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    LÉVY AREA ANALYSIS AND PARAMETER ESTIMATION FOR FOU PROCESSES VIA NON-GEOMETRIC ROUGH PATH THEORY*
    Zhongmin Qian, Xingcheng Xu
    Acta mathematica scientia,Series B    2024, 44 (5): 1609-1638.   DOI: 10.1007/s10473-024-0501-8
    Abstract150)            Save
    This paper addresses the estimation problem of an unknown drift parameter matrix for a fractional Ornstein-Uhlenbeck process in a multi-dimensional setting. To tackle this problem, we propose a novel approach based on rough path theory that allows us to construct pathwise rough path estimators from both continuous and discrete observations of a single path. Our approach is particularly suitable for high-frequency data. To formulate the parameter estimators, we introduce a theory of pathwise Itô integrals with respect to fractional Brownian motion. By establishing the regularity of fractional Ornstein-Uhlenbeck processes and analyzing the long-term behavior of the associated Lévy area processes, we demonstrate that our estimators are strongly consistent and pathwise stable. Our findings offer a new perspective on estimating the drift parameter matrix for fractional Ornstein-Uhlenbeck processes in multi-dimensional settings, and may have practical implications for fields including finance, economics, and engineering.
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    THE GRADIENT ESTIMATE OF SUBELLIPTIC HARMONIC MAPS WITH A POTENTIAL
    Han Luo
    Acta mathematica scientia,Series B    2024, 44 (4): 1189-1199.   DOI: 10.1007/s10473-024-0401-y
    Abstract143)            Save
    In this paper, we investigate subelliptic harmonic maps with a potential from noncompact complete sub-Riemannian manifolds corresponding to totally geodesic Riemannian foliations. Under some suitable conditions, we give the gradient estimates of these maps and establish a Liouville type result.
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    MEAN SENSITIVITY AND BANACH MEAN SENSITIVITY FOR LINEAR OPERATORS
    Quanquan Yao, Peiyong Zhu
    Acta mathematica scientia,Series B    2024, 44 (4): 1200-1228.   DOI: 10.1007/s10473-024-0402-x
    Abstract112)            Save
    Let $(X,T)$ be a linear dynamical system, where $X$ is a Banach space and $T:X \to X$ is a bounded linear operator. This paper obtains that $(X,T)$ is sensitive (Li-Yorke sensitive, mean sensitive, syndetically mean sensitive, respectively) if and only if $(X,T)$ is Banach mean sensitive (Banach mean Li-Yorke sensitive, thickly multi-mean sensitive, thickly syndetically mean sensitive, respectively). Several examples are provided to distinguish between different notions of mean sensitivity, syndetic mean sensitivity and mean Li-Yorke sensitivity.
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    STARLIKENESS ASSOCIATED WITH THE SINE HYPERBOLIC FUNCTION
    Mohsan Raza, Hadiqa Zahid, Jinlin Liu
    Acta mathematica scientia,Series B    2024, 44 (4): 1244-1270.   DOI: 10.1007/s10473-024-0404-8
    Abstract109)            Save
    Let $q_{\lambda }\left( z\right) =1+\lambda \sinh (\zeta ),\ 0<\lambda <1/\sinh \left( 1\right) $ be a non-vanishing analytic function in the open unit disk. We introduce a subclass $\mathcal{S}^{\ast }\left( q_{\lambda }\right) $ of starlike functions which contains the functions $\mathfrak{f}$ such that $z\mathfrak{f}^{\prime }/\mathfrak{f}$ is subordinated by $q_{\lambda }$. We establish inclusion and radii results for the class $\mathcal{S}^{\ast }\left( q_{\lambda }\right) $ for several known classes of starlike functions. Furthermore, we obtain sharp coefficient bounds and sharp Hankel determinants of order two for the class $\mathcal{S}^{\ast }\left( q_{\lambda }\right) $. We also find a sharp bound for the third Hankel determinant for the case $\lambda =1/2$.
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    GENERALIZED COUNTING FUNCTIONS AND COMPOSITION OPERATORS ON WEIGHTED BERGMAN SPACES OF DIRICHLET SERIES
    Min He, Maofa Wang, Jiale Chen
    Acta mathematica scientia,Series B    2025, 45 (2): 291-309.   DOI: 10.1007/s10473-025-0201-z
    Abstract104)            Save
    In this paper, we study composition operators on weighted Bergman spaces of Dirichlet series. We first establish some Littlewood-type inequalities for generalized mean counting functions. Then we give sufficient conditions for a composition operator with zero characteristic to be bounded or compact on weighted Bergman spaces of Dirichlet series. The corresponding sufficient condition for compactness in the case of positive characteristics is also obtained.
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    ESTIMATION OF AVERAGE DIFFERENTIAL ENTROPY FOR A STATIONARY ERGODIC SPACE-TIME RANDOM FIELD ON A BOUNDED AREA*
    Zhanjie SONG, Jiaxing ZHANG
    Acta mathematica scientia,Series B    2024, 44 (5): 1984-1996.   DOI: 10.1007/s10473-024-0521-4
    Abstract101)            Save
    In this paper, we mainly discuss a discrete estimation of the average differential entropy for a continuous time-stationary ergodic space-time random field. By estimating the probability value of a time-stationary random field in a small range, we give an entropy estimation and obtain the average entropy estimation formula in a certain bounded space region. It can be proven that the estimation of the average differential entropy converges to the theoretical value with a probability of 1. In addition, we also conducted numerical experiments for different parameters to verify the convergence result obtained in the theoretical proofs.
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    APPROXIMATION PROBLEMS ON THE SMOOTHNESS CLASSES*
    Yongping LIU, Man LU
    Acta mathematica scientia,Series B    2024, 44 (5): 1721-1734.   DOI: 10.1007/s10473-024-0505-4
    Abstract97)            Save
    This paper investigates the relative Kolmogorov $n$-widths of $2\pi$-periodic smooth classes in $\widetilde{L}_{q}$. We estimate the relative widths of $\widetilde{W}^{r} H_{p}^{\omega}$ and its generalized class $K_{p}H^{\omega}(P_{r})$, where $K_{p}H^{\omega}(P_{r})$ is defined by a self-conjugate differential operator $P_{r}(D)$ induced by $P_{r}(t):= t^{\sigma} \Pi_{j=1}^{l}(t^{2}- t_{j}^{2}),~t_{j} > 0,~j=1, 2 ,\cdots, l,~l \geq 1,~\sigma \geq 1,~r=2l+\sigma .$ Also, the modulus of continuity of the $r$-th derivative, or $r$-th self-conjugate differential, does not exceed a given modulus of continuity $\omega$. Then we obtain the asymptotic results, especially for the case $p=\infty , 1\leq q \leq \infty$.
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    HEAT KERNEL ON RICCI SHRINKERS (II)*
    Yu Li, Bing Wang
    Acta mathematica scientia,Series B    2024, 44 (5): 1639-1695.   DOI: 10.1007/s10473-024-0502-7
    Abstract92)            Save
    This paper is the sequel to our study of heat kernel on Ricci shrinkers [29]. In this paper, we improve many estimates in [29] and extend the recent progress of Bamler [2]. In particular, we drop the compactness and curvature boundedness assumptions and show that the theory of $\mathbb{F}$-convergence holds naturally on any Ricci flows induced by Ricci shrinkers.
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    THE STABILITY OF BOUSSINESQ EQUATIONS WITH PARTIAL DISSIPATION AROUND THE HYDROSTATIC BALANCE
    Saiguo Xu, Zhong Tan
    Acta mathematica scientia,Series B    2024, 44 (4): 1466-1486.   DOI: 10.1007/s10473-024-0415-5
    Abstract91)            Save
    This paper is devoted to understanding the stability of perturbations around the hydrostatic equilibrium of the Boussinesq system in order to gain insight into certain atmospheric and oceanographic phenomena. The Boussinesq system focused on here is anisotropic, and involves only horizontal dissipation and thermal damping. In the 2D case $\mathbb{R}^2$, due to the lack of vertical dissipation, the stability and large-time behavior problems have remained open in a Sobolev setting. For the spatial domain $\mathbb{T}\times\mathbb{R}$, this paper solves the stability problem and gives the precise large-time behavior of the perturbation. By decomposing the velocity $u$ and temperature $\theta$ into the horizontal average $(\bar{u},\bar{\theta})$ and the corresponding oscillation $(\tilde{u},\tilde{\theta})$, we can derive the global stability in $H^2$ and the exponential decay of $(\tilde{u},\tilde{\theta})$ to zero in $H^1$. Moreover, we also obtain that $(\bar{u}_2,\bar{\theta})$ decays exponentially to zero in $H^1$, and that $\bar{u}_1$ decays exponentially to $\bar{u}_1(\infty)$ in $H^1$ as well; this reflects a strongly stratified phenomenon of buoyancy-driven fluids. In addition, we establish the global stability in $H^3$ for the 3D case $\mathbb{R}^3$.
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    THE $\rm BSE$ PROPERTY FOR SOME VECTOR-VALUED BANACH FUNCTION ALGEBRAS*
    Fatemeh Abtahi, Ali Rejali, Farshad Sayaf
    Acta mathematica scientia,Series B    2024, 44 (5): 1945-1954.   DOI: 10.1007/s10473-024-0518-z
    Abstract88)            Save
    In this paper, $X$ is a locally compact Hausdorff space and ${\mathcal A}$ is a Banach algebra. First, we study some basic features of $C_0(X,\mathcal A)$ related to $\rm BSE$ concept, which are gotten from ${\mathcal A}$. In particular, we prove that if $C_0(X,\mathcal A)$ has the $\rm BSE$ property then $\mathcal A$ has so. We also establish the converse of this result, whenever $X$ is discrete and $\mathcal A$ has the BSE-norm property. Furthermore, we prove the same result for the $\rm BSE$ property of type I. Finally, we prove that $C_0(X,{\mathcal A})$ has the BSE-norm property if and only if $\mathcal A$ has so.
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    THE STABLE RECONSTRUCTION OF STRONGLY-DECAYING BLOCK SPARSE SIGNALS*
    Yifang yang, Jinping wang
    Acta mathematica scientia,Series B    2024, 44 (5): 1787-1800.   DOI: 10.1007/s10473-024-0509-0
    Abstract85)            Save
    In this paper, we reconstruct strongly-decaying block sparse signals by the block generalized orthogonal matching pursuit (BgOMP) algorithm in the $l_2$-bounded noise case. Under some restraints on the minimum magnitude of the nonzero elements of the strongly-decaying block sparse signal, if the sensing matrix satisfies the the block restricted isometry property (block-RIP), then arbitrary strongly-decaying block sparse signals can be accurately and steadily reconstructed by the BgOMP algorithm in iterations. Furthermore, we conjecture that this condition is sharp.
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    ON MONOTONE TRAVELING WAVES FOR NICHOLSON'S BLOWFLIES EQUATION WITH DEGENERATE $p$-LAPLACIAN DIFFUSION
    Rui Huang, Yong Wang, Zhuo Yin
    Acta mathematica scientia,Series B    2024, 44 (4): 1550-1571.   DOI: 10.1007/s10473-024-0420-8
    Abstract83)            Save
    We study the existence and stability of monotone traveling wave solutions of Nicholson's blowflies equation with degenerate $p$-Laplacian diffusion. We prove the existence and nonexistence of non-decreasing smooth traveling wave solutions by phase plane analysis methods. Moreover, we show the existence and regularity of an original solution via a compactness analysis. Finally, we prove the stability and exponential convergence rate of traveling waves by an approximated weighted energy method.
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    VARIATIONAL ANALYSIS FOR THE MAXIMAL TIME FUNCTION IN NORMED SPACES*
    Ziyi Zhou, Yi Jiang
    Acta mathematica scientia,Series B    2024, 44 (5): 1696-1706.   DOI: 10.1007/s10473-024-0503-6
    Abstract83)            Save
    For a general normed vector space, a special optimal value function called a maximal time function is considered. This covers the farthest distance function as a special case, and has a close relationship with the smallest enclosing ball problem. Some properties of the maximal time function are proven, including the convexity, the lower semicontinuity, and the exact characterizations of its subdifferential formulas.
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    MULTIPLE SOLUTIONS TO CRITICAL MAGNETIC SCHRÖDINGER EQUATIONS
    Ruijiang Wen, Jianfu Yang
    Acta mathematica scientia,Series B    2024, 44 (4): 1373-1393.   DOI: 10.1007/s10473-024-0411-9
    Abstract78)            Save

    In this paper, we are concerned with the existence of multiple solutions to the critical magnetic Schrödinger equation

    $\begin{matrix}(-{\rm i}\nabla-A(x))^2u+\lambda V(x)u=\mu |u|^{p-2}u+\Big(\int_{\mathbb{R^N}}\frac{|u(y)|^{2^*_\alpha}}{|x-y|^\alpha}{\rm d}y\Big)|u|^{2^*_\alpha-2}u\quad {\rm in}\ \mathbb{R}^N,\end{matrix}$ (0.1)

    where $N\geq4$, $2\leq p<2^*$, $2^*_\alpha=\frac{2N-\alpha}{N-2}$ with $0<\alpha<4$, $\lambda>0$, $\mu\in\mathbb{R}$, $A(x)= (A_1(x), A_2(x),\cdots , A_N(x))$ is a real local Hölder continuous vector function, $i$ is the imaginary unit, and $V(x)$ is a real valued potential function on $\mathbb{R}^N$.Supposing that $\Omega={\rm int}\,V^{-1}(0)\subset\mathbb{R}^N$ is bounded, we show that problem (0.1) possesses at least cat$_\Omega(\Omega)$ nontrivial solutions if $\lambda$ is large.

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    REFINEMENTS OF THE NORM OF TWO ORTHOGONAL PROJECTIONS
    Xiaohui Li, Meiqi Liu, Chunyuan Deng
    Acta mathematica scientia,Series B    2024, 44 (4): 1229-1243.   DOI: 10.1007/s10473-024-0403-9
    Abstract77)            Save
    In this paper, some refinements of norm equalities and inequalities of combination of two orthogonal projections are established. We use certain norm inequalities for positive contraction operator to establish norm inequalities for combination of orthogonal projections on a Hilbert space. Furthermore, we give necessary and sufficient conditions under which the norm of the above combination of orthogonal projections attains its optimal value.
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    TOEPLITZ OPERATORS BETWEEN WEIGHTED BERGMAN SPACES OVER THE HALF-PLANE*
    Lixia Feng, Yan Li, Zhiyu Wang, Liankuo Zhao
    Acta mathematica scientia,Series B    2024, 44 (5): 1707-1720.   DOI: 10.1007/s10473-024-0504-5
    Abstract76)            Save
    In this paper, by characterizing Carleson measures, we investigate a class of bounded Toeplitz operator between weighted Bergman spaces with Békollé weights over the half-plane for all index choices.
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    STABILITY OF TRANSONIC SHOCKS TO THE EULER-POISSON SYSTEM WITH VARYING BACKGROUND CHARGES
    Yang Cao, Yuanyuan Xing, Na Zhang
    Acta mathematica scientia,Series B    2024, 44 (4): 1487-1506.   DOI: 10.1007/s10473-024-0416-4
    Abstract74)            Save
    This paper is devoted to studying the stability of transonic shock solutions to the Euler-Poisson system in a one-dimensional nozzle of finite length. The background charge in the Poisson equation is a piecewise constant function. The structural stability of the steady transonic shock solution is obtained by the monotonicity argument. Furthermore, this transonic shock is proved to be dynamically and exponentially stable with respect to small perturbations of the initial data. One of the crucial ingredients of the analysis is to establish the global well-posedness of a free boundary problem for a quasilinear second order equation with nonlinear boundary conditions.
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    ON THE EMPTY BALLS OF A CRITICAL OR SUBCRITICAL BRANCHING RANDOM WALK*
    Shuxiong Zhang, Jie Xiong
    Acta mathematica scientia,Series B    2024, 44 (5): 2051-2072.   DOI: 10.1007/s10473-024-0525-0
    Abstract74)            Save
    Let $\{Z_n\}_{n\geq 0 }$ be a critical or subcritical $d$-dimensional branching random walk started from a Poisson random measure whose intensity measure is the Lebesugue measure on $\mathbb{R}^d$. Denote by $R_n:=\sup\{u>0:Z_n(\{x\in\mathbb{R}^d:|x|<u\})=0\}$ the radius of the largest empty ball centered at the origin of $Z_n$. In this work, we prove that after suitable renormalization, $R_n$ converges in law to some non-degenerate distribution as $n\to\infty$. Furthermore, our work shows that the renormalization scales depend on the offspring law and the dimension of the branching random walk. This completes the results of Révész [13] for the critical binary branching Wiener process.
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    ON THE CAUCHY PROBLEM FOR THE GENERALIZED BOUSSINESQ EQUATION WITH A DAMPED TERM*
    Xiao SU, Shubin WANG
    Acta mathematica scientia,Series B    2024, 44 (5): 1766-1786.   DOI: 10.1007/s10473-024-0508-1
    Abstract72)            Save
    This paper is devoted to the Cauchy problem for the generalized damped Boussinesq equation with a nonlinear source term in the natural energy space. With the help of linear time-space estimates, we establish the local existence and uniqueness of solutions by means of the contraction mapping principle. The global existence and blow-up of the solutions at both subcritical and critical initial energy levels are obtained. Moreover, we construct the sufficient conditions of finite time blow-up of the solutions with arbitrary positive initial energy.
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    GLOBAL UNIQUE SOLUTIONS FOR THE INCOMPRESSIBLE MHD EQUATIONS WITH VARIABLE DENSITY AND ELECTRICAL CONDUCTIVITY*
    Xueli KE
    Acta mathematica scientia,Series B    2024, 44 (5): 1747-1765.   DOI: 10.1007/s10473-024-0507-2
    Abstract71)            Save
    We study the global unique solutions to the 2-D inhomogeneous incompressible MHD equations, with the initial data $(u_{0},B_{0})$ being located in the critical Besov space $\dot{B}_{p,1}^{-1+\frac{2}{p}}(\mathbb{R}^{2}) \,\, (1<p<2)$ and the initial density $\rho_{0}$ being close to a positive constant. By using weighted global estimates, maximal regularity estimates in the Lorentz space for the Stokes system, and the Lagrangian approach, we show that the 2-D MHD equations have a unique global solution.
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