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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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Global Stability of a Timedelayed Malaria Model with Standard Incidence Rate
Songbai GUO, Min HE, Jingan CUI
Acta Mathematicae Applicatae Sinica(English Series). 2023, 39(2): 211221.
DOI:
10.1007/s102550231042y
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A fourdimensional delay differential equations (DDEs) model of malaria with standard incidence rate is proposed. By utilizing the limiting system of the model and Lyapunov direct method, the global stability of equilibria of the model is obtained with respect to the basic reproduction number ${R}_{0}$. Specifically, it shows that the diseasefree equilibrium ${E}^{0}$ is globally asymptotically stable (GAS) for ${R}_{0}<1$, and globally attractive (GA) for ${R}_{0}=1$, while the endemic equilibrium $E^{\ast}$ is GAS and ${E}^{0}$ is unstable for ${R}_{0}>1$. Especially, to obtain the global stability of the equilibrium $E^{\ast}$ for $R_{0}>1$, the weak persistence of the model is proved by some analysis techniques.
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Mild Solutions of a Class of Conformable Fractional Differential Equations with Nonlocal Conditions
Mohamed BOUAOUID
Acta Mathematicae Applicatae Sinica(English Series). 2023, 39(2): 249261.
DOI:
10.1007/s1025502210156
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This paper deals with the existence, uniqueness and continuous dependence of mild solutions for a class of conformable fractional differential equations with nonlocal initial conditions. The results are obtained by means of the classical fixed point theorems combined with the theory of cosine family of linear operators.
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Best Linear Unbiased Estimators of Location and Scale Ranked Set Parameters under Moving Extremes Sampling Design
Yanfei DONG, Wangxue CHEN, Minyu XIE
Acta Mathematicae Applicatae Sinica(English Series). 2023, 39(2): 222231.
DOI:
10.1007/s102550231043x
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In the current paper, the best linear unbiased estimators (BLUEs) of location and scale parameters from locationscale family will be respectively proposed in cases when one parameter is known and when both are unknown under moving extremes ranked set sampling (MERSS). Explicit mathematical expressions of these estimators and their variances are derived. Their relative efficiencies with respect to the minimum variance unbiased estimators (MVUEs) under simple random sampling (SRS) are compared for the cases of some usual distributions. The numerical results show that the BLUEs under MERSS are significantly more efficient than the MVUEs under SRS.
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Energy Decay for von KármánGurtinPipkin System
Hanni DRIDI
Acta Mathematicae Applicatae Sinica(English Series). 2023, 39(2): 306319.
DOI:
10.1007/s1025502310458
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This paper aims to prove the asymptotic behavior of the solution for the thermoelastic von Karman system where the thermal conduction is given by GurtinPipkins law. Existence and uniqueness of the solution are proved within the semigroup framework and stability is achieved thanks to a suitable Lyapunov functional. Therefore, the stability result clarified that the solutions energy functional decays exponentially at infinite time.
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On the Existence of Ground State Solutions to a Quasilinear Schrödinger Equation involving
p
Laplacian
Jixiu WANG, Qi GAO
Acta Mathematicae Applicatae Sinica(English Series). 2023, 39(2): 381395.
DOI:
10.1007/s1025502310538
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We consider the following quasilinear Schrödinger equation involving $p$Laplacian \begin{align*} \Delta_p u +V(x)u^{p2}u\Delta_p(u^{2\eta})u^{2\eta2}u=\lambda\frac{u^{q2}u}{x^{\mu}}+\frac{u^{2\eta p^*(\nu)2}u}{x^\nu}\quad\text{in}\ \mathbb{R}^N, \end{align*} where $ N> p>1,\ \eta\ge \frac{p}{2(p1)}$, $p< q<2\eta p^*(\mu)$, $p^*(s)=\frac{p(Ns)}{Np}$, and $\lambda, \mu, \nu$ are parameters with $\lambda>0$, $\mu, \nu \in [0,p)$. Via the Mountain Pass Theorem and the Concentration Compactness Principle, we establish the existence of nontrivial ground state solutions for the above problem.
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Solutions of the CamassaHolm Equation Near the Soliton
Danping DING, Wei LU
Acta Mathematicae Applicatae Sinica(English Series). 2023, 39(2): 450464.
DOI:
10.1007/s1025502310565
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In this paper, solutions of the CamassaHolm equation near the soliton $Q$ is decomposed by pseudoconformal transformation as follows: $\lambda^{1/2}(t)u(t,\lambda(t)y+x(t))=Q(y)+\varepsilon(t,y)$, and the estimation formula with respect to $\varepsilon(t,y)$ is obtained: $\varepsilon(t,y)\leq Ca_3Te^{\thetay}+\lambda^{1/2}(t)\varepsilon_0$. For the CH equation, we prove that the solution of the Cauchy problem and the soliton $Q$ is sufficiently close as $y\rightarrow\infty$, and the approximation degree of the solution and $Q$ is the same as that of initial data and $Q$, besides the energy distribution of $\varepsilon$ is consistent with the distribution of the soliton $Q$ in $H^2$.
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Approximate Analytical Solution of the Generalized KolmogorovPetrovskyPiskunov Equation with Cubic Nonlinearity
Weiguo ZHANG, Xiekui HU, Xingqian LING, Wenxia LI
Acta Mathematicae Applicatae Sinica(English Series). 2023, 39(2): 424449.
DOI:
10.1007/s1025502310547
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In this paper, the approximate analytical oscillatory solutions to the generalized KolmogorovPetrovskyPiskunov equation (gKPPE for short) are discussed by employing the theory of dynamical system and hypothesis undetermined method. According to the corresponding dynamical system of the bounded traveling wave solutions to the gKPPE, the number and qualitative properties of these bounded solutions are received. Furthermore, pulses (bellshaped) and waves fronts (kinkshaped) of the gKPPE are given. In particular, two types of approximate analytical oscillatory solutions are constructed. Besides, the error estimations between the approximate analytical oscillatory solutions and the exact solutions of the gKPPE are obtained by the homogeneity principle. Finally, the approximate analytical oscillatory solutions are compared with the numerical solutions, which shows the two types of solutions are similar.
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Path Factors and Neighborhoods of Independent Sets in Graphs
Sizhong ZHOU
Acta Mathematicae Applicatae Sinica(English Series). 2023, 39(2): 232238.
DOI:
10.1007/s1025502210962
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A pathfactor is a spanning subgraph $F$ of $G$ such that every component of $F$ is a path with at least two vertices. Let $k\geq2$ be an integer. A $P_{\geq k}$factor of $G$ means a path factor in which each component is a path with at least $k$ vertices. A graph $G$ is a $P_{\geq k}$factor covered graph if for any $e\in E(G)$, $G$ has a $P_{\geq k}$factor including $e$. Let $\beta$ be a real number with $\frac{1}{3}\leq\beta\leq1$ and $k$ be a positive integer. We verify that (\romannumeral1) a $k$connected graph $G$ of order $n$ with $n\geq5k+2$ has a $P_{\geq3}$factor if $N_G(I)>\beta(n3k1)+k$ for every independent set $I$ of $G$ with $I=\lfloor\beta(2k+1)\rfloor$; (\romannumeral2) a $(k+1)$connected graph $G$ of order $n$ with $n\geq5k+2$ is a $P_{\geq3}$factor covered graph if $N_G(I)>\beta(n3k1)+k+1$ for every independent set $I$ of $G$ with $I=\lfloor\beta(2k+1)\rfloor$.
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Estimation of Treatment Effects in Nonlinear Models with Unobserved Confounding
Yuling LI, Jun WANG
Acta Mathematicae Applicatae Sinica(English Series). 2023, 39(2): 320336.
DOI:
10.1007/s102550231051x
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Estimation of treatment effects is one of the crucial mainstays in economics and sociology studies. The problem will become more serious and complicated if the treatment variable is endogenous for the presence of unobserved confounding. The estimation and conclusion are likely to be biased and misleading if the endogeny of treatment variable is ignored. In this article, we propose the pseudo maximum likelihood method to estimate treatment effects in nonlinear models. The proposed method allows the unobserved confounding and random error terms to exist in an arbitrary relationship (such as, add or multiply), and the unobserved confounding have different influence directions on treatment variables and outcome variables. The proposed estimator is consistent and asymptotically normally distributed. Simulation studies show that the proposed estimator performs better than the special regression estimator, and the proposed method is stable for various distribution of error terms. Finally, the proposed method is applied to the real data that studies the influence of individuals have health insurance on an individual’s decision to visit a doctor.
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The Spread Speed of Multiple Catalytic Branching Random Walks
Rongli LIU
Acta Mathematicae Applicatae Sinica(English Series). 2023, 39(2): 262292.
DOI:
10.1007/s1025502310467
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In this paper we study the asymptotic behavior of the maximal position of a supercritical multiple catalytic branching random walk $(X_n)$ on $\mathbb Z$. If $M_n$ is its maximal position at time $n$, we prove that there is a constant $\alpha>0$ such that $M_n/n$ converges to $\alpha$ almost surely on the set of infinite number of visits to the set of catalysts. We also derive the asymptotic law of the centered process $M_n\alpha n$ as $n\to \infty$. Our results are similar to those in [13]. However, our results are proved under the assumption of finite $L\log L$ moment instead of finite second moment. We also study the limit of $(X_n)$ as a measurevalued Markov process. For any function $f$ with compact support, we prove a strong law of large numbers for the process $X_n(f)$.
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Controllability of Nonlinear Discrete Systems with Degeneracy
Yue LYU, Xueli TAN, Xue YANG, Yong LI
Acta Mathematicae Applicatae Sinica(English Series). 2023, 39(2): 293305.
DOI:
10.1007/s1025502310476
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This paper concerns the controllability of autonomous and nonautonomous nonlinear discrete systems, in which linear parts might admit certain degeneracy. By introducing Fredholm operators and coincidence degree theory, sufficient conditions for nonlinear discrete systems to be controllable are presented. In addition, applications are given to illustrate main results.
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Starfactorization of the Complete Bipartite Multigraphs
Jing SHI, Jian WANG, Beiliang DU
Acta Mathematicae Applicatae Sinica(English Series). 2023, 39(2): 239248.
DOI:
10.1007/s1025502310449
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Let $\lambda K_{m,n}$ be a complete bipartite multigraph with two partite sets having $m$ and $n$ vertices, respectively. A $K_{p,q}$factorization of $\lambda K_{m,n}$ is a set of $K_{p,q}$factors of $\lambda K_{m,n}$ which partition the set of edges of $\lambda K_{m,n}$. When $\lambda =1$, Martin, in [Complete bipartite factorizations by complete bipartite graphs, Discrete Math., 167/168 (1997), 461480], gave simple necessary conditions for such a factorization to exist, and conjectured those conditions are always sufficient. In this paper, we will study the $K_{p,q}$factorization of $\lambda K_{m,n}$ for $p=1$, to show that the necessary conditions for such a factorization are always sufficient whenever related parameters are sufficiently large.
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A.S. Convergence Rate and $L^{p}$Convergence of Bisexual Branching Processes in a Random Environment and Varying Environment
Sheng XIAO, Xiangdong LIU, Yingqiu LI
Acta Mathematicae Applicatae Sinica(English Series). 2023, 39(2): 337353.
DOI:
10.1007/s1025502310529
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Let $(Z_{n})$ be a supercritical bisexual branching process in a random environment $\xi$. We study the almost sure (a.s.) convergence rate of the submartingale $\overline{W}_{n} =Z_{n}/I_{n}$ to its limit $\overline{W}$, where $(I_n)$ is an usually used norming sequence. We prove that under a moment condition of order $p \in (1,2),\overline{W}\overline{W}_{n}=o(e^{na})$ a.s. for some $a>0$ that we find explicitly; assuming the logarithmic moment condition holds, we have $\overline{W}\overline{W}_{n}=o(n^{\alpha})$ a.s.. In order to obtain these results, we provide the $L^{p}$ convergence of $(\overline{W}_{n})$; similar conclusions hold for a bisexual branching process in a varying environment.
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A Bregmanstyle Partially Symmetric Alternating Direction Method of Multipliers for Nonconvex Multiblock Optimization
Pengjie LIU, Jinbao JIAN, Guodong MA
Acta Mathematicae Applicatae Sinica(English Series). 2023, 39(2): 354380.
DOI:
10.1007/s1025502310485
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The alternating direction method of multipliers (ADMM) is one of the most successful and powerful methods for separable minimization optimization. Based on the idea of symmetric ADMM in twoblock optimization, we add an updating formula for the Lagrange multiplier without restricting its position for multiblock one. Then, combining with the Bregman distance, in this work, a Bregmanstyle partially symmetric ADMM is presented for nonconvex multiblock optimization with linear constraints, and the Lagrange multiplier is updated twice with different relaxation factors in the iteration scheme. Under the suitable conditions, the global convergence, strong convergence and convergence rate of the presented method are analyzed and obtained. Finally, some preliminary numerical results are reported to support the correctness of the theoretical assertions, and these show that the presented method is numerically effective.
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Defined Contribution Pension Planning with the Return of Premiums Clauses and HARA Preference in Stochastic Environments
Hao CHANG, Xingjiang CHEN
Acta Mathematicae Applicatae Sinica(English Series). 2023, 39(2): 396423.
DOI:
10.1007/s102550231050y
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This paper studies a defined contribution (DC) pension fund investment problem with return of premiums clauses in a stochastic interest rate and stochastic volatility environment. In practice, most of pension plans were subject to the return of premiums clauses to protect the rights of pension members who died before retirement. In the mathematical modeling, we assume that a part of pension members could withdraw their premiums if they died before retirement and surviving members could equally share the difference between accumulated contributions and returned premiums. We suppose that the financial market consists of a riskfree asset, a stock, and a zerocoupon bond. The interest rate is driven by a stochastic affine interest rate model and the stock price follows the Heston’s stochastic volatility model with stochastic interest rates. Different fund managers have different risk preferences, and the hyperbolic absolute risk aversion (HARA) utility function is a general one including a power utility, an exponential utility, and a logarithm utility as special cases. We are concerned with an optimal portfolio to maximize the expected utility of terminal wealth by choosing the HARA utility function in the analysis. By using the principle of dynamic programming and Legendre transformdual theory, we obtain explicit solutions of optimal strategies. Some special cases are also derived in detail. Finally, a numerical simulation is provided to illustrate our results.
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