacciNarayana sequence and characterize some types of graphs with minimum and maximum
iz
index, respectively.
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A Note on Harnack Type Inequality for the Gaussian Curvature Flow
Caipeng CHEN, Hongxin GUO, Chengzhe ZHU
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(1): 14.
DOI:
10.1007/s1025502210668
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In this short note we present a new Harnack expression for the Gaussian curvature flow, which is modeled from the shrinking self similiar solutions. As applications we give alternate proofs of Chow's Harnack inequality and entropy estimate.
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Projectionbased Consistent Test for Linear Regression Model with Missing Response and Covariates
Sujin ZHENG, Siyu GAO, Zhihua SUN
Acta Mathematicae Applicatae Sinica(English Series). 2020, 36(4): 917935.
DOI:
10.1007/s1025502009766
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89
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In recent years, there has been a large amount of literature on missing data. Most of them focus on situations where there is only missingness in response or covariate.
In this paper, we consider the adequacy check for the linear regression model with the response and covariates missing simultaneously.
We apply model adjustment and inverse probability weighting methods to deal with the missingness of response and covariate, respectively. In order to avoid the curse of dimension, we propose an empirical process test with the linear indicator weighting function. The asymptotic properties of the proposed test under the null, local and global alternative hypothetical models are rigorously investigated. A consistent wild bootstrap method is developed to approximate the critical value. Finally, simulation studies and real data analysis are performed to show that the proposed method performed well.
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List Edge Coloring of Outer1planar Graphs
Xin ZHANG
Acta Mathematicae Applicatae Sinica(English Series). 2020, 36(3): 737752.
DOI:
10.1007/s1025502009405
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110
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A graph is outer1planar if it can be drawn in the plane so that all vertices are on the outer face and each edge is crossed at most once. It is known that the list edge chromatic number
χ
'
_{l}
(
G
) of any outer1planar graph
G
with maximum degree △(
G
) ≥ 5 is exactly its maximum degree. In this paper, we prove
χ
'
_{l}
(
G
)=△(
G
) for outer1planar graphs
G
with △(
G
)=4 and with the crossing distance being at least 3.
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Semiparametric Likelihoodbased Inference for Censored Data with Auxiliary Information from External Massive Data Sources
Yuexin FANG, Yong ZHOU
Acta Mathematicae Applicatae Sinica(English Series). 2020, 36(3): 642656.
DOI:
10.1007/s102550200948x
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83
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Published auxiliary information can be helpful in conducting statistical inference in a new study. In this paper, we synthesize the auxiliary information with semiparametric likelihoodbased inference for censoring data with the total sample size is available. We express the auxiliary information as constraints on the regression coefficients and the covariate distribution, then use empirical likelihood method for general estimating equations to improve the efficiency of the interested parameters in the specified model. The consistency and asymptotic normality of the resulting regression parameter estimators established. Also numerical simulation and application with different supposed conditions show that the proposed method yields a substantial gain in efficiency of the interested parameters.
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The Stability of Nontrivial Positive Steady States for the SKT Model with Large Cross Diffusion
Qing LI, Qian XU
Acta Mathematicae Applicatae Sinica(English Series). 2020, 36(3): 657669.
DOI:
10.1007/s1025502009512
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101
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This paper is concerned with the existence and stability of steady state solutions for the SKT biological competition model with crossdiffusion.
By applying the detailed spectral analysis and in virtue of the bifurcating direction to the limiting system as the cross diffusion rate tends to infinity, it is proved the stability/instability of the nontrivial positive steady states with some special bifurcating structure.
Further, the existence and stability/instability of the corresponding nontrivial positive steady states for the original crossdiffusion system are proved by applying perturbation argument.
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Perfect Matching in
k
partite
k
graphs and 3uniform HMbipartite Hypergraphs
Chunqiu FANG, Mei LU
Acta Mathematicae Applicatae Sinica(English Series). 2020, 36(3): 636641.
DOI:
10.1007/s000000200962z
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103
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Let
H
=(
V, E
) be an nbalanced
k
partite
k
graph with partition classes
V
_{1}
, …,
V
_{k}
. Suppose for every legal (
k
1)tuple
f
contained in
V
\
V
_{1}
and for every legal (
k
1)tuple
g
contained in
V
\
V
_{k}
such that
f
∪
g
∉
E
(
H
), we have
d
(
f
) +
d
(
g
) ≥
n
+ 1. In this paper, we prove that under this condition
H
must have a perfect matching. Another result of this paper is about the perfect matching in 3uniform hmbipartite hypergraphs. Let
G
be a 3uniform hmbipartite hypergraph with one of whose sides
V
_{1}
has the size
n
, the another side
V
_{2}
has size 2
n
. If for all the legal 2tuple
f
with
f
∩
V
_{1}
=1 and for all the legal 2tuple
g
with
g
∩
V
_{1}
=0, we have
d
(
f
) ≥
n
2 and
d
(
g
) >
n
/2, then
G
has a perfect matching.
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On the General Stability of a Viscoelastic Wave Equation with an Integral Condition
FARIDA BELHANNACHE, SALIM A. MESSAOUDI
Acta Mathematicae Applicatae Sinica(English Series). 2020, 36(4): 857869.
DOI:
10.1007/s1025502009793
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We consider a nonlocal boundary value problem for a viscoelastic equation with a Bessel operator and a weighted integral condition and we prove a general decay result. We also give an example to show that our general result gives the optimal decay rate for ceratin polynomially decaying relaxation functions. This result improves some other results in the literature.
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Rainbow
k
connectivity of Random Bipartite Graphs
Xiaolin CHEN, Xueliang LI, Huishu LIAN
Acta Mathematicae Applicatae Sinica(English Series). 2020, 36(4): 879890.
DOI:
10.1007/s102550200970z
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A path in an edgecolored graph
G
is called a
rainbow
path
if no two edges of the path are colored the same color. The minimum number of colors required to color the edges of
G
such that every pair of vertices are connected by at least
k
internally vertexdisjoint rainbow paths is called the
rainbow kconnectivity
of the graph
G
, denoted by
rc
_{k}
(
G
). For the random graph
G
(
n, p
), He and Liang got a sharp threshold function for the property
rc
_{k}
(
G
(
n, p
)) ≤
d
. For the random equibipartite graph
G
(
n, n, p
), Fujita et. al. got a sharp threshold function for the property
rc
_{k}
(
G
(
n, n, p
)) ≤ 3. They also posed the following problem: For
d
≥ 2, determine a sharp threshold function for the property
rc
_{k}
(
G
) ≤
d
, where
G
is another random graph model. This paper is to give a solution to their problem in the general random bipartite graph model
G
(
m, n, p
).
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Uniform Lipschitz Estimates of Homogenization of Elliptic Systems in Divergence Form with Dini Conditions
Rong DONG, Dongsheng LI, Hailiang ZHANG
Acta Mathematicae Applicatae Sinica(English Series). 2021, 37(1): 4868.
DOI:
10.1007/s1025502110014
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The paper is devoted to the homogenization of elliptic systems in divergence form. We obtain uniform interior as well as boundary Lipschitz estimates in a bounded
C
^{1,γ}
domain when the coefficients are Dini continuous, inhomogeneous terms are divergence of Dini continuous functions and the boundary functions have Dini continuous derivatives. The results extend Avellaneda and Lin's work[Comm. Pure Appl. Math., 40:803847 (1987)], where Hölder continuity is the main assumption on smoothness of the data.
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Isolated Toughness and
k
Hamiltonian[
a, b
]factors
Zhiren SUN, Sizhong ZHOU
Acta Mathematicae Applicatae Sinica(English Series). 2020, 36(3): 539544.
DOI:
10.1007/s102550200963y
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Let
a, b
and
k
be nonnegative integers with
a
≥ 2 and
b
≥
a
(
k
+ 1) + 2. A graph
G
is called a
k
Hamiltonian graph if after deleting any
k
vertices of
G
the remaining graph of
G
has a Hamiltonian cycle. A graph
G
is said to have a
k
Hamiltonian[
a, b
]factor if after deleting any
k
vertices of
G
the remaining graph of
G
admits a Hamiltonian[
a, b
]factor. Let
G
is a
k
Hamiltonian graph of order
n
with
n
≥
a
+
k
+ 2. In this paper, it is proved that
G
contains a
k
Hamiltonian[
a, b
]factor if
δ
(
G
) ≥
a
+
k
and
δ
(
G
) ≥
I
(
G
) ≥
a
1 + ((
a
(
k
+1))/
b
2).
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Modelling the Effects of Pest Control with Development of Pesticide Resistance
Bing LIU, Baolin KANG, Fengmei TAO, Gang HU
Acta Mathematicae Applicatae Sinica(English Series). 2021, 37(1): 109125.
DOI:
10.1007/s102550210988x
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In this paper, we assume that the pest population is divided into susceptible pests and infected pests, and only susceptible pests do harm to crops. Considering the two methods of spraying pesticides and releasing infected pests and natural enemies to control susceptible pests (the former is applied more frequently), and assuming that only susceptible pests develop resistance to pesticides, a pest control model with resistance development is established. By using the basic theory of impulsive differential systems and analytical methods, the sufficient condition for the global attractiveness of the susceptible pest eradication periodic solution is given. Combined with numerical simulations, the effects of spraying frequency of pesticides on critical threshold conditions for eradicating susceptible pests are discussed. The results confirm that it is not that the more frequently the pesticides are sprayed, the better the result of the pest control is. Two control strategies for eradicating susceptible pests are proposed:switching pesticides and releasing natural enemies elastically. Finally, the parameters in the critical threshold are analyzed from the following two aspects:(1) The key factors affecting pest control are determined by parameter sensitivity analyses. The results indicate that the correlation of the critical threshold concerning the killing efficiency rate and the decay rate of pesticides to susceptible pests varies due to the resistance development of susceptible pests. (2) Threedimensional graphs and contours of susceptible pest eradication critical threshold with two parameters are simulated, and the effects of the main parameters on the critical threshold are analyzed.
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Existence and Regularity of Solution of the Liquid
^{4}
He Model Coupling with an Applied Magnetic Field
Chen PENG
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(2): 497511.
DOI:
10.1007/s1025502210917
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14
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In this paper, we derive a timedependent GinzburgLandau model for liquid
^{4}
He coupling with an applied magnetic field basing on the Le Châtlier principle.
We also obtain the existence and uniqueness of global weak solution for this model. In addition, by utilizing the regularity estimates for linear semigroup, we prove that the model possesses a global classical solution.
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Construction of Multilevel Spacefilling Designs via Code Mappings
Huili XUE, Xingyou HUANG, Hongyi LI
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(1): 2436.
DOI:
10.1007/s102550221055y
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29
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Spacefilling designs are widely used in various fields because of their nice spacefilling properties. Uniform designs are one of spacefilling designs, which desires the experimental points to scatter uniformly over the experimental area. For practical need, the construction and their properties of ninelevel uniform designs are discussed via two code mappings in this paper. Firstly, the algorithm of constructing ninelevel uniform designs is presented from an initial threelevel design by the TypeI code mapping and tripling technique. Secondly, the algorithm of constructing ninelevel uniform designs is presented from a threelevel base design by the TypeII code mapping and generalized orthogonal arrays. Moreover, relative properties are discussed based on the two code mappings. Finally, some numerical examples are given out for supporting our theoretical results.
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1planar Graphs without 4cycles or 5cycles are 5colorable
Lili SONG, Lei SUN
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(1): 169176.
DOI:
10.1007/s1025502210739
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24
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A graph is 1planar if it can be drawn on the Euclidean plane so that each edge is crossed by at most one other edge. A proper vertex
k
coloring of a graph
G
is defined as a vertex coloring from a set of
k
colors such that no two adjacent vertices have the same color. A graph that can be assigned a proper
k
coloring is
k
colorable. A cycle is a path of edges and vertices wherein a vertex is reachable from itself. A cycle contains
k
vertices and
k
edges is a
k

cycle
. In this paper, it is proved that 1planar graphs without 4cycles or 5cycles are 5colorable.
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Oscillatory Behavior of Thirdorder Nonlinear Differential Equations with a Sublinear Neutral Term
Wenjuan LI, Yuanhong YU
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(2): 484496.
DOI:
10.1007/s1025502210891
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10
)
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The authors present some new criteria for oscillation and asymptotic behavior of solutions of thirdorder nonlinear differential equations with a sublinear neutral term of the form $$\left(r(t)(z''(t))^{\alpha}\right)'+\int^{d}_{c}q(t,\xi)f\left(x\left(\sigma(t,\xi)\right)\right)d\xi=0, \qquad t\geq t_{0}$$ where $z(t)=x(t)+\int^{b}_{a}p(t,\xi)x^{\gamma}\left(\tau(t,\xi)\right)d\xi,~0<\gamma\leq1.$ Under the conditions $\int^{\infty}_{t_{0}}r^{\frac{1}{\alpha}}(t)dt=\infty$ or $\int^{\infty}_{t_{0}}r^{\frac{1}{\alpha}}(t)dt<\infty.$ The results obtained here extend, improve and complement to some known results in the literature. Examples are provided to illustrate the theorems.
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A New Lower Bound on the PotentialRamsey Number of Two Graphs
Jinzhi DU, Jianhua YIN
Acta Mathematicae Applicatae Sinica(English Series). 2021, 37(1): 176182.
DOI:
10.1007/s1025502109997
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117
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A nonincreasing sequence
π
=(
d
_{1}
, …,
d
_{n}
) of nonnegative integers is a
graphic sequence
if it is realizable by a simple graph
G
on
n
vertices. In this case,
G
is referred to as a
realization
of
π
. Given a graph
H
, a graphic sequence
π
is
potentially
Hgraphic
if
π
has a realization containing
H
as a subgraph. For graphs
G
_{1}
and
G
_{2}
, the
potentialRamsey number
r
_{pot}
(
G
_{1}
,
G
_{2}
) is the smallest integer
k
such that for every
k
term graphic sequence
π
, either
π
is potentially
G
_{1}
graphic or the complementary sequence
π
=(
k
1
d
_{k}
, …,
k
1
d
_{1}
) is potentially
G
_{2}
graphic. For 0 ≤
k
≤ ⎣
t
/2 」, denote
K
_{t}
^{k}
to be the graph obtained from
K
_{t}
by deleting
k
independent edges. If
k
=0, Busch et al. (Graphs Combin., 30(2014)847859) present a lower bound on
r
_{pot}
(
G, K
_{t}
) by using the 1dependence number of
G
. In this paper, we utilize
i
dependence number of
G
for
i
≥ 1 to give a new lower bound on rpot(
G
,
K
_{t}
^{k}
) for any
k
with 0 ≤
k
≤ ⎣
t
/2 」. Moreover, we also determine the exact values of
r
_{pot}
(
K
_{n}
,
K
_{t}
^{k}
) for 1 ≤
k
≤ 2.
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Multiple Solutions for the KleinGordonMaxwell System with Steep Potential Well
Xiaoqi LIU, Chunlei TANG
Acta Mathematicae Applicatae Sinica(English Series). 2021, 37(1): 155165.
DOI:
10.1007/s102550210986z
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204
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In this paper, we concern the KleinGordonMaxwell system with steep potential well
???20210112???
Without global and local compactness, we can tell the difference of multiple solutions from their norms in
L
^{p}
(R
^{3}
).
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