Acta Mathematica Scientia (Series A)
Measurment Science and Technology ,CAS
Edited by  Editorial Committee of Acta Mathematica
Scientia
Tel: 027-87199206(Series A & Series B)
027-87199087(Series B)
E-mail: actams@wipm.ac.cn
ISSN 1003-3998
CN 　42-1226/O
 The General Inverse Bonnesen-Style Inequalities in $\mathbb{R}^n$ Xu Dong,Yan Zhang,Chunna Zeng,Xingxing Wang Acta mathematica scientia,Series A. 2022, 42 (3):  641-650.  Abstract ( 42 )   RICH HTML PDF(337KB) ( 39 )   The isoperimetric problem plays an important role in integral geometry. In this paper we mainly investigate the inverse form of the isoperimetric inequality, i.e. the general inverse Bonnesen-type inequalities. The upper bounds of several new general isoperimetric genus are obtained. Futhermore, as corollaries, we get a series of classical inverse Bonnesen-type inequalities in the plane. Finally, the best estimate between the results of three upper bounds is given.
 Multiple Spectra of Self-Similar Measures with Three Digits Haixiong Li,Xinlin Wu,Daoxin Ding Acta mathematica scientia,Series A. 2022, 42 (3):  705-715.  Abstract ( 6 )   RICH HTML PDF(390KB) ( 7 )   It is well known that the self-similar measure $\mu_{k, a, b}$ defined by $\mu_{k, a, b}(\cdot)=\frac{1}{3}\sum\limits_{i=0}^{2}\mu_{k, a, b}(3k(\cdot)-ki)$ is a spectral measure with a spectrum $\Lambda(3k, C)=\left\{\sum\limits_{j=0}^{{\rm finite}}(3k)^jc_j:c_j\in C=\{0, 1, 2\}\right\}.$ In this paper, by applying the properties of congruences and the order of elements in the finite group, we obtain some conditions on the integer $p$ such that the set $p\Lambda(3k, C)$ is also a spectrum for $\mu_{k, a, b}$. Moreover, an example is given to explain our theory.
 Research on the Lowest Energy Solution ofChern-Simons-Schrödinger Equation with Trapping Potential Ying Yang,Liejun Shen Acta mathematica scientia,Series A. 2022, 42 (3):  716-729.  Abstract ( 11 )   RICH HTML PDF(394KB) ( 14 )   In this paper, we mainly study the existence of solutions with prescribed $L^{2}$-norm to the Chern-Simons-Schrödinger (CSS) equation. This type problem can be transformed into look for the minimizer of the corresponding energy functional $E^\beta_{p} (u)$ under the constraint $\|u\|_{L^{ 2}(\mathbb{R}^2)}=1$. Concerning the subcritical mass case, that is, $p\in(0,2)$, no matter whether the potential function $V(x)$ equals to $0$, we prove that the constraint minimization can be achieved by some simple methods. We are also concerned with the critical mass case of $p=2$:if $V(x)\equiv0$, there exist two constants $\beta^*>\beta_*>0$ which can be explicitly determined such that the constraint minimization cannot achieved for any $\beta\in(0,\beta_{*}]\cup(\beta^{*},+\infty)$; if $V(x)\not\equiv0$, the constraint minimization cannot be achieved for $\beta>\beta^{*}$, but can be achieved for $\beta\in(0,\beta_{*}]$. In addition, we discuss the limit behavior of the mass subcritical constrained minimum energy when $p\nearrow2$.
 Ground State Solutions for Quasilinear Schrödinger Equation of Choquard Type Yanan Wang,Kaimin Teng Acta mathematica scientia,Series A. 2022, 42 (3):  730-748.  Abstract ( 6 )   RICH HTML PDF(399KB) ( 17 )   In this paper, we consider the following quasilinear Schrödinger equations of Choquard type $\begin{eqnarray*} -\triangle u+\frac{k}{2}u\triangle u^2+V(x)u=(I_{\alpha}\ast|u|^{p})|u|^{p-2}u, \, \, x\in\mathbb{R}^N, \end{eqnarray*}$ where $N\geq3$, 0 < $\alpha$ < $N$, $ Existence of Positive Ground State Solutions for the Choquard Equation Xudong Shang,Jihui Zhang Acta mathematica scientia,Series A. 2022, 42 (3): 749-759. Abstract ( 13 ) RICH HTML PDF(355KB) ( 16 ) In this paper we study the following nonlinear Choquard equation$ $$-\Delta u + V(x)u= (I_{\alpha}* F(u))f(u), \hskip0.5cm x\in{{\Bbb R}} ^{N} ,$$ $where$N \geq 3$,$\alpha \in (0, N)$,$I_{\alpha}$is the Riesz potential,$V(x):\mathbb{R} ^{N} \rightarrow \mathbb{R} $is a given potential function, and$F\in {\cal C}^{1}(\mathbb{R}, \mathbb{R})$with$F'(s)=f(s)$. Under assumptions on$V$and$f$, we do not require the$(AR)$condition of$f$, the existence of ground state solutions are obtained via variational methods.  Existence and Multiplicity of Solutions for a 2$n$th-Order Discrete Boundary Value Problems with a Parameter Zhenguo Wang Acta mathematica scientia,Series A. 2022, 42 (3): 760-766. Abstract ( 12 ) RICH HTML PDF(313KB) ( 17 ) In this paper, we consider the existence and multiplicity of solutions for a$2n$th-order discrete boundary value problems depending on a parameter$\lambda$. When$\lambda\in\left(\frac{p(T)}{2B}, \frac{1}{2A}\right)$, we obtain a sufficient condition for the existence of solutions of a discrete boundary value problems by means of critical point theory. Finally, one example is given to illustrate our main result.  Infinitely Solutions for a Class of Nonlocal Quasilinear Elliptic Equations Qian Wang,Lin Chen,Nan Tang Acta mathematica scientia,Series A. 2022, 42 (3): 767-774. Abstract ( 10 ) RICH HTML PDF(281KB) ( 11 ) In this paper, we study the existence of multiple solutions for a class of nonlocal quasilinear elliptic problem$\left\{\begin{array}{ll} M\Big(\int_{\mathbb{R} ^{N}}(|\nabla u|^{p}+V(x)|u|^{p}){\rm d}x\Big)(-\Delta_{p}u+V(x)|u|^{p-2}u)=\sigma d^{-1}F_{u}(x, u, v)+\lambda|u|^{q-2}u, \nonumber\\ M\Big(\int_{\mathbb{R} ^{N}}(|\nabla v|^{p}+V(x)|v|^{p}){\rm d}x\Big)(-\Delta_{p}v+V(x)|v|^{p-2}v)=\sigma d^{-1}F_{v}(x, u, v)+\mu|v|^{q-2}v, \nonumber\\ u, v\in W^{1, p}(\mathbb{R} ^{N}), x\in\mathbb{R} ^{N}\nonumber \end{array}\right. $where$M(s)=s^{k}, k>0, N\geq3, 10, \sigma\in\mathbb{R} ^{N}$, and in which$p^{\ast}=\frac{Np}{N-p}, $and$p^{\ast}=\infty$if$p=N.$The weight function$V(x)\in C(\mathbb{R} ^{N})$satisfy some conditions.  Breather Wave Solutions, Lump Solutions and Semi-Rational Solutions of a Reduced (3+1)Dimensional Hirota Equation Chunmei Fang,Shoufu Tian Acta mathematica scientia,Series A. 2022, 42 (3): 775-783. Abstract ( 7 ) RICH HTML PDF(1085KB) ( 7 ) In this paper, the long wave limit method is used to study the exact solutions of the (3+1)dimensional Hirota equation under dimensional reduction$z$=$x$. First, the bilinear form is constructed by using Bell polynomials. Based on the bilinear form, the$n$-order breather wave solutions are obtained under some parameter constraints on the$N$-order soliton solution. Secondly, by using the long wave limit method, high order lump wave solutions are obtained. Finally, the combined solutions of the first-order, second-order lump wave solutions and single solitary wave solutions are derived, i.e. semi-rational solutions. All the obtained solutions were analyzed with Maple software for physical characteristics.  Statistical Solutions and Its Limiting Behavior for the Impulsive Discrete Ginzburg-Landau Equations Caidi Zhao,Huite Jiang,Chunqiu Li,Caraballo Tomás Acta mathematica scientia,Series A. 2022, 42 (3): 784-806. Abstract ( 5 ) RICH HTML PDF(497KB) ( 10 ) In this article we first prove the global well-posedness of the impulsive discrete Ginzburg-Landau equations. Then we establish that the generated process by the solution operators possesses a pullback attractor and a family of invariant Borel probability measures. Further, we formulate the definition of statistical solution for the addressed impulsive system and prove the existence. Our results reveal that the statistical solution of the impulsive system satisfies merely the Liouville type theorem piecewise, which implies that the Liouville type equation for impulsive system will not always hold true on the interval containing any impulsive point. Finally, we prove that the statistical solution of the impulsive discrete Ginzburg-Landau equations converges to that of the impulsive discrete Schrödinger equations.  Asymptotic Stability Analysis of Solutions to Transport Equations in Structured Bacterial Population Growth Hongxing Wu,Dengbin Yuan,Shenghua Wang Acta mathematica scientia,Series A. 2022, 42 (3): 807-817. Abstract ( 10 ) RICH HTML PDF(377KB) ( 10 ) With the help of linear operator theory, the transport equation with more general boundary condition for the structured equation with bacterial population as background is discussed. By means of resolving operator and comparison operator, it is proved that the corresponding transfer operator spectrum of the transfer equation consists of only a finite number of discrete eigenvalues with finite algebraic multiplicity in band domain$\Gamma_{\alpha, \beta}$. It is proved that the solution of the transfer equation is asymptotically stable when$\psi_0 \in D (A_{H_{\alpha, \beta}})$.  A Dynamic Model for a Class of New Generalized Absolute Value Equations Wenli Zheng,Jia Tang,Cairong Chen Acta mathematica scientia,Series A. 2022, 42 (3): 818-825. Abstract ( 10 ) RICH HTML PDF(352KB) ( 13 ) In this paper, a dynamic model for solving a class of new generalized absolute value equations (GAVE) is proposed. Under suitable conditions, it can be proved that the solution of the GAVE is equivalent to the equilibrium point of the dynamic model and that the equilibrium point of the dynamic model is asymptotically stable. Numerical experiments show that the proposed dynamic model is feasible and effective.  Oscillation Analysis of a Kind of Systems with Piecewise Continuous Arguments Ying Liu,Jianfang Gao Acta mathematica scientia,Series A. 2022, 42 (3): 826-838. Abstract ( 4 ) RICH HTML PDF(378KB) ( 7 ) In this paper, we mainly use$\theta$-method to analyze the oscillation of differential equations with piecewise continuous arguments of retarded type, and discuss the oscillation and non-oscillation of analytic solution and numerical solution. The sufficient conditions for the numerical methods to preserve the oscillation of the equation under the condition of the analytic solution oscillation are obtained. Meanwhile, some numerical experiments are given.  Oscillation of Second-Order Nonlinear Nonautonomous Delay Dynamic Equations on Time Scales Ping Zhang,Jiashan Yang,Guijiang Qin Acta mathematica scientia,Series A. 2022, 42 (3): 839-850. Abstract ( 8 ) RICH HTML PDF(418KB) ( 8 ) The oscillatory behavior of a class of second-order nonlinear nonautonomous variable delay damped dynamic equations are studied on a time scale T, where the equations are noncanonical form. By using the generalized Riccati transformation, and the time scales theory and the classical inequality, we establish some new oscillation criteria for the equation. The results fully reflect the influential actions of delay functions and damping terms in system oscillation. Finally, some examples are given to show that our results extend, improve and enrich those reported in the literature, and that they have good effectiveness and practicability.  Dynamics of an Anthrax Epidemiological Model with Time Delay and Seasonality Tailei Zhang,Junli Liu,Mengjie Han Acta mathematica scientia,Series A. 2022, 42 (3): 851-866. Abstract ( 10 ) RICH HTML PDF(480KB) ( 13 ) In this paper, we developed a time-delayed epidemiological model to describe the anthrax transmission, which incorporates seasonality and the incubation period of the animal population. The basic reproduction number$R_{0}$can be obtained. It is shown that the threshold dynamics is completely determined by the basic reproduction number. If$R_{0}<1$, the disease-free periodic solution is globally attractive and the disease will die out; if$R_{0} >1$, then there exists at least one positive periodic solution and the disease persists. We further investigate the corresponding autonomous system, the global stability of the disease-free equilibrium and the positive equilibrium is established in terms of$[R_0]$. Numerical simulations are carried out to investigate the sensitivity of$R_0$about the parameters, the effects of vaccination and carcass disposal on controlling the spread of anthrax is also analyzed.  Optimal Boundary Control for a Hierarchical Size-Structured Population Model with Delay Zerong He,Yimeng Dou,Mengjie Han Acta mathematica scientia,Series A. 2022, 42 (3): 867-880. Abstract ( 5 ) RICH HTML PDF(438KB) ( 12 ) In this article, we formulate a population control model, which is based upon the hierarchical size-structure and the incubation delay. For a given ideal population distribution, we investigate the optimal input problem: How to choose a inflow way such that the sum of the deviation between the terminal state and the given one and the total costs is minimal. The well-posedness is established by the method of characteristics, the existence of unique optimal policy is shown by the Ekeland variational principle, and the optimal policy is exactly described by a normal cone and an adjoint system. These results set a foundational framework for practical applications.  Regular Kernel Method for State Space Model Chao Wang,Bo Li,Lei Wang Acta mathematica scientia,Series A. 2022, 42 (3): 881-890. Abstract ( 9 ) RICH HTML PDF(465KB) ( 7 ) State space models(SSMs) provide a general framework for studying stochastic processes, which has been applied in revealing the true underlying economic processes of an economy, recognizing cellphone signals, detecting the loaction of an airplane on a radar screen, et al. In this paper, we study the Markov state space models by modeling the space transformation with reproducing kernel Hilbert space. Not only the existence and uniqueness of solutions are given, but also the error is estimate in L2 spaces. We applied our method in Visibility prediction at airport in National Post-Graduate Mathematical Contest in Modeling supported by China Academic Degrees & Graduate Education Development Center.  Moving-Water Equilibria Preserving Central Scheme for the Saint-Venant System Yiming Luo,Dingfang Li,Min Liu,Jian Dong Acta mathematica scientia,Series A. 2022, 42 (3): 891-903. Abstract ( 6 ) RICH HTML PDF(5011KB) ( 9 ) In this paper, we propose a second-order unstaggered central finite volume scheme for the Saint-Venant system. Classical central scheme can preserve still-water steady state solution by reconstructing conservative variables and the water level, but generates enormous numerical oscillation when considering moving-water steady state. We choose to reconstruct conservative variables and the energy, and design a new discretization of the source term to preserve moving-water equilibria and capture its small perturbations. In the end, several classical numerical experiments are performed to verify the proposed scheme which is convergent, well-balanced and robust.  A New Projection Algorithm for Solving Pseudo-Monotone Variational Inequality and Fixed Point Problems Jing Yang,Xianjun Long Acta mathematica scientia,Series A. 2022, 42 (3): 904-919. Abstract ( 16 ) RICH HTML PDF(694KB) ( 13 ) In this paper, we propose a new projection algorithm for finding a common element of psedomonotone variational inequality problems and fixed point set of demicontractive mappings in Hilbert spaces. We prove that this new algorithm converges strongly to the common element for a psedomonotone and uniformly continuous mapping. Finally, we provide some numerical experiments to illustrate the efficiency and advantages of the new projection algorithm.  The Greedy Simplex Algorithm for Double Sparsity Constrained Optimization Problems Tingwei Pan,Suxiang He Acta mathematica scientia,Series A. 2022, 42 (3): 920-933. Abstract ( 8 ) RICH HTML PDF(400KB) ( 12 ) In view of the shortcomings of the alternating minimization method in which it needs to calculate the Lipschitz constant of the objective function gradient and the Lipschitz condition is needed to construct the L-stable point of the problem, this paper proposes a greedy simplex algorithm to solve the optimization problems with double sparse constraints. The CW optimality condition for double sparse constrained optimization problems is described. Based on the CW optimality condition, the iterative steps of the algorithm are designed, and the global convergence of the sequence of iterative points generated by the algorithm to the optimal solution of the problem is proved under the weaker assumptions.  Precise Large Deviations for a Bidimensional Risk Model with the Regression Dependent Structure Zhenlong Chen,Yang Liu,Ke-ang Fu Acta mathematica scientia,Series A. 2022, 42 (3): 934-942. Abstract ( 7 ) RICH HTML PDF(374KB) ( 21 ) In this paper, we consider a non-standard bidimensional risk model, in which the claim sizes$ \{\vec{X}_k=(X_{1k}, X_{2k})^T, k\ge 1\}\$ form a sequence of independent and identically distributed random vectors with nonnegative components that are allowed to be dependent on each other, and there exists a regression dependent structure between these vectors and the inter-arrival times. By assuming that the univariate marginal distributions of claim vectors have consistently varying tails, we obtain the precise large deviation formulas for the bidimensional risk model with the regression dependent structure.