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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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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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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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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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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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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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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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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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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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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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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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18
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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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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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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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14
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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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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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15
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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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Global Asymptotic Stability in a Delay Differential Equation Model for Mosquito Population Suppression
Mugen HUANG, Jianshe YU
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(4): 882901.
DOI:
10.1007/s1025502210218
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A biosafe dengue control strategy is to use
Wolbachia
, which can induce incomplete cytoplasmic incompatibility (CI) and reduce the mating competitiveness of infected males. In this work, we formulate a delay differential equation model, including both the larval and adult stages of wild mosquitoes, to assess the impacts of CI intensity
ξ
and mating competitiveness
θ
of infected males on the suppression efficiency. Our analysis identifies a CI intensity threshold
ξ
^{*}
below which a successful suppression is impossible. When
ξ
≥
ξ
^{*}
, the wild population will be eliminated ultimately if the releasing level exceeds the release amount threshold
R
^{*}
uniformly. The dependence of
R
^{*}
on
ξ
and
θ
, and the impact of temperature on suppression are further exhibited through numerical examples. Our analyses indicate that a slight reduction of
ξ
is more devastating than significantly decrease of
θ
in the suppression efficiency. To suppress more than 95% wild mosquitoes during the peak season of dengue in Guangzhou, the optimal starting date for releasing is sensitive to
ξ
but almost independent of
θ
. One percent reduction of
ξ
from 1 requires at least one week earlier in the optimal releasing starting date from 7 weeks ahead of the peak season of dengue.
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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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Almost Periodic Solutions by the Harmonic Balance Method
Jifeng CHU, Zuohuan ZHENG, Zhe ZHOU
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(4): 943954.
DOI:
10.1007/s1025502210263
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11
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We consider nonautonomous ordinary differential equations in two cases. One is the onedimensional case that admits a condition of hyperbolicity, and the other one is the higherdimensional case that admits an exponential dichotomy. For differential equations of this kind, we give a rigorous treatment of the Harmonic Balance Method to look for almost periodic solutions.
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The Existence and Uniqueness of a New Boundary Value Problem (Type of Problem “E”) for a Class of Semi Linear (Power Type Nonlinearities) Mixed HyperbolicElliptic System Equations of Keldysh Type with Changing Time Direction
Mahammad A. NURMAMMADOV
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(4): 763777.
DOI:
10.1007/s1025502210165
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In present work studied the new boundary value problem for semi linear (Powertype nonlinearities) system equations of mixed hyperbolic elliptic Keldysh type in the multivariate dimension with the changing time direction. Considered problem and equation belongs to the modern level partial differential equations. Applying methods of functional analysis, topological methods, “
ε
” regularizing. and continuation by the parameter at the same time with aid of a prior estimates, under assumptions conditions on coefficients of equations of system, the existence and uniqueness of generalized and regular solutions of a boundary problem are established in a weighted Sobolev’s space. In this work one of main idea consists of show that the new boundary value problem which investigated in case of linear system equations can be wellposed when added nonlinear terms to this linear system equations, moreover in this case constructed new weithged spaces, the identity between of strong and weak solutions is established.
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A Note on the Dynamics of the Logistic Family Modified by Fuzzy Numbers
J.S. CÁNOVAS
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(4): 741752.
DOI:
10.1007/s1025502210855
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In this paper, we consider a modification of the wellknown logistic family using a family of fuzzy numbers. The dynamics of this modified logistic map is studied by computing its topological entropy with a given accuracy. This computation allows us to characterize when the dynamics of the modified family is chaotic. Besides, some attractors that appear in bifurcation diagrams are explained. Finally, we will show that the dynamics induced by the logistic family on the fuzzy numbers need not be complicated at all.
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Insider Trading with a Random Deadline under Partial Observations: Maximal Principle Method
Kai XIAO, Yonghui ZHOU
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(4): 753762.
DOI:
10.1007/s1025502211126
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For a revised model of Caldentey and Stacchetti (Econometrica, 2010) in continuoustime insider trading with a random deadline which allows market makers to observe some information on a risky asset, a closed form of its market equilibrium consisting of optimal insider trading intensity and market liquidity is obtained by maximum principle method. It shows that in the equilibrium, (i) as time goes by, the optimal insider trading intensity is exponentially increasing even up to infinity while both the market liquidity and the residual information are exponentially decreasing even down to zero; (ii) the more accurate information observed by market makers, the stronger optimal insider trading intensity is such that the total expect profit of the insider is decreasing even go to zero while both the market liquidity and the residual information are decreasing; (iii) the longer the mean of random time, the weaker the optimal insider trading intensity is while the more both the residual information and the expected profit are, but there is a threshold of trading time, half of the mean of the random time, such that if and only if after it the market liquidity is increasing with the mean of random time increasing.
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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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Stability and Instability of SchwarzschildAdS for the Nonlinear EinsteinKleinGordon System
FengXia LIU, BoLing GUO
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(4): 778812.
DOI:
10.1007/s1025502211019
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22
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In this paper, we study the global behavior of solutions to the spherically symmetric(it means that the problem is 2+2 framework)coupled NonlinearEinsteinKleinGordon (NLEKG) system in the presence of a negative cosmological constant. We prove the well posedness of the NLEKG system in the SchwarzschildAdS spacetimes and that the SchwarzschildAdS spacetimes (the trivial black hole solutions of the EKG system for which
ϕ
= 0 identically) are asymptotically stable. Small perturbations of SchwarzschildAdS initial data again lead to regular black holes, with the metric on the black hole exterior approaching a SchwarzschildAdS spacetime. Bootstrap argument on the black hole exterior, with the redshift effect and weighted Hardy inequalities playing the fundamental role in the analysis. Both integrated decay and pointwise decay estimates are obtained.
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Statistical Inferences in a Partially Linear Model with Autoregressive Errors
Xiaohui LIU, Yu WANG, Yawen FAN, Yuzi LIU
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(4): 822842.
DOI:
10.1007/s1025502210174
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In this paper, we consider the statistical inferences in a partially linear model when the model error follows an autoregressive process. A twostep procedure is proposed for estimating the unknown parameters by taking into account of the special structure in error. Since the asymptotic matrix of the estimator for the parametric part has a complex structure, an empirical likelihood function is also developed. We derive the asymptotic properties of the related statistics under mild conditions. Some simulations, as well as a real data example, are conducted to illustrate the finite sample performance.
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PMcompact Graphs and Vertexdeleted Subgraphs
Yipei ZHANG, Xiumei WANG, Jinjiang YUAN
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(4): 955965.
DOI:
10.1007/s1025502210183
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The perfect matching polytope of a graph $G$ is the convex hull of the incidence vectors of all perfect matchings of $G$. A graph $G$ is PMcompact if the 1skeleton graph of the prefect matching polytope of $G$ is complete. Equivalently, a matchable graph $G$ is PMcompact if and only if for each even cycle $C$ of $G$, $GV(C)$ has at most one perfect matching. This paper considers the class of graphs from which deleting any two adjacent vertices or nonadjacent vertices, respectively, the resulting graph has a unique perfect matching. The PMcompact graphs in this class of graphs are presented.
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Eulerian and Bipartite Binary Deltamatroids
Qi YAN, Xianan JIN
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(4): 813821.
DOI:
10.1007/s1025502210147
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Deltamatroid theory is often thought of as a generalization of topological graph theory. It is wellknown that an orientable embedded graph is bipartite if and only if its Petrie dual is orientable. In this paper, we first introduce the concepts of Eulerian and bipartite deltamatroids and then extend the result from embedded graphs to arbitrary binary deltamatroids. The dual of any bipartite embedded graph is Eulerian. We also extend the result from embedded graphs to the class of deltamatroids that arise as twists of binary matroids. Several related results are also obtained.
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Integrability and Exact Solutions of the (2+1)dimensional KdV Equation with Bell Polynomials Approach
Juncai PU, Yong CHEN
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(4): 861881.
DOI:
10.1007/s1025502210209
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In this paper, the bilinear formalism, bilinear Bäcklund transformations and Lax pair of the (2+1)dimensional KdV equation are constructed by the Bell polynomials approach. The
N
soliton solution is derived directly from the bilinear form. Especially, based on the twosoliton solution, the lump solution is given out analytically by taking special parameters and using Taylor expansion formula. With the help of the multidimensional Riemann theta function, multiperiodic (quasiperiodic) wave solutions for the (2+1)dimensional KdV equation are obtained by employing the Hirota bilinear method. Moreover, the asymptotic properties of the one and twoperiodic wave solution, which reveal the relations with the single and twosoliton solution, are presented in detail.
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Rainbow and Monochromatic Vertexconnection of Random Graphs
Wenjing LI, Hui JIANG, Jiabei HE
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(4): 966972.
DOI:
10.1007/s1025502210272
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A vertexcolored path $P$ is
rainbow
if its internal vertices have distinct colors; whereas $P$ is
monochromatic
if its internal vertices are colored the same. For a vertexcolored connected graph $G$, the rainbow vertexconnection number ${\rm rvc} (G)$ is the minimum number of colors used such that there is a rainbow path joining any two vertices of $G$; whereas the monochromatic vertexconnection number ${\rm mvc} (G)$ is the maximum number of colors used such that any two vertices of $G$ are connected by a monochromatic path. These two opposite concepts are the vertexversions of rainbow connection number ${\rm rc} (G)$ and monochromatic connection number ${\rm mc} (G)$ respectively. The study on ${\rm rc} (G)$ and ${\rm mc} (G)$ of random graphs drew much attention, and there are few results on the rainbow and monochromatic vertexconnection numbers. In this paper, we consider these two vertexconnection numbers of random graphs and establish sharp threshold functions for them, respectively.
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Starcritical Ramsey Numbers of Wheels Versus Odd Cycles
Yuchen LIU, Yaojun CHEN
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(4): 916924.
DOI:
10.1007/s1025502210236
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5
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Let $K_{1,k}$ be a star of order $k+1$ and $K_n\sqcup K_{1,k}$ the graph obtained from a complete graph $K_n$ and an additional vertex $v$ by joining $v$ to $k$ vertices of $K_n$. For graphs $G$ and $H$, the starcritical Ramsey number $r_*(G,H)$ is the minimum integer $k$ such that any red/blue edgecoloring of $K_{r1}\sqcup K_{1,k}$ contains a red copy of $G$ or a blue copy of $H$, where $r$ is the classical Ramsey number $R(G,H)$. Let $C_m$ denote a cycle of order $m$ and $W_n$ a wheel of order $n+1$. Hook (2010) proved that $r_*(W_n,C_3)=n+3$ for $n\geq6$. In this paper, we show that $r_*(W_n,C_m)=n+3$ for $m$ odd, $m\geq5$ and $n\geq 3(m1)/2+2$.
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A Class of Singular Coupled Systems of Superlinear MongeAmpère Equations
Meiqiang FENG
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(4): 925942.
DOI:
10.1007/s1025502210245
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In this paper, we analyze the existence, multiplicity and nonexistence of nontrivial radial convex solutions to the following system coupled by singular MongeAmpère equations $$ \left \{ \begin{array}{l} \text{det}\ D^2u_1=\lambda h_1(x)f_1(u_2), \qquad \text{in} \ \ \Omega,\\ \text{det}\ D^2u_2=\lambda h_2(x)f_2(u_1), \qquad \text{in} \ \ \Omega,\\ u_1=u_2=0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{on} \ \ \partial \Omega \end{array} \right. $$ for a certain range of $\lambda >0$, $h_i$ are weight functions, $f_i$ are continuous functions with possible singularity at $0$ and satisfy a combined $N$superlinear growth at $\infty$, where $i\in \{1,2\}$, $\Omega$ is the unit ball in $\mathbb{R}^N$. We establish the existence of a nontrivial radial convex solution for small $\lambda$, multiplicity results of nontrivial radial convex solutions for certain ranges of $\lambda$, and nonexistence results of nontrivial radial solutions for the case $\lambda\gg 1$. The asymptotic behavior of nontrivial radial convex solutions is also considered.
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The Characterization of Graphs with No 2connected Spanning Subgraph of
V
_{8}
as a Minor
Xiaomin ZHOU, Xiaxia GUAN, Chengfu QIN, Weihua YANG
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(4): 902915.
DOI:
10.1007/s1025502210227
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It is difficult to characterize graphs which contain no a 2connected graph as a minor in graph theory. Let
V
_{8}
be a graph constructed from an 8cycle by connecting the antipodal vertices. There are thirteen 2connected spanning subgraphs of
V
_{8}
. In particular, one of them is obtained from the Petersen graph by deleting two vertices and it is also a hard problem to characterize Petersenminorfree graphs. In this paper, we characterize internally 4connected graphs which contain a 2connected spanning subgraph of
V
_{8}
as a forbidden minor.
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Analyzing Multiple Phenotypes Based on Principal Component Analysis
Deliang BU, Sanguo ZHANG, Na LI
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(4): 843860.
DOI:
10.1007/s1025502210192
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Joint analysis of multiple phenotypes can have better interpretation of complex diseases and increase statistical power to detect more significant single nucleotide polymorphisms (SNPs) compare to traditional single phenotype analysis in genomewide association analysis. Principle component analysis (PCA), as a popular dimension reduction method, has been broadly used in the analysis of multiple phenotypes. Since PCA transforms the original phenotypes into principal components (PCs), it is natural to think that by analyzing these PCs, we can combine information across phenotypes. Existing PCAbased methods can be divided into two categories, either selecting one particular PC manually or combining information from all PCs. In this paper, we propose an adaptive principle component test (APCT) which selects and combines the PCs adaptively by using Cauchy combination method. Our proposed method can be seen as a generalization of traditional PCA based method since it contains two existing methods as special situation. Extensive simulation shows that our method is robust and can generate powerful result in various situations. The real data analysis of stock mice data also demonstrate that our proposed APCT can identify significant SNPs that are missed by traditional methods.
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