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26 October 2023, Volume 43 Issue 5 Previous Issue    Next Issue

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The Schödinger Uncertainty Relation in the Fock-Type Spaces Li Wenxin,Lian Pan,Liang Yuxia Acta mathematica scientia,Series A. 2023, 43 (5):  1321-1332.  Abstract ( 225 )   RICH HTML PDF (660KB) ( 449 )   Save In this paper, the Schödinger uncertainty relation for the unilateral weighted shift operators on Fock space is established, and the explicit expression when the equality attained is given, which further extends the Heisenberg uncertainty relation on Fock space established in [4] and overcomes the difficulty in [16]. In addition, we generalize the uncertainty relation to the multiple operators case. A new uncertainty inequality in the form of non-self adjoint operators is obtained as well. References | Related Articles | Metrics

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The Reproducing Kernel of Bergman Space and the Eigenvectors of Toeplitz Operator Ding Xuanhao,Hou Lin,Li Yongning Acta mathematica scientia,Series A. 2023, 43 (5):  1333-1340.  Abstract ( 104 )   RICH HTML PDF (614KB) ( 178 )   Save In the Bergman space, it is well-known that $ T_{\varphi}K_{z}=\varphi(z)K_{z} $ for $ \varphi\in \overline{H^{\infty}} $, that is, $ K_{z} $ is the eigenvector of $ T_{\varphi} $ corresponding the eigenvalue $ \varphi(z) $, where $ K_{z} $ is the reproducing kernel of Bergman space. Conversely, if $ \varphi $ is a bounded harmonic function and if there is $ z\in \mathbb{D} $ (or for every $ z\in\mathbb{D} $), $ K_{z} $ is a eigenvector of $ T_{\varphi} $, whether there must be $ \varphi\in \overline{H^{\infty}} $ ? In view of the above questions, in this paper we give a complete characterization of the Toeplitz operator with the bounded harmonic symbol which have the reproducing kernels $ K_{z} $ as their eigenvectors. Moreover, we partially describe the Toeplitz operators with the bounded harmonic symbol whose eigenvalues are all $ \varphi(z) (z\in \mathbb{D}) $. References | Related Articles | Metrics

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Exact Multiplicity of Positive Solutions for a Semipositone Mean Curvature Problem with Concave Nonlinearity Li Xiaodong, Gao Hongliang, Xu Jing Acta mathematica scientia,Series A. 2023, 43 (5):  1341-1349.  Abstract ( 70 )   RICH HTML PDF (770KB) ( 322 )   Save In this paper, we study the exact multiplicity and bifurcation diagrams of positive solutions for the prescribed mean curvature problem in one-dimensional Minkowski space in the form of $ \left\{\begin{array}{ll} -\left(\frac{u'}{\sqrt{1-u'^{2}}}\right)'=\lambda f(u), x\in(-L,L),\\ u(-L)=0=u(L), \end{array} \right. $ where $\lambda>0$ is a bifurcation parameter and $L>0$ is an evolution parameters, $f\in C^{2}([0,\infty), \mathbb{R})$ satisfies $f(0)<0$ and $f$ is concave for $0. In two different cases, we obtain that the above problem has zero, exactly one, or exactly two positive solutions according to different ranges of $\lambda$. The arguments are based upon a detailed analysis of the time map. Figures and Tables | References | Related Articles | Metrics

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On the Blow-Up Solutions of Inhomogeneous Nonlinear Schrödinger Equation with a Partial Confinement Jian Hui, Gong Min, Wang Li Acta mathematica scientia,Series A. 2023, 43 (5):  1350-1372.  Abstract ( 102 )   RICH HTML PDF (797KB) ( 342 )   Save This paper is devoted to the Cauchy problem of inhomogeneous nonlinear Schrödinger equation in the presence of a partial confinement, which is an important model in Bose-Einstein condensates. Combining the variational characterization of the ground state of a nonlinear elliptic equation and the conservations of mass and energy, we first obtain a global solution and show the existence of blow-up solutions for some special initial data by scaling techniques. Then, we study the $L^2$-concentration phenomenon for the blow-up solutions. Finally, we apply the variational arguments connected to the above ground state to investigate the dynamics of $L^2$-minimal blow-up solutions, i.e., the limiting profile, mass-concentration and blow-up rate of the blow-up solutions with minimal mass. We extend the global existence and blow-up results of Zhang[34] to the case of inhomogeneous nonlinearities and improve partial results of Pan and Zhang[23] to space dimensions $N\geq2$ in the inhomogeneous case. References | Related Articles | Metrics

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Existence of Positive Solutions for a Class of Schrödinger-Newton Systems with Critical Exponent Cheng Qingfang,Liao Jiafeng,Yuan Yanxiang Acta mathematica scientia,Series A. 2023, 43 (5):  1373-1381.  Abstract ( 77 )   RICH HTML PDF (629KB) ( 312 )   Save In this paper, we study the existence of positive solutions for a class of Schrödinger-Newton system with critical exponents on bounded domain, and obtain two positive solutions by the variational method. References | Related Articles | Metrics

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On Second Order Complex Differential Equations with Coefficients of Period $ {2\pi{\rm i}}$ Zhang Jie,Zhao Donghai Acta mathematica scientia,Series A. 2023, 43 (5):  1382-1390.  Abstract ( 101 )   RICH HTML PDF (633KB) ( 254 )   Save This paper mainly learned classic book `Nevanlinna theory and complex differential equations' due to Laine and considered the second order complex differential equation $f ''(z) + A (z) f(z)=0, \lambda(f)<\infty$ with coefficient $A (z) $ whose period is $2 \pi{\rm i}$. It found a possible error in the original proof and gave its partial correction, and also it gave a slightly weaker conclusion than its possibly controversial result in the original literature. References | Related Articles | Metrics

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Global Bifurcation for the Yamabe Equation on the Unit Sphere Dai Guowei,Gao Siyu,Ma Ruyun Acta mathematica scientia,Series A. 2023, 43 (5):  1391-1396.  Abstract ( 65 )   RICH HTML PDF (629KB) ( 244 )   Save We study the Yamabe equation on the $N$-dimensional unit sphere $\mathbb{S}^N$ $\begin{equation} -\Delta_{\mathbb{S}^N} v+\lambda v=v^{\frac{N+2}{N-2}}.\nonumber \end{equation}$ By bifurcation technique, for each $k\geq1$, we prove that this equation has at least one non-constant solution $v_k$ for any $\lambda>\lambda_k:=(k+N-1)(N-2)/4$ such that $v_k-\lambda^{1/(N^{*}-1)}$ has exactly $k$ zeroes, all of them are in $(-1, 1)$ and are simple, where $N^{*}$ is the sobolev critical exponent. As application, we obtain the existence of non-radial solutions of a nonlinear elliptic equation on $\mathbb{R}^N$ with $n\geq4$. Moreover, we also obtain the global bifurcation results of the Yamabe problem in product manifolds with one of the manifold is the unit sphere. References | Related Articles | Metrics

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Time Decay Rate for Large-Solution About 3D Compressible MHD Equations Chen Fei,Wang Shuai,Zhao Yongye,Wang Chuanbao Acta mathematica scientia,Series A. 2023, 43 (5):  1397-1408.  Abstract ( 82 )   RICH HTML PDF (697KB) ( 345 )   Save This paper focus on time decay rate for large-solution about compressible magnetohydrodynamic equations in $\mathbb{R}^3$. Provided that $(\sigma_{0}-1,u_{0},M_{0})\in L^1\cap H^2$, based on the work of Chen et al.[1], $\|\nabla(\sigma-1,u,M)\|_{H^1}\leqslant C(1+t)^{-\frac{5}{4}}$ is obtained in reference [2], obviously, time decay rate of the 2nd-order derivative of the solution in [2] is not ideal. Here, we improve that of $\|\nabla^2 (\sigma-1,u,M)\|_{L^2}$ to be $(1+t)^{-\frac{7}{4}}$ by the frequency decomposition method[3]. References | Related Articles | Metrics

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The Radial Symmetry and Monotonicity of Entire Solutions for Fractional Parabolic Equations Tang Yanjuan Acta mathematica scientia,Series A. 2023, 43 (5):  1409-1416.  Abstract ( 55 )   RICH HTML PDF (581KB) ( 299 )   Save This paper mainly develops the radial symmetry and monotonicity of entire solutions for fractional parabolic equations. To obtain the symmetry and monotonicity of entire solutions, the narrow region principle and maximum principle for antisymmetric functions in [9] are needed. Furthermore, to circumvent the difficulty from nonlocality for the fractional Laplacian, a fractional parabolic version of the method of moving planes will be adopted. References | Related Articles | Metrics

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Blow-Up Conditions of Porous Medium Systems with Gradient Source Terms and Nonlinear Boundary Conditions Shen Xuhui,Ding Juntang Acta mathematica scientia,Series A. 2023, 43 (5):  1417-1426.  Abstract ( 42 )   RICH HTML PDF (602KB) ( 309 )   Save In this paper, we consider the blow-up of solutions to the following porous medium systems: $ \left\{ \begin{array}{ll} u_{t} =\Delta u^l+f(u,v,|\nabla u|^2,t), & \\\displaystyle v_{t} =\Delta v^m+g(u,v,|\nabla v|^2,t),&x\in\Omega, \ t\in(0,t^*), \\\displaystyle \frac{\partial u}{\partial\nu}=p(u), \ \frac{\partial v}{\partial\nu}=q(v), &x\in\partial\Omega, \ t\in(0,t^*), \\\displaystyle u(x,0)=u_{0}(x), \ v(x,0)=v_{0}(x), &x\in\overline{\Omega}, \end{array} \right. $ where $l,m>1, \ \Omega\subset\mathbb{R}^N \ (N\geq2)$ is a bounded domain with smooth boundary $\partial\Omega$. Using the differential inequality techniques and the maximum principles, we give a sufficient condition to ensure that the positive solution $(u,v)$ of the above problem is a blow-up solution that blows up at a certain finite time $t^*$. An upper estimate of $t^*$ and an upper estimate of the blow-up rate of $(u,v)$ are also obtained. References | Related Articles | Metrics

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Symmetric and Periodic Solutions for a Class of Weakly Coupled Systems Composed of Two Particles with Obstacles Wang Zihuan,Wang Chao Acta mathematica scientia,Series A. 2023, 43 (5):  1427-1439.  Abstract ( 39 )   RICH HTML PDF (686KB) ( 283 )   Save The problems of the existence and multiplicity of symmetric periodic solutions with impact for a class of weakly coupled systems of two degrees of freedom with obstacles are concerned. Under some superlinear assumption on time-mapping, the existence of infinite symmetric harmonic solutions and symmetric subharmonic solutions with impacts of the equation are proved. Furthermore, a sufficient condition for the existence of even and periodic bouncing solution is given for the coupled symmetric impact equations of two degrees of freedom. References | Related Articles | Metrics

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Comparison on the Criticality Parameters for Two Supercritical Branching Processes in Random Environments Fan Xiequan,Hu Haijuan,Wu Hao,Ye Yinna Acta mathematica scientia,Series A. 2023, 43 (5):  1440-1470.  Abstract ( 60 )   RICH HTML PDF (839KB) ( 242 )   Save Let $\{Z_{1,n}, n\geq 0\}$ and $\{Z_{2,n}, n\geq 0\}$ be two supercritical branching processes in different random environments, with criticality parameters $\mu_1$ and $\mu_2$ respectively. It is known that with certain conditions, $\frac{1}{n} \ln Z_{1,n} \rightarrow \mu_1$ and $\frac{1}{m} \ln Z_{2,m} \rightarrow \mu_2$ in probability as $m, n \rightarrow \infty.$ In this paper, we are interested in the comparison on the two criticality parameters, which can be regarded as two-sample $U$-statistic. To this end, we prove a non-uniform Berry-Esseen's bound and Cramér's moderate deviations for $\frac{1}{n} \ln Z_{1,n} - \frac{1}{m} \ln Z_{2,m}$ as $m, n \rightarrow \infty.$ An application is also given for constructing confidence intervals of $\mu_1-\mu_2$. References | Related Articles | Metrics

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The PDP Boundary Control for a Class of 2$\times$2 Hyperbolic Partial Differential System Pang Yuting,Zhao Dongxia,Zhao Xin,Gao Caixia Acta mathematica scientia,Series A. 2023, 43 (5):  1471-1482.  Abstract ( 39 )   RICH HTML PDF (1065KB) ( 215 )   Save This paper studies the exponential stability of a single open-channel system with constant slope and bottom friction, which is described by a $2\times2$ hyperbolic partial differential equation. The position feedback and delayed position feedback (PDP for short) boundary controller is designed to solve the problem of feedback stabilization. Firstly, the well-posedness of the system is proved by using operator semigroup theory. Then, the exponential stability of the closed-loop system is analyzed by constructing an appropriate Lyapunov function, and sufficient conditions for feedback parameters and time-delay are obtained. In addition, the asymptotic expressions of the eigenvalues and the eigenfunctions of the system operator are given by spectral analysis method. Finally, a numerical example is used to evaluate the performance of the PDP controller. Figures and Tables | References | Related Articles | Metrics

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Parameter Estimation for an Ornstein-Uhlenbeck Process Driven by a Type of Gaussian Noise with Hurst Parameter $H\in (0,\frac{1}{2})$ Chen Yong,Li Ying,Sheng Ying,Gu Xiangmeng Acta mathematica scientia,Series A. 2023, 43 (5):  1483-1518.  Abstract ( 51 )   RICH HTML PDF (867KB) ( 266 )   Save In 2021, Chen and Zhou consider an inference problem for an Ornstein-Uhlenbeck process driven by a type of centered fractional Gaussian process $(G_t)_{t\ge 0}$. The second order mixed partial derivative of the covariance function $ R(t,\, s)=\mathbb{E}[G_t G_s]$ can be decomposed into two parts, one of which coincides with that of fractional Brownian motion and the other is bounded by $(ts)^{H-1}$ with $H\in (\frac12,\,1)$, up to a constant factor. In this paper, we investigate the same problem but with the assumption of $H\in (0,\,\frac12)$. It is well known that there is a significant difference between the Hilbert space associated with the fractional Gaussian processes in the case of $H\in (\frac12, 1)$ and that of $H\in (0, \frac12)$. The starting point of this paper is a quantitative relation between the inner product of $\mathfrak{H}$ associated with the Gaussian process $(G_t)_{t\ge 0}$ and that of the Hilbert space $\mathfrak{H}_1$ associated with the fractional Brownian motion $(B^{H}_t)_{t\ge 0}$. We prove the strong consistency with $H\in (0, \frac12)$, and the asymptotic normality and the Berry-Esséen bounds with $H\in (0,\frac38)$ for both the least squares estimator and the moment estimator of the drift parameter based on the continuous observations. References | Related Articles | Metrics

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Berry-Esseen Bound of Wavelet Estimator for Regression Model with Linear Process Errors Generated by LENQD Sequence Li Yongming,Pang Weicai,Li Naiyi Acta mathematica scientia,Series A. 2023, 43 (5):  1519-1528.  Abstract ( 33 )   RICH HTML PDF (646KB) ( 262 )   Save Based on linear process random errors generated by LENQD random sequence, the wavelet estimator of nonparametric fixed design regression model is considered. By the characteristic function inequality and moment inequality of LENQD random sequence, the Berry-Esseen bounds of the wavelet estimator for unknown regression function are obtained. And by choosing some suitable constants, their bounds can reach $O(n^{-\frac{1}{6}})$. The obtained results generalize the corresponding results in recent literature. References | Related Articles | Metrics

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Asymptotic Finite-Time Ruin Probability for a Bidimensional Perturbed Risk Model with General Investment Returns and Time-Dependent Claim Sizes Cheng Ming,Wang Dingcheng Acta mathematica scientia,Series A. 2023, 43 (5):  1529-1558.  Abstract ( 50 )   RICH HTML PDF (810KB) ( 293 )   Save The paper considers a bi-dimensional perturbed insurance risk model with general investment returns. Assume that the investment return is described by a càdlàg process, and two classes of claims and the inter-arrival times follow the Sarmanov dependence structure. When the claim-size distribution has a regularly varying tail, the paper derives the asymptotic formula of the finite-time ruin probability. When the càdlàg process describing investment returns is chosen as the Lévy process, Vasicek interest rate model, Cox-Ingersoll-Ross (CIR) interest rate model, or Heston model, the paper derives the asymptotic estimates for ruin probabilities under the corresponding investment returns. References | Related Articles | Metrics

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Statistical Solutions and Kolmogorov Entropy for First-Order Lattice Systems in Weighted Spaces Zou Tianfang,Zhao Caidi Acta mathematica scientia,Series A. 2023, 43 (5):  1559-1574.  Abstract ( 45 )   RICH HTML PDF (704KB) ( 259 )   Save This article studies the statistical solution and Kolmogorov entropy for first-order lattice systems in weighted spaces. The authors first establish that the initial value problem is global well-posed in weighted spaces and that the continuous process associated to the solution operators possesses a family of invariant Borel probability measures. Then they prove that this family of invariant Borel probability measures meets the Liouville theorem and is a statistical solution of the addressed systems. Finally, they prove the upper bound of the Kolmogorov entropy of the statistical solution. References | Related Articles | Metrics

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Application of VG Distortion Operator in Option Pricing Yao Luogen,Liu Huan,Chen Qiqiong Acta mathematica scientia,Series A. 2023, 43 (5):  1575-1584.  Abstract ( 30 )   RICH HTML PDF (717KB) ( 252 )   Save Using the idea of probability transformation, VG distortion operator is proposed based on the distribution of VG. It is shown that option prices obtained by VG distortion operator are consistent with option prices obtained under the mean correcting martingale measure in the VG model. The numerical results show that option prices obtained by VG distortion operator are very accurate. Figures and Tables | References | Related Articles | Metrics

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Doubling Weights Which are Piecewise Monotonic or Piecewise Doubling on $\mathbb{R}$ Dang Yungui,Dai Yuxia,Yu Ximei Acta mathematica scientia,Series A. 2023, 43 (5):  1585-1594.  Abstract ( 27 )   RICH HTML PDF (584KB) ( 270 )   Save The paper will study several kinds of doubling weights on $\mathbb{R}$. We firstly give sufficient and necessary conditions on which monotonic weights are doubling. Secondly we describe doubling piecewise monotonic weights. At last, we discuss doubling weights which are piecewise doubling on $\mathbb{R}$. References | Related Articles | Metrics

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Survival Analysis of an SVIR Epidemic Model with Media Coverage Li Dan,Wei Fengying,Mao Xuerong Acta mathematica scientia,Series A. 2023, 43 (5):  1595-1606.  Abstract ( 64 )   RICH HTML PDF (1602KB) ( 301 )   Save We consider the long-term properties of a stochastic SVIR epidemic model with media coverage and the logistic growth in this paper. We firstly derive the fitness of a unique global positive solution. Then we construct appropriate Lyapunov functions and obtain the existence of ergodic stationary distribution when ${R}_{0}^{s}>1$ is valid, and also derive sufficient conditions for persistence in the mean. Moreover, the exponential extinction to the density of the infected is figured out when ${R}_{0}^{e}<1$ holds. Figures and Tables | References | Related Articles | Metrics

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A Normalized Gradient Flow with Lagrange Multipliers for Computing Ground States of Spin-Orbit Coupled Spin-2 Bose-Einstein Condensates Yuan Yongjun Acta mathematica scientia,Series A. 2023, 43 (5):  1607-1619.  Abstract ( 38 )   RICH HTML PDF (2438KB) ( 226 )   Save In this paper, a normalized gradient flow with Lagrange multipliers is designed to compute ground states of spin-orbit coupled Spin-2 Bose-Einstein condensates. By excavating the implicit relation between projection coefficients, the difficult that the existed conditions (the conservation of total mass and magnetization) of the model problem is insufficient to determine all the projection coefficients, is overcome. Extensive numerical experiments are done to compute the ground states of spin-orbit coupled Spin-2 BECs with cyclic/ferromagenetic interactions. As a result, the effectiveness of the two algorithms is verified, and the phase transformation law about how the stripe pattern ground states and the square-lattice pattern ground states of spin-orbit coupled Spin-2 BECs change to each other with the spin-orbit coupling parameter,are revealed. Figures and Tables | References | Related Articles | Metrics

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Multi-Scale Approach for Diffeomorphic Multi-Modality Image Registration Ding Zijuan,Han Huan Acta mathematica scientia,Series A. 2023, 43 (5):  1620-1640.  Abstract ( 70 )   RICH HTML PDF (2647KB) ( 236 )   Save Multi-modality image registration is widely used in remote sensing, clinical medicine and other fields. Many models for multi-modality image registration have been proposed in the past few decades. Concerning this problem, there are two major challenges: (1) the existence of physical mesh folding; (2) the ill-posedness of similarity measure minimization/maximization problem. In order to address those problems, a multi-scale approach for diffeomorphic image registration based on Rényi's statistical dependence measure is proposed, which can avoid estimating joint probability density function, and obtain a smooth minimizer of the energy functional without mesh folding and prior regularization. In addition, the existence of solution for the proposed model and the convergence of the multi-scale approach are proved. And numerical experiments are performed to show the efficiency of the proposed algorithm in the monomodality image registration and multi-modality image registration. Figures and Tables | References | Related Articles | Metrics