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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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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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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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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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Equilibrium Strategies in an Unobservable Onoff Fluid Queue
Xiuli XU, Shuo WANGy
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(2): 324336.
DOI:
10.1007/s102550221080x
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This paper considers an onoff fluid queue model. The on and off states of the system appear alternately, and the sojourn times at these two different states are independent, and each one follows an exponential distribution. The fluid flows into the system buffer with some strategies to wait for the system service under the firstcome firstserved discipline. Here the system can process the fluid in the buffer only when the system is on state. With given utility functions such as an expected average social profit, and an individual expected profit, the equilibrium strategies are characterized under both the fully unobservable case and the partially observable case.
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Analytical Expressions to Counterparty Credit Risk Exposures for Interest Rate Derivatives
Shuang LI, Cheng PENG, Ying BAO, Yanlong ZHAO, Zhen CAO
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(2): 254270.
DOI:
10.1007/s1025502210748
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This paper proposes an approximate analytical solution method to calculate counterparty credit risk exposures. Compared with the Standard Approach for measuring Counterparty Credit Risk and the Internal Modeling Method provided by Basel Committee, the proposed method significantly improves the calculation efficiency based on sacrificing a little accuracy. Taking Forward Rate Agreement as an example, this article derives the exact expression for Expected Exposure. By approximating the distribution of Forward Rate Agreement’s future value to a normal distribution, the approximate analytical expression for Potential Future Exposure is derived. Numerical results show that this method is reliable and is robust under different parameters.
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Positive Solutions for a Class of Fractional
p
Laplacian Equation with Critical Sobolev Exponent and Decaying Potentials
Na LI, Xiaoming HE
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(2): 463483.
DOI:
10.1007/s1025502210908
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In this paper, we study the existence of positive solution for the $p$Laplacian equations with fractional critical nonlinearity \[ \begin{cases} (\Delta)^{s}_{p}u+V(x)u^{p2}u=K(x)f(u)+P(x)u^{p^{*}_{s}2}u, \qquad x\in \mathbb{R}^{N}, \\ u\in \mathcal {D}^{s,p}(\mathbb{R}^{N}), \end{cases} \] where $s\in(0,1), \ p^{*}_{s}=\frac{Np}{Nsp}, \ N>sp, \ p>1$ and $ V(x),K(x)$ are positive continuous functions which vanish at infinity, $f$ is a function with a subcritical growth, and $P(x)$ is bounded, nonnegative continuous function. By using variational method in the weighted spaces, we prove the above problem has at least one positive solution.
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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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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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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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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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Generalized Lagrangian Duality in Setvalued Vector Optimization via Abstract Subdifferential
Yanfei CHAI, Sanyang LIU, Siqi WANG
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(2): 337351.
DOI:
10.1007/s1025502210793
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In this paper, we investigate dual problems for nonconvex setvalued vector optimization via abstract subdifferential. We first introduce a generalized augmented Lagrangian function induced by a coupling vectorvalued function for setvalued vector optimization problem and construct related setvalued dual map and dual optimization problem on the basic of weak efficiency, which used by the concepts of supremum and infimum of a set. We then establish the weak and strong duality results under this augmented Lagrangian and present sufficient conditions for exact penalization via an abstract subdifferential of the object map. Finally, we define the suboptimal path related to the dual problem and show that every cluster point of this suboptimal path is a primal optimal solution of the object optimization problem. In addition, we consider a generalized vector variational inequality as an application of abstract subdifferential.
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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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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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On Disjoint Cycles of the Same Length in Tournaments
On Disjoint Cycles of the Same Length in Tournaments
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(2): 271281.
DOI:
10.1007/s102550221072x
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A tournament is an orientation of the complete graph. Tournaments form perhaps the most interesting class of digraphs and it has a great potential for application. Tournaments provide a model of the statistical technique called the method of paired comparisons and they have also been studied in connection with sociometric relations in small groups. In this paper, we investigate disjoint cycles of the same length in tournaments. In 2010, Lichiardopol conjectured that for given integers
l
≥ 3 and
k
≥ 1, any tournament with minimum outdegree at least (
l
 1)
k
 1 contains
k
disjoint
l
cycles, where an
l
cycle is a cycle of order
l
. BangJensen et al. verified the conjecture for
l
= 3 and Ma et al. proved that it also holds for
l
≥ 10. This paper provides a proof of the conjecture for the case of 9 ≥
l
≥ 4
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General Decay for a Thermoelastic Problem of a Microbeam with GurtinPipkin Thermal Law
Dongqin CHEN, Wenjun LIU, Zhijing CHEN
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(2): 426440.
DOI:
10.1007/s1025502210873
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In this paper, we study the wellposedness and the asymptotic stability of a onedimensional thermoelastic microbeam system, where the heat conduction is given by GurtinPipkin thermal law. We first establish the wellposedness of the system by using the semigroup arguments and LumerPhillips theorem. We then obtain an explicit and general formula for the energy decay rates through perturbed energy method and some properties of the convex functions.
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Diffusioninduced Spatiotemporal Oscillations in an Epidemic Model with Two Delays
Yanfei DU, Ben NIU, Junjie WEI
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(1): 128153.
DOI:
10.1007/s102550221062z
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We investigate a diffusive, stagestructured epidemic model with the maturation delay and freelymoving delay. Choosing delays and diffusive rates as bifurcation parameters, the only possible way to destabilize the endemic equilibrium is through Hopf bifurcation. The normal forms of Hopf bifurcations on the center manifold are calculated, and explicit formulae determining the criticality of bifurcations are derived. There are two different kinds of stable oscillations near the first bifurcation: on one hand, we theoretically prove that when the diffusion rate of infected immature individuals is sufficiently small or sufficiently large, the first branch of Hopf bifurcating solutions is always spatially homogeneous; on the other, fixing this diffusion rate at an appropriate size, stable oscillations with different spatial profiles are observed, and the conditions to guarantee the existence of such solutions are given by calculating the corresponding eigenfunction of the Laplacian at the first Hopf bifurcation point. These bifurcation behaviors indicate that spatial diffusion in the epidemic model may lead to spatially inhomogeneous distribution of individuals.
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On a Damped Vibration Problem Involving
p
Laplacian Operator: Fast Homoclinic Orbits
Peng CHEN, Xianhua TANG, Yuanyuan ZHANG
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(2): 368387.
DOI:
10.1007/s1025502210837
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n this paper, we deal with the nonlinear secondorder differential equation with damped vibration term involving
p
Laplacian operator. Of particular interest is the resolution of an open problem. An interesting outcome from our result is that we can obtain the fast homoclinic solution with general superlinear growth assumption in suitable Sobolev space. To our knowledge, our theorems appear to be the first such result about damped vibration problem with
p
Laplacian operator.
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Discussion on Fractional (
a
,
b
,
k
)critical Covered Graphs
Wei ZHANG, Sufang WANG
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(2): 304311.
DOI:
10.1007/s1025502210766
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A graph G is called a fractional [
a
,
b
]covered graph if for each
e
∈ E(G), G contains a fractional [
a
,
b
]factor covering e. A graph G is called a fractional (
a
,
b
,
k
)critical covered graph if for any
W
⊆
V
(
G
) with 
W
 =
k
,
G

W
is fractional [
a
,
b
]covered, which was first defined and investigated by Zhou, Xu and Sun [S. Zhou, Y. Xu, Z. Sun, Degree conditions for fractional (
a
,
b
,
k
)critical covered graphs, Information Processing Letters 152(2019)105838]. In this work, we proceed to study fractional (
a
,
b
,
k
)critical covered graphs and derive a result on fractional (
a
,
b
,
k
)critical covered graphs depending on minimum degree and neighborhoods of independent sets.
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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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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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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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Uniform Regularity for the Isentropic Compressible Magnetohydrodynamic System
Jishan FAN, Fucai LI, GEN NAKAMURA
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(2): 410416.
DOI:
10.1007/s1025502210846
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In this paper we first establish the uniform regularity of smooth solutions with respect to the viscosity coefficients to the isentropic compressible magnetohydrodynamic system in a periodic domain $\mathbb{T}^n$. We then apply our result to obtain the isentropic compressible magnetohydrodynamic system with zero viscosity.
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NavierStokes/AllenCahn System with Generalized Navier Boundary Condition
Yazhou CHEN, Qiaolin HE, Bin HUANG, Xiaoding SHI
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(1): 98115.
DOI:
10.1007/s1022502210687
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In this paper, we focus on the immiscible compressible twophase flow described by the coupled compressible NavierStokes system and the modified AllenCahn equations. The generalized Navier boundary condition and the relaxation boundary condition are established in order to solve the problem of moving contact lines on the solid boundary by using the principle of minimum energy dissipation. The existence and uniqueness for local strong solution in three dimensional bounded domain for this type of boundary value problem is obtained by the elementary energy method and the maximum principle.
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A Delayed HIV Infection Model with the Homeostatic Proliferation of CD4
^{+}
T Cells
Qianghui XU, Jicai HUANG, Yueping DONG, Yasuhiro TAKEUCHI
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(2): 441462.
DOI:
10.1007/s1025502210882
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In this paper, we investigate a delayed HIV infection model that considers the homeostatic proliferation of CD4
^{+}
T cells. The existence and stability of uninfected equilibrium and infected equilibria (smaller and larger ones) are studied by analyzing the characteristic equation of the system. The intracellular delay does not affect the stability of uninfected equilibrium, but it can change the stability of larger positive equilibrium and Hopf bifurcation appears inducing stable limit cycles. Furthermore, direction and stability of Hopf bifurcation are well investigated by using the central manifold theorem and the normal form theory. The numerical simulation results show that the stability region of larger positive equilibrium becomes smaller as the increase of time delay. Moreover, when the maximum homeostatic growth rate is very small, the larger positive equilibrium is always stable. On the contrary, when the rate of supply of T cells is very small, the larger positive equilibrium is always unstable.
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Dynamical Behavior of SEIRSVS Epidemic Models with Nonlinear Incidence and Vaccination
Xiaomei FENG, Lili LIU, Fengqin ZHANG
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(2): 282303.
DOI:
10.1007/s1025502210757
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For some infectious diseases such as mumps, HBV, there is evidence showing that vaccinated individuals always lose their immunity at different rates depending on the inoculation time. In this paper, we propose an agestructured epidemic model using a step function to describe the rate at which vaccinated individuals lose immunity and reduce the agestructured epidemic model to the delay differential model. For the agestructured model, we consider the positivity, boundedness, and compactness of the semiflow and study global stability of equilibria by constructing appropriate Lyapunov functionals. Moreover, for the reduced delay differential equation model, we study the existence of the endemic equilibrium and prove the global stability of equilibria. Finally, some numerical simulations are provided to support our theoretical results and a brief discussion is given.
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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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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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Discussions on Orthogonal Factorizations in Digraphs
Sizhong ZHOU, Hongxia LIU
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(2): 417425.
DOI:
10.1007/s1025502210864
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Let $m$, $t$, $r$ and $k_i$ $(1\leq i\leq m)$ be positive integers with $k_i\geq\frac{(t+3)r}{2}$, and $G$ be a digraph with vertex set $V(G)$ and arc set $E(G)$. Let $H_1,H_2,\cdots,H_t$ be $t$ vertexdisjoint subdigraphs of $G$ with $mr$ arcs. In this article, it is verified that every $[0,k_1+k_2+\cdots+k_m(m1)r]$digraph $G$ has a $[0,k_i]_1^{m}$factorization $r$orthogonal to every $H_i$ for $1\leq i\leq t$.
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A Line Search SQPtype Method with Biobject Strategy for Nonlinear Semidefinite Programming
Wenhao FU, Zhongwen CHEN
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(2): 388409.
DOI:
10.1007/s1025502210819
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We propose a line search exact penalty method with biobject strategy for nonlinear semidefinite programming. At each iteration, we solve a linear semidefinite programming to test whether the linearized constraints are consistent or not. The search direction is generated by a piecewise quadraticlinear model of the exact penalty function. The penalty parameter is only related to the information of the current iterate point. The line search strategy is a penaltyfree one. Global and local convergence are analyzed under suitable conditions. We finally report some numerical experiments to illustrate the behavior of the algorithm on various degeneracy situations.
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Highdimensional Tests for Mean Vector: Approaches without Estimating the Mean Vector Directly
Bo CHEN, Haimeng WANG
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(1): 7886.
DOI:
10.1007/s102550221070z
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22
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Several tests for multivariate mean vector have been proposed in the recent literature. Generally, these tests are directly concerned with the mean vector of a highdimensional distribution. The paper presents two new test procedures for testing mean vector in large dimension and small samples. We do not focus on the mean vector directly, which is a different framework from the existing choices. The first test procedure is based on the asymptotic distribution of the test statistic, where the dimension increases with the sample size. The second test procedure is based on the permutation distribution of the test statistic, where the sample size is fixed and the dimension grows to infinity. Simulations are carried out to examine the finitesample performance of the tests and to compare them with some popular nonparametric tests available in the literature.
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Local Stability to the EinsteinEuler System in Schwarzschild Spacetime
Jun WU, Boling GUO
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(2): 352367.
DOI:
10.1007/s1025502210809
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We study the Schwarzschild spacetime solutions to the EinsteinEuler equations. In our analysis, we aim to show local stability under small perturbations. To resolve this problem, we use the NashMoser (Hamilton) theorem. The work was originally developed for the nonrelativistic EulerPoisson equations.
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A New Sufficient Degree Condition for a Graphic Sequence to Be Forcibly
k
EdgeConnected
Jianhua YIN, Jiyun GUO
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(1): 223228.
DOI:
10.1007/s1025502210579
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A graphic sequence
π
=(
d
_{1}
,
d
_{2}
, ...,
d
_{n}
) is said to be
forcibly
k
edgeconnected if every realization of
π
is
k
edgeconnected. In this paper, we obtain a new sufficient degree condition for
π
to be forcibly
k
edgeconnected. We also show that this new sufficient degree condition implies a strongest monotone degree condition for
π
to be forcibly 2edgeconnected and a conjecture about a strongest monotone degree condition for
π
to be forcibly 3edgeconnected due to Bauer et al. (Networks, 54(2) (2009) 9598), and also implies a strongest monotone degree condition for
π
to be forcibly 4edgeconnected.
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Influence of Discretization Step on Asymptotic Stability of a Certain Class of Twodimensional Continuousdiscrete Fractional Linear Systems
Souad BOUGUESSA, Djillali BOUAGADA, Mohammed Amine GHEZZAR
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(1): 6877.
DOI:
10.1007/s1025502210601
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The effectiveness of this paper lies in the influence of the discretization step on the asymptotic stability of the positive twodimensional fractional linear systems. It aims at investigating whether, how and when this step affects the asymptotically stable twodimensional positive fractional linear continuousdiscrete systems. To accomplish this study, a new test was outlined and used so that the asymptotic stability of the system was measured both before and after being exposed to the sampling step. Furthermore, the conditions of that stability were assessed. As a result, the outcome of the approximation shows that the stability is preserved under a particular set of conditions. On this basis, the newly proposed approach is recommended for testing the intended stability of such systems. A numerical example is tested to show the accuracy and the applicability of the proposed tests.
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Multiple Positive Solutions for One Dimensional Third Order
p
Laplacian Equations with Integral Boundary Conditions
Youyuan YANG, Qiru WANG
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(1): 116127.
DOI:
10.1007/s1025502210659
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In this paper, we consider the one dimensional third order
p
Laplacian equation (Φ
_{p}
(
u
"))'+
h
(
t
)
f
(
t
,
u
(
t
))=0 with integral boundary conditions
u
(0)
α
u
'(0)= ∫
t
_{0}
^{1}
g
_{1}
(
s
)
u
(
s
)
ds
,
u
(1)+
β
u
'(1)= ∫
t
_{0}
^{1}
g
_{2}
(
s
)
u
(
s
)
ds
,
u
"(0)=0. By using kernel functions and the AveryPeterson fixed point theorem, we establish the existence of at least three positive solutions.
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The Distance Matching Extension in
K
_{1, k}
free Graphs with High Local Connectedness
Weichan LIU, Guiying YAN
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(1): 3743.
DOI:
10.1007/s1022502210696
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A matching is extendable in a graph
G
if
G
has a perfect matching containing it. A distance q matching is a matching such that the distance between any two distinct matching edges is at least
q
. In this paper, we prove that any distance 2
k
3 matching is extendable in a connected and locally (
k
1)connected
K
_{1, k}
free graph of even order. Furthermore, we also prove that any distance
q
matching
M
in an rconnected and locally (
k
1)connected
K
_{1, k}
free graph of even order is extendable provided that
M
is bounded by a function on
r
,
k
and
q
. Our results improve some results in [J. Graph Theory 93 (2020), 5C20].
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Weighted Estimates for a Class of Global Maximal Operators Associated with Dispersive Equation
Yong DING, Yaoming NIU
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(1): 187208.
DOI:
10.1007/s102550221071y
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For a function
?
satisfying some suitable growth conditions, consider the following general dispersive equation defined by{
i
?
_{t}
u
+
?
(√
?
)
u
= 0, (
x
,
t
)∈ R
^{n}
×R,
u
(
x
, 0)=
f
(
x
),
f
∈
S
(R
^{n}
), (*)where
?
(√
?
) is a pseudodifferential operator with symbol
?
(
ξ
). In the present paper, when the initial data
f
belongs to Sobolev space, we give the local and global weighted
L
^{q}
estimate for the global maximal operator
S
_{?}
^{**}
defined by
S
_{?}
^{**}
f
(
x
)=sup
_{t∈R}
S
_{t, ?}
f
(
x
), where
S
_{t, ?}
f
(
x
) = (2
π
)
^{n}
∫
_{Rn}
e
^{ix·ξ+it?(ξ)}
f
(
ξ
)
dξ
is a formal solution of the equation (*).
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Empirical Likelihood in Generalized Linear Models with Working Covariance Matrix
Xiuqing ZHOU, Qibing GAO, Chunhua ZHU, Xiuli DU, Liuliu MAO
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(1): 8797.
DOI:
10.1007/s1025502210677
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23
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Empirical likelihood in generalized linear models with multivariate responses and working covariance matrix is discussed. Under the weakest assumption on eigenvalues of Fisher's information matrix and some other regular conditions, we prove that the nonparametric Wilk's property still holds, that is, the empirical loglikelihood ratio at the true parameter values converges to the standard chisquare distribution. Numerical simulations are given to verify our theoretical result.
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Lower Bounds of Size Ramsey Number for Graphs with Small Independence Number
Chunlin YOU
Acta Mathematicae Applicatae Sinica(English Series). 2021, 37(4): 841846.
DOI:
10.1007/s1025502110473
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20
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Let
r
≥ 3 be an integer such that
r
2 is a prime power and let
H
be a connected graph on
n
vertices with average degree at least
d
and
α
(
H
) ≤
β
n
, where 0 <
β
< 1 is a constant. We prove that the size Ramsey number
R
(
H
;
r
)>
nd
/2(
r
2)
^{2}

C
√
n
for all sufficiently large
n
, where
C
is a constant depending only on
r
,
d
and
β
. In particular, for integers
k
≥ 1, and
r
≥ 3 such that
r
2 is a prime power, we have that there exists a constant
C
depending only on
r
and
k
such that
R
(
P
_{n}
^{k}
;
r
)>
kn
(
r
 2)
^{2}

C
√
n
(
k
^{2}
+
k
)/2(
r
 2)
^{2}
for all sufficiently large
n
, where
P
_{n}
^{k}
is the
kth
power of
P
_{n}
.
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Hypergraph Turán Numbers of Vertex Disjoint Cycles
Ran GU, Xueliang LI, Yongtang SHI
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(1): 229234.
DOI:
10.1007/s102550221056x
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The Turán number of a
k
uniform hypergraph
H
, denoted by
e
x
_{k}
(
n
;
H
), is the maximum number of edges in any
k
uniform hypergraph
F
on
n
vertices which does not contain
H
as a subgraph. Let
C
_{e}
^{(k)}
denote the family of all
k
uniform minimal cycles of length
e
,
S
(
e
_{1}
, ...,
e
_{r}
) denote the family of hypergraphs consisting of unions of
r
vertex disjoint minimal cycles of length
e
_{1}
, ...,
e
_{r}
, respectively, and
C
_{e}
^{(k)}
denote a
k
uniform linear cycle of length
e
. We determine precisely
e
x
_{k}
(
n
;
S
(
e
_{1}
, ...,
e
_{r}
)) and
e
x
_{k}
(
n
;
C
_{e1}
^{(k)}
, ...,
C
_{er}
^{(k)}
) for sufficiently large
n
. Our results extend recent results of Füredi and Jiang who determined the Turán numbers for single
k
uniform minimal cycles and linear cycles.
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A Result on Fractional (
a
,
b
,
k
)critical Covered Graphs
Sizhong ZHOU
Acta Mathematicae Applicatae Sinica(English Series). 2021, 37(4): 657664.
DOI:
10.1007/s1025502110348
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A fractional [
a
,
b
]factor of a graph
G
is a function
h
from
E
(
G
) to [0, 1] satisfying
a
≤
d
_{G}
^{h}
(
v
) ≤
b
for every vertex
v
of
G
, where
d
_{G}
^{h}
(
v
)=Σ
_{e∈ E(v)}
h
(
e
) and
E
(
v
)={
e
=
uv
:
u
∈
V
(
G
)}. A graph
G
is called fractional [
a
,
b
]covered if
G
contains a fractional [
a
,
b
]factor
h
with
h
(
e
)=1 for any edge
e
of
G
. A graph
G
is called fractional (
a
,
b
,
k
)critical covered if
G

Q
is fractional [
a
,
b
]covered for any
Q
⊂
eq
V
(
G
) with
Q
=
k
. In this article, we demonstrate a neighborhood condition for a graph to be fractional (
a
,
b
,
k
)critical covered. Furthermore, we claim that the result is sharp.
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Novel Kernel Function With a Hyperbolic Barrier Term to Primaldual Interior Point Algorithm for SDP Problems
Imene TOUIL, Wided CHIKOUCHE
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(1): 4467.
DOI:
10.1007/s1025502210610
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In this paper, we introduce for the first time a new eligible kernel function with a hyperbolic barrier term for semidefinite programming (SDP). This add a new type of functions to the class of eligible kernel functions. We prove that the interiorpoint algorithm based on the new kernel function meets O(
n
^{3/4}
log
n
/
ε
) iterations as the worst case complexity bound for the largeupdate method. This coincides with the complexity bound obtained by the first kernel function with a trigonometric barrier term proposed by El Ghami et al. in 2012, and improves with a factor
n
^{1/4}
the obtained iteration bound based on the classic kernel function. We present some numerical simulations which show the effectiveness of the algorithm developed in this paper.
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The Characterization of
p
factorcritical Graphs
Shaohui ZHAI, Erling WEI, Fuji ZHANG
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(1): 154158.
DOI:
10.1007/s102550221064x
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20
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A graph
G
is said to be
p
factorcritical if
G

u
_{1}

u
_{2}
...
u
_{p}
has a perfect matching for any
u
_{1}
,
u
_{2}
, ...,
u
_{p}
∈
V
(
G
). The concept of
p
factorcritical is a generalization of the concepts of factorcritical and bicritical for
p
=1 and
p
=2, respectively. Heping Zhang and Fuji Zhang[
Construction for bicritical graphs and kextendable bipartite graphs
, Discrete Math., 306(2006) 1415–1423] gave a concise structure characterization of bicritical graphs. In this paper, we present the characterizations of
p
factorcritical graphs and minimal
p
factorcritical graphs for
p
≥ 2. As an application, we also obtain a class of graphs which are minimal
p
factorcritical for
p
≥ 1.
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Construction of Orthogonal Arrays without Interaction Columns
Shanqi PANG, Yaping WANG, Mingyao AI
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(1): 159168.
DOI:
10.1007/s102550221063y
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25
)
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In this paper a new class of orthogonal arrays (OAs), i.e., OAs without interaction columns, are proposed which are applicable in factor screening, interaction detection and other cases. With the tools of difference matrices, we present some general recursive methods for constructing OAs of such type. Several families of OAs with high percent saturation are constructed. In particular, for any integer
λ
≥ 3, such a twolevel OA of run 4
λ
can always be obtained if the corresponding Hadamard matrix exists.
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Ground State Solutions for a Class of Choquard Equations Involving Doubly Critical Exponents
Yongyong LI, Guidong LI, Chunlei TANG
Acta Mathematicae Applicatae Sinica(English Series). 2021, 37(4): 820840.
DOI:
10.1007/s1025502110464
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25
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In this paper, we are concerned with the autonomous Choquard equation
Δ
u
+
u
=(
I
_{α}
*
u
^{α/N+1}
)
u
^{α/N1}
u
+
u
^{2*2u}
+
f
(
u
) in R
^{N}
,
where
N
≥ 3,
I
_{α}
denotes the Riesz potential of order
α
∈(0,
N
), the exponents
α
/
N
+1 and 2
^{*}
=2
N
/
N
2 are critical with respect to the HardyLittlewoodSobolev inequality and Sobolev embedding, respectively. Based on the variational methods, by using the minimax principles and the Pohožaev manifold method, we prove the existence of ground state solution under some suitable assumptions on the perturbation
f
.
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Global Stability of a Mumps Transmission Model with Quarantine Measure
Yuzhen BAI, Xiaojing WANG, Songbai GUO
Acta Mathematicae Applicatae Sinica(English Series). 2021, 37(4): 665672.
DOI:
10.1007/s1025502110357
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In this paper, a model of mumps transmission with quarantine measure is proposed and then the control reproduction number
R
_{c}
of the model is obtained. This model admits a unique endemic equilibrium
P
^{*}
if and only if
R
_{c}
> 1, while the diseasefree equilibrium
P
^{0}
always exists. By using the technique of constructing Lyapunov functions and the generalized LyapunovLaSalle theorem, we first show that the equilibrium
P
^{0}
is globally asymptotically stable (GAS) if
R
_{c}
≤ 1; second, we prove that the equilibrium
P
^{*}
is GAS if
R
_{c}
> 1. Our results reveal that mumps can be eliminated from the community for
R
_{c}
≤ 1 and it will be persistent for
R
_{c}
> 1, and quarantine measure can also effectively control the mumps transmission.
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Vertextransitive Diameter Two Graphs
Wei JIN, Li TAN
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(1): 209222.
DOI:
10.1007/s1025502210588
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We investigate the family of vertextransitive graphs with diameter 2. Let Γ be such a graph. Suppose that its automorphism group is transitive on the set of ordered nonadjacent vertex pairs. Then either Γ is distancetransitive or Γ has girth at most 4. Moreover, if Γ has valency 2, then Γ ?
C
_{4}
or
C
_{5}
; and for any integer
n
≥ 3, there exist such graphs Γ of valency
n
such that its automorphism group is not transitive on the set of arcs. Also, we determine this family of graphs of valency less than 5. Finally, the family of diameter 2 circulants is characterized.
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