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    25 April 2024, Volume 44 Issue 2 Previous Issue   
    FROM WAVE FUNCTIONS TO TAU-FUNCTIONS FOR THE VOLTERRA LATTICE HIERARCHY
    Ang FU, Mingjin LI, Di YANG
    Acta mathematica scientia,Series B. 2024, 44 (2):  405-419.  DOI: 10.1007/s10473-024-0201-4
    Abstract ( 43 )   RICH HTML PDF (442KB) ( 28 )   Save
    For an arbitrary solution to the Volterra lattice hierarchy, the logarithmic derivatives of the tau-function of the solution can be computed by the matrix-resolvent method. In this paper, we define a pair of wave functions of the solution and use them to give an expression of the matrix resolvent; based on this we obtain a new formula for the $k$-point functions for the Volterra lattice hierarchy in terms of wave functions. As an application, we give an explicit formula of $k$-point functions for the even GUE (Gaussian Unitary Ensemble) correlators.
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    SHARP MORREY REGULARITY THEORY FOR A FOURTH ORDER GEOMETRICAL EQUATION
    Changlin XIANG, Gaofeng ZHENG
    Acta mathematica scientia,Series B. 2024, 44 (2):  420-430.  DOI: 10.1007/s10473-024-0202-3
    Abstract ( 32 )   RICH HTML PDF (395KB) ( 19 )   Save
    This paper is a continuation of recent work by Guo-Xiang-Zheng[10]. We deduce the sharp Morrey regularity theory for weak solutions to the fourth order nonhomogeneous Lamm-Rivière equation $\begin{equation*} \Delta^{2}u=\Delta(V\nabla u)+{\rm div}(w\nabla u)+(\nabla\omega+F)\cdot\nabla u+f\qquad\text{in }B^{4},\end{equation*}$ under the smallest regularity assumptions of $V,w,\omega, F$, where $f$ belongs to some Morrey spaces. This work was motivated by many geometrical problems such as the flow of biharmonic mappings. Our results deepens the $L^p$ type regularity theory of [10], and generalizes the work of Du, Kang and Wang [4] on a second order problem to our fourth order problems.
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    ON THE SOBOLEV DOLBEAULT COHOMOLOGY OF A DOMAIN WITH PSEUDOCONCAVE BOUNDARIES
    Jian CHEN
    Acta mathematica scientia,Series B. 2024, 44 (2):  431-444.  DOI: 10.1007/s10473-024-0203-2
    Abstract ( 27 )   RICH HTML PDF (420KB) ( 13 )   Save
    In this note, we mainly make use of a method devised by Shaw [15] for studying Sobolev Dolbeault cohomologies of a pseudoconcave domain of the type $\Omega=\widetilde{\Omega} \backslash \overline{\bigcup_{j=1}^{m}\Omega_j}$, where $\widetilde{\Omega}$ and $\{\Omega_j\}_{j=1}^m\Subset\widetilde{\Omega}$ are bounded pseudoconvex domains in $\mathbb{C}^n$ with smooth boundaries, and $\overline{\Omega}_1,\cdots,\overline{\Omega}_m$ are mutually disjoint. The main results can also be quickly obtained by virtue of [5].
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    THREE KINDS OF DENTABILITIES IN BANACH SPACES AND THEIR APPLICATIONS
    Zihou ZHANG, Jing ZHOU
    Acta mathematica scientia,Series B. 2024, 44 (2):  445-454.  DOI: 10.1007/s10473-024-0204-1
    Abstract ( 36 )   RICH HTML PDF (338KB) ( 23 )   Save
    In this paper, we study some dentabilities in Banach spaces which are closely related to the famous Radon-Nikodym property. We introduce the concepts of the weak$^*$-weak denting point and the weak$^*$-weak$^*$ denting point of a set. These are the generalizations of the weak$^*$ denting point of a set in a dual Banach space. By use of the weak$^*$-weak denting point, we characterize the very smooth space, the point of weak$^*$-weak continuity, and the extreme point of a unit ball in a dual Banach space. Meanwhile, we also characterize an approximatively weak compact Chebyshev set in dual Banach spaces. Moreover, we define the nearly weak dentability in Banach spaces, which is a generalization of near dentability. We prove the necessary and sufficient conditions of the reflexivity by nearly weak dentability. We also obtain that nearly weak dentability is equivalent to both the approximatively weak compactness of Banach spaces and the $w$-strong proximinality of every closed convex subset of Banach spaces.
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    THE SMOOTHING EFFECT IN SHARP GEVREY SPACE FOR THE SPATIALLY HOMOGENEOUS NON-CUTOFF BOLTZMANN EQUATIONS WITH A HARD POTENTIAL
    Lvqiao LIU, Juan ZENG
    Acta mathematica scientia,Series B. 2024, 44 (2):  455-473.  DOI: 10.1007/s10473-024-0205-0
    Abstract ( 40 )   RICH HTML PDF (406KB) ( 19 )   Save
    In this article, we study the smoothing effect of the Cauchy problem for the spatially homogeneous non-cutoff Boltzmann equation for hard potentials. It has long been suspected that the non-cutoff Boltzmann equation enjoys similar regularity properties as to whose of the fractional heat equation. We prove that any solution with mild regularity will become smooth in Gevrey class at positive time, with a sharp Gevrey index, depending on the angular singularity. Our proof relies on the elementary $L^2$ weighted estimates.
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    A STABILITY RESULT FOR TRANSLATING SPACELIKE GRAPHS IN LORENTZ MANIFOLDS
    Ya GAO, Jing MAO, Chuanxi WU
    Acta mathematica scientia,Series B. 2024, 44 (2):  474-483.  DOI: 10.1007/s10473-024-0206-z
    Abstract ( 2 )   RICH HTML PDF (376KB) ( 3 )   Save
    In this paper, we investigate spacelike graphs defined over a domain $\Omega\subset M^{n}$ in the Lorentz manifold $M^{n}\times\mathbb{R}$ with the metric $-{\rm d}s^{2}+\sigma$, where $M^{n}$ is a complete Riemannian $n$-manifold with the metric $\sigma$, $\Omega$ has piecewise smooth boundary, and $\mathbb{R}$ denotes the Euclidean $1$-space. We prove an interesting stability result for translating spacelike graphs in $M^{n}\times\mathbb{R}$ under a conformal transformation.
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    MAXIMAL FUNCTION CHARACTERIZATIONS OF HARDY SPACES ASSOCIATED WITH BOTH NON-NEGATIVE SELF-ADJOINT OPERATORS SATISFYING GAUSSIAN ESTIMATES AND BALL QUASI-BANACH FUNCTION SPACES
    Xiaosheng LIN, Dachun YANG, Sibei YANG, Wen YUAN
    Acta mathematica scientia,Series B. 2024, 44 (2):  484-514.  DOI: 10.1007/s10473-024-0207-y
    Abstract ( 3 )   RICH HTML PDF (577KB) ( 0 )   Save
    Assume that $L$ is a non-negative self-adjoint operator on $L^2(\mathbb{R}^n)$ with its heat kernels satisfying the so-called Gaussian upper bound estimate and that $X$ is a ball quasi-Banach function space on $\mathbb{R}^n$ satisfying some mild assumptions. Let $H_{X,\,L}(\mathbb{R}^n)$ be the Hardy space associated with both $X$ and $L,$ which is defined by the Lusin area function related to the semigroup generated by $L$. In this article, the authors establish various maximal function characterizations of the Hardy space $H_{X,\,L}(\mathbb{R}^n)$ and then apply these characterizations to obtain the solvability of the related Cauchy problem. These results have a wide range of generality and, in particular, the specific spaces $X$ to which these results can be applied include the weighted space, the variable space, the mixed-norm space, the Orlicz space, the Orlicz-slice space, and the Morrey space. Moreover, the obtained maximal function characterizations of the mixed-norm Hardy space, the Orlicz-slice Hardy space, and the Morrey-Hardy space associated with $L$ are completely new.
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    THE ABSENCE OF SINGULAR CONTINUOUS SPECTRUM FOR PERTURBED JACOBI OPERATORS
    Zhengqi FU, Xiong LI
    Acta mathematica scientia,Series B. 2024, 44 (2):  515-531.  DOI: 10.1007/s10473-024-0208-x
    Abstract ( 4 )   RICH HTML PDF (429KB) ( 0 )   Save
    This paper is mainly about the spectral properties of a class of Jacobi operators $ (H_{c,b}u)(n)=c_{n}u(n+1)+c_{n-1}u(n-1)+b_{n}u(n), $ where $|c_{n}-1|=O(n^{-\alpha})$ and $b_{n}=O(n^{-1})$. We will show that, for $\alpha\ge1$, the singular continuous spectrum of the operator is empty.
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    THE WEIGHTED KATO SQUARE ROOT PROBLEM OF ELLIPTIC OPERATORS HAVING A BMO ANTI-SYMMETRIC PART
    Wenxian MA, Sibei YANG
    Acta mathematica scientia,Series B. 2024, 44 (2):  532-550.  DOI: 10.1007/s10473-024-0209-9
    Abstract ( 3 )   RICH HTML PDF (490KB) ( 1 )   Save
    Let $n\ge2$ and let $L$ be a second-order elliptic operator of divergence form with coefficients consisting of both an elliptic symmetric part and a $\mathrm{BMO}$ anti-symmetric part in $\mathbb{R}^n$. In this article, we consider the weighted Kato square root problem for $L$. More precisely, we prove that the square root $L^{1/2}$ satisfies the weighted $L^p$ estimates $\|L^{1/2}(f)\|_{L^p_\omega(\mathbb{R}^n)}\le C\|\nabla f\|_{L^p_\omega (\mathbb{R}^n;\mathbb{R}^n)}$ for any $p\in(1,\infty)$ and $\omega\in A_p{(\mathbb{R}^n)}$ (the class of Muckenhoupt weights), and that $\|\nabla f\|_{L^p_\omega(\mathbb{R}^n;\mathbb{R}^n)}\le C\|L^{1/2}(f)\|_{L^p_\omega(\mathbb{R}^n)}$ for any $p\in(1,2+\varepsilon)$ and $\omega\in A_p(\mathbb{R}^n)\cap RH_{(\frac{2+\varepsilon}{p})'}(\mathbb{R}^n)$ (the class of reverse Hölder weights), where $\varepsilon\in(0,\infty)$ is a constant depending only on $n$ and the operator $L$, and where $(\frac{2+\varepsilon}{p})'$ denotes the Hölder conjugate exponent of $\frac{2+\varepsilon}{p}$. Moreover, for any given $q\in(2,\infty)$, we give a sufficient condition to obtain that $\|\nabla f\|_{L^p_\omega(\mathbb{R}^n;\mathbb{R}^n)} \le C\|L^{1/2}(f)\|_{L^p_\omega(\mathbb{R}^n)}$ for any $p\in(1,q)$ and $\omega\in A_p(\mathbb{R}^n)\cap RH_{(\frac{q}{p})'}(\mathbb{R}^n)$. As an application, we prove that when the coefficient matrix $A$ that appears in $L$ satisfies the small $\mathrm{BMO}$ condition, the Riesz transform $\nabla L^{-1/2}$ is bounded on $L^p_\omega(\mathbb{R}^n)$ for any given $p\in(1,\infty)$ and $\omega\in A_p(\mathbb{R}^n)$. Furthermore, applications to the weighted $L^2$-regularity problem with the Dirichlet or the Neumann boundary condition are also given.
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    STRONGLY CONVERGENT INERTIAL FORWARD-BACKWARD-FORWARD ALGORITHM WITHOUT ON-LINE RULE FOR VARIATIONAL INEQUALITIES
    Yonghong YAO, Abubakar ADAMU, Yekini SHEHU
    Acta mathematica scientia,Series B. 2024, 44 (2):  551-566.  DOI: 10.1007/s10473-024-0210-3
    Abstract ( 2 )   RICH HTML PDF (389KB) ( 0 )   Save
    This paper studies a strongly convergent inertial forward-backward-forward algorithm for the variational inequality problem in Hilbert spaces. In our convergence analysis, we do not assume the on-line rule of the inertial parameters and the iterates, which have been assumed by several authors whenever a strongly convergent algorithm with an inertial extrapolation step is proposed for a variational inequality problem. Consequently, our proof arguments are different from what is obtainable in the relevant literature. Finally, we give numerical tests to confirm the theoretical analysis and show that our proposed algorithm is superior to related ones in the literature.
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    THE SPARSE REPRESENTATION RELATED WITH FRACTIONAL HEAT EQUATIONS
    Wei QU, Tao QIAN, Ieng Tak LEONG, Pengtao LI
    Acta mathematica scientia,Series B. 2024, 44 (2):  567-582.  DOI: 10.1007/s10473-024-0211-2
    Abstract ( 4 )   RICH HTML PDF (490KB) ( 0 )   Save
    This study introduces a pre-orthogonal adaptive Fourier decomposition (POAFD) to obtain approximations and numerical solutions to the fractional Laplacian initial value problem and the extension problem of Caffarelli and Silvestre (generalized Poisson equation). As a first step, the method expands the initial data function into a sparse series of the fundamental solutions with fast convergence, and, as a second step, makes use of the semigroup or the reproducing kernel property of each of the expanding entries. Experiments show the effectiveness and efficiency of the proposed series solutions.
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    THE LIMITING PROFILE OF SOLUTIONS FOR SEMILINEAR ELLIPTIC SYSTEMS WITH A SHRINKING SELF-FOCUSING CORE
    Ke JIN, Ying SHI, Huafei XIE
    Acta mathematica scientia,Series B. 2024, 44 (2):  583-608.  DOI: 10.1007/s10473-024-0212-1
    Abstract ( 3 )   RICH HTML PDF (484KB) ( 0 )   Save
    In this paper, we consider the semilinear elliptic equation systems $ \left\{\begin{array}{ll} -\Delta u+u=\alpha Q_{n}(x)|u|^{\alpha-2}|v|^{\beta}u &\mbox{in}\hspace{1.14mm} \mathbb{R}^{N},\\ -\Delta v+v=\beta Q_{n}(x)|u|^{\alpha}|v|^{\beta-2}v &\mbox{in}\hspace{1.14mm} \mathbb{R}^{N}, \end{array} \right. $ where $N\geqslant 3$, $\alpha$, $\beta>1$, $\alpha+\beta<2^{*}$, $2^{*}=\frac{2N}{N-2}$ and $Q_{n}$ are bounded given functions whose self-focusing cores $\{x\in\mathbb{R}^N|Q_n(x)>0\}$ shrink to a set with finitely many points as $n\rightarrow\infty$. Motivated by the work of Fang and Wang [13], we use variational methods to study the limiting profile of ground state solutions which are concentrated at one point of the set with finitely many points, and we build the localized concentrated bound state solutions for the above equation systems.
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    BLOW-UP CONDITIONS FOR A SEMILINEAR PARABOLIC SYSTEM ON LOCALLY FINITE GRAPHS
    Yiting WU
    Acta mathematica scientia,Series B. 2024, 44 (2):  609-631.  DOI: 10.1007/s10473-024-0213-0
    Abstract ( 2 )   RICH HTML PDF (416KB) ( 0 )   Save
    In this paper, we investigate a blow-up phenomenon for a semilinear parabolic system on locally finite graphs. Under some appropriate assumptions on the curvature condition $CDE'(n,0)$, the polynomial volume growth of degree $m$, the initial values, and the exponents in absorption terms, we prove that every non-negative solution of the semilinear parabolic system blows up in a finite time. Our current work extends the results achieved by Lin and Wu (Calc Var Partial Differ Equ, 2017, 56: Art 102) and Wu (Rev R Acad Cien Serie A Mat, 2021, 115: Art 133).
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    FLOCKING OF A THERMODYNAMIC CUCKER-SMALE MODEL WITH LOCAL VELOCITY INTERACTIONS
    Chunyin JIN, Shuangzhi LI
    Acta mathematica scientia,Series B. 2024, 44 (2):  632-649.  DOI: 10.1007/s10473-024-0214-z
    Abstract ( 2 )   RICH HTML PDF (474KB) ( 0 )   Save
    In this paper, we study the flocking behavior of a thermodynamic Cucker-Smale model with local velocity interactions. Using the spectral gap of a connected stochastic matrix, together with an elaborate estimate on perturbations of a linearized system, we provide a sufficient framework in terms of initial data and model parameters to guarantee flocking. Moreover, it is shown that the system achieves a consensus at an exponential rate.
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    A GENERALIZED SCALAR AUXILIARY VARIABLE METHOD FOR THE TIME-DEPENDENT GINZBURG-LANDAU EQUATIONS
    Zhiyong SI
    Acta mathematica scientia,Series B. 2024, 44 (2):  650-670.  DOI: 10.1007/s10473-024-0215-y
    Abstract ( 2 )   RICH HTML PDF (1341KB) ( 0 )   Save
    This paper develops a generalized scalar auxiliary variable (SAV) method for the time-dependent Ginzburg-Landau equations. The backward Euler method is used for discretizing the temporal derivative of the time-dependent Ginzburg-Landau equations. In this method, the system is decoupled and linearized to avoid solving the non-linear equation at each step. The theoretical analysis proves that the generalized SAV method can preserve the maximum bound principle and energy stability, and this is confirmed by the numerical result, and also shows that the numerical algorithm is stable.
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    THE LONG TIME BEHAVIOR OF THE FRACTIONAL ORNSTEIN-UHLENBECK PROCESS WITH LINEAR SELF-REPELLING DRIFT
    Xiaoyu XIA, Litan YAN, Qing YANG
    Acta mathematica scientia,Series B. 2024, 44 (2):  671-685.  DOI: 10.1007/s10473-024-0216-x
    Abstract ( 2 )   RICH HTML PDF (422KB) ( 0 )   Save
    Let $B^{H} $ be a fractional Brownian motion with Hurst index $\frac{1}{2}\leq H< 1$. In this paper, we consider the equation (called the Ornstein-Uhlenbeck process with a linear self-repelling drift) $\begin{equation*} {\rm d}X_{t}^{H}={\rm d}B_{t}^{H}+\sigma X_t^{H}{\rm d}t+\nu {\rm d}t-\theta \left(\int_{0}^{t}(X_t^{H}-X_{s}^{H}){\rm d}s\right){\rm d}t, \end{equation*} $ where $\theta<0$, $\sigma,\nu \in \mathbb{R}$. The process is an analogue of {self-attracting} diffusion (Cranston, Le Jan. Math Ann, 1995, 303: 87-93). Our main aim is to study the large time behaviors of the process. We show that the solution $X^H$ diverges to infinity as $t$ tends to infinity, and obtain the speed at which the process $X^H$ diverges to infinity as $t$ tends to infinity.
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    THE EXTREMES OF DEPENDENT CHI-PROCESSESATTRACTED BY THE BROWN-RESNICK PROCESS
    Junjie SUN, Zhongquan TAN
    Acta mathematica scientia,Series B. 2024, 44 (2):  686-701.  DOI: 10.1007/s10473-024-0217-9
    Abstract ( 2 )   RICH HTML PDF (407KB) ( 0 )   Save
    Motivated by some recent works on the topic of the Brown-Resnick process, we study the functional limit theorem for normalized pointwise maxima of dependent chi-processes. It is proven that the properly normalized pointwise maxima of those processes are attracted by the Brown-Resnick process.
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    A FLEXIBLE OBJECTIVE-CONSTRAINT APPROACH AND A NEW ALGORITHM FOR CONSTRUCTING THE PARETO FRONT OF MULTIOBJECTIVE OPTIMIZATION PROBLEMS
    N. HOSEINPOOR, M. GHAZNAVI
    Acta mathematica scientia,Series B. 2024, 44 (2):  702-720.  DOI: 10.1007/s10473-024-0218-8
    Abstract ( 2 )   RICH HTML PDF (451KB) ( 1 )   Save
    In this article, a novel scalarization technique, called the improved objective-constraint approach, is introduced to find efficient solutions of a given multiobjective programming problem. The presented scalarized problem extends theobjective-constraint problem. It is demonstrated that how adding variables to the scalarized problem, can lead to find conditions for (weakly, properly) Pareto optimal solutions. Applying the obtained necessary and sufficient conditions, two algorithms for generating the Pareto front approximation of bi-objective and three-objective programming problems are designed. These algorithms are easy to implement and can achieve an even approximation of (weakly, properly) Pareto optimal solutions. These algorithms can be generalized for optimization problems with more than three criterion functions, too. The effectiveness and capability of the algorithms are demonstrated in test problems.
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    AN OPTIMAL CONTROL PROBLEM FOR A LOTKA-VOLTERRA COMPETITION MODEL WITH CHEMO-REPULSION
    Diana I. HERNÁNDEZ, Diego A. RUEDA-GÓMEZ, Élder J. VILLAMIZAR-ROA
    Acta mathematica scientia,Series B. 2024, 44 (2):  721-751.  DOI: 10.1007/s10473-024-0219-7
    Abstract ( 2 )   RICH HTML PDF (1461KB) ( 0 )   Save
    In this paper we study a bilinear optimal control problem for a diffusive Lotka-Volterra competition model with chemo-repulsion in a bounded domain of $\mathbb{R^N},$ $N=2,3$. This model describes the competition of two species in which one of them avoid encounters with rivals through a chemo-repulsion mechanism. We prove the existence and uniqueness of weak-strong solutions, and then we analyze the existence of a global optimal solution for a related bilinear optimal control problem, where the control is acting on the chemical signal. Posteriorly, we derive first-order optimality conditions for local optimal solutions using the Lagrange multipliers theory. Finally, we propose a discrete approximation scheme of the optimality system based on the gradient method, which is validated with some computational experiments.
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    A NOVEL STOCHASTIC HEPATITIS B VIRUS EPIDEMIC MODEL WITH SECOND-ORDER MULTIPLICATIVE α-STABLE NOISE AND REAL DATA
    Anwarud DIN, Yassine SABBAR, Peng WU
    Acta mathematica scientia,Series B. 2024, 44 (2):  752-788.  DOI: 10.1007/s10473-024-0220-1
    Abstract ( 3 )   RICH HTML PDF (2713KB) ( 0 )   Save
    This work presents an advanced and detailed analysis of the mechanisms of hepatitis B virus (HBV) propagation in an environment characterized by variability and stochasticity. Based on some biological features of the virus and the assumptions, the corresponding deterministic model is formulated, which takes into consideration the effect of vaccination. This deterministic model is extended to a stochastic framework by considering a new form of disturbance which makes it possible to simulate strong and significant fluctuations. The long-term behaviors of the virus are predicted by using stochastic differential equations with second-order multiplicative $\alpha$-stable jumps. By developing the assumptions and employing the novel theoretical tools, the threshold parameter responsible for ergodicity (persistence) and extinction is provided. The theoretical results of the current study are validated by numerical simulations and parameters estimation is also performed. Moreover, we obtain the following new interesting findings: (a) in each class, the average time depends on the value of $\alpha$; (b) the second-order noise has an inverse effect on the spread of the virus; (c) the shapes of population densities at stationary level quickly changes at certain values of $\alpha$. The last three conclusions can provide a solid research base for further investigation in the field of biological and ecological modeling.
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