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BIG HANKEL OPERATORS ON HARDY SPACES OF STRONGLY PSEUDOCONVEX DOMAINS Boyong Chen, Liangying Jiang Acta mathematica scientia,Series B    2024, 44 (3): 789-809.   DOI: 10.1007/s10473-024-0301-1 Abstract （108）            Save In this article, we investigate the (big) Hankel operator $H_f$ on the Hardy spaces of bounded strongly pseudoconvex domains $\Omega$ in $\mathbb{C}^n$. We observe that $H_f$ is bounded on $H^p(\Omega)$ ($1< p<\infty$) if $f$ belongs to BMO and we obtain some characterizations for $H_f$ on $H^2(\Omega)$ of other pseudoconvex domains. In these arguments, Amar's $L^p$-estimations and Berndtsson's $L^2$-estimations for solutions of the $\bar{\partial}_b$-equation play a crucial role. In addition, we solve Gleason's problem for Hardy spaces $H^p(\Omega)$ ($1\le p\le\infty$) of bounded strongly pseudoconvex domains. Reference | Related Articles | Metrics

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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 Accepted: 16 October 2023 Online available: 06 December 2023 Abstract （69）      PDF       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. Reference | Related Articles | Metrics

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CLASSIFICATIONS OF DUPIN HYPERSURFACES IN LIE SPHERE GEOMETRY* Thomas E. Cecil Acta mathematica scientia,Series B    2024, 44 (1): 1-36.   DOI: 10.1007/s10473-024-0101-7 Abstract （56）      PDF       Save This is a survey of local and global classification results concerning Dupin hypersurfaces in $S^n$ (or ${\bf R}^n$) that have been obtained in the context of Lie sphere geometry. The emphasis is on results that relate Dupin hypersurfaces to isoparametric hypersurfaces in spheres. Along with these classification results, many important concepts from Lie sphere geometry, such as curvature spheres, Lie curvatures, and Legendre lifts of submanifolds of $S^n$ (or ${\bf R}^n$), are described in detail. The paper also contains several important constructions of Dupin hypersurfaces with certain special properties. Reference | Related Articles | Metrics

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THE RIEMANN PROBLEM FOR ISENTROPIC COMPRESSIBLE EULER EQUATIONS WITH DISCONTINUOUS FLUX* Yinzheng Sun, Aifang Qu, Hairong Yuan Acta mathematica scientia,Series B    2024, 44 (1): 37-77.   DOI: 10.1007/s10473-024-0102-6 Abstract （53）      PDF       Save We consider the singular Riemann problem for the rectilinear isentropic compressible Euler equations with discontinuous flux, more specifically, for pressureless flow on the left and polytropic flow on the right separated by a discontinuity $x=x(t)$. We prove that this problem admits global Radon measure solutions for all kinds of initial data. The over-compressing condition on the discontinuity $x=x(t)$ is not enough to ensure the uniqueness of the solution. However, there is a unique piecewise smooth solution if one proposes a slip condition on the right-side of the curve $x=x(t)+0$, in addition to the full adhesion condition on its left-side. As an application, we study a free piston problem with the piston in a tube surrounded initially by uniform pressureless flow and a polytropic gas. In particular, we obtain the existence of a piecewise smooth solution for the motion of the piston between a vacuum and a polytropic gas. This indicates that the singular Riemann problem looks like a control problem in the sense that one could adjust the condition on the discontinuity of the flux to obtain the desired flow field. Reference | Related Articles | Metrics

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SUMS OF DUAL TOEPLITZ PRODUCTS ON THE ORTHOGONAL COMPLEMENTS OF FOCK-SOBOLEV SPACES Yong CHEN, Young Joo LEE Acta mathematica scientia,Series B    2024, 44 (3): 810-822.   DOI: 10.1007/s10473-024-0302-0 Abstract （53）            Save We consider dual Toeplitz operators on the orthogonal complements of the ock-Sobolev spaces of all nonnegative real orders. First, for symbols in a certain class containing all bounded functions, we study the problem of when an operator which is finite sums of the dual Toeplitz products is compact or zero. Next, for bounded symbols, we construct a symbol map and exhibit a short exact sequence associated with the $C^*$-algebra generated by all dual Toeplitz operators with bounded symbols. Reference | Related Articles | Metrics

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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 Accepted: 16 October 2023 Online available: 06 December 2023 Abstract （48）      PDF       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. Reference | Related Articles | Metrics

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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 Accepted: 16 October 2023 Online available: 06 December 2023 Abstract （45）      PDF       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. Reference | Related Articles | Metrics

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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 Accepted: 16 October 2023 Online available: 06 December 2023 Abstract （41）      PDF       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. Reference | Related Articles | Metrics

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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 Accepted: 16 October 2023 Abstract （35）      PDF       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]. Reference | Related Articles | Metrics

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ENERGY CONSERVATION FOR THE WEAK SOLUTIONS TO THE 3D COMPRESSIBLE NEMATIC LIQUID CRYSTAL FLOW Zhong Tan, Xinliang Li, Hui Yang Acta mathematica scientia,Series B    2024, 44 (3): 851-864.   DOI: 10.1007/s10473-024-0305-x Abstract （25）            Save In this paper, we establish some regularity conditions on the density and velocity fields to guarantee the energy conservation of the weak solutions for the three-dimensional compressible nematic liquid crystal flow in the periodic domain. Reference | Related Articles | Metrics

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THE BOUNDARY SCHWARZ LEMMA AND THE RIGIDITY THEOREM ON REINHARDT DOMAINS $B_{p}^{n}$ OF $\mathbb{C}^{n}$p OF Cn Jianfei WANG, Yanhui ZHANG Acta mathematica scientia,Series B    2024, 44 (3): 839-850.   DOI: 10.1007/s10473-024-0304-y Abstract （25）            Save By introducing the Carathéodory metric, we establish the Schwarz lemma at the boundary for holomorphic self-mappings on the unit $p$-ball $B_{p}^{n}$ of $\mathbb{C}^n$. Furthermore, the boundary rigidity theorem for holomorphic self-mappings defined on $B_{p}^{n}$ is obtained. These results cover the boundary Schwarz lemma and rigidity result for holomorphic self-mappings on the unit ball for $p=2$, and the unit polydisk for $p=\infty$, respectively. Reference | Related Articles | Metrics

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GLOBAL WEAK SOLUTIONS FOR AN ATTRACTION-REPULSION CHEMOTAXIS SYSTEM WITH $p$-LAPLACIAN DIFFUSION AND LOGISTIC SOURCE Xiaoshan Wang, Zhongqian Wang, Zhe Jia Acta mathematica scientia,Series B    2024, 44 (3): 909-924.   DOI: 10.1007/s10473-024-0308-7 Abstract （23）            Save This paper is concerned with the following attraction-repulsion chemotaxis system with $p$-Laplacian diffusion and logistic source: $$\left\{\begin{array}{ll}u_{t}=\nabla\cdot(|\nabla u|^{p-2}\nabla u)-\chi \nabla\cdot(u \nabla v)+\xi \nabla\cdot(u \nabla w)+f(u),\;\;\;x\in \Omega,\;t>0,\\v_{t}=\triangle v-\beta v+\alpha u^{k_{1}},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x\in \Omega,\;t>0,\\0=\triangle w-\delta w+\gamma u^{k_{2}},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x\in \Omega,\;t>0,\\u(x,0)=u_{0}(x),\;\;v(x,0)=v_{0}(x),\;\;w(x,0)=w_{0}(x), \;\;\;\;\;\;\;\;\;\;\;\;\;x\in \Omega.\end{array}\right.$$ The system here is under a homogenous Neumann boundary condition in a bounded domain $ \Omega \subset \mathbb{R}^{n}(n\geq2) $, with $ \chi, \xi, \alpha,\beta,\gamma,\delta, k_{1}, k_{2} >0, p\geq 2$. In addition, the function $f$ is smooth and satisfies that $f(s)\leq\kappa-\mu s^{l}$ for all $s\geq0$, with $\kappa\in \mathbb{R}, \mu>0, l>1$. It is shown that (i) if $l>\max\{ 2k_{1}, \frac{2k_{1}n}{2+n}+\frac{1}{p-1}\}$, then system possesses a global bounded weak solution and (ii) if $k_{2}>\max\{2k_{1}-1, \frac{2k_{1}n}{2+n}+\frac{2-p}{p-1}\}$ with $l>2$, then system possesses a global bounded weak solution. Reference | Related Articles | Metrics

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COMPLETE KAHLER METRICS WITH POSITIVE HOLOMORPHIC SECTIONAL CURVATURES ON CERTAIN LINE BUNDLES (RELATED TO A COHOMOGENEITY ONE POINT OF VIEW ON A YAU CONJECTURE)* Xiaoman Duan, Zhuangdan Guan Acta mathematica scientia,Series B    2024, 44 (1): 78-102.   DOI: 10.1007/s10473-024-0103-5 Abstract （23）            Save In this article, we study Kähler metrics on a certain line bundle over some compact Kähler manifolds to find complete Kähler metrics with positive holomorphic sectional (or bisectional) curvatures. Thus, we apply a strategy to a famous Yau conjecture with a co-homogeneity one geometry. Reference | Related Articles | Metrics

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ON A UNIVERSAL INEQUALITY FOR APPROXIMATE PHASE ISOMETRIES Duanxu Dai, Haixin Que, Longfa Sun, Bentuo Zheng Acta mathematica scientia,Series B    2024, 44 (3): 823-838.   DOI: 10.1007/s10473-024-0303-z Abstract （22）            Save Let $X$ and $Y$ be two normed spaces. Let $\mathcal{U}$ be a non-principal ultrafilter on $\mathbb{N}$. Let $g: X\rightarrow Y$ be a standard $\varepsilon$-phase isometry for some $\varepsilon\geq 0$, i.e., $g(0)=0$, and for all $u,v\in X$, $$|\; |\|g(u)+g(v)\|\pm \|g(u)-g(v)\||-|\|u+v\|\pm\|u-v\||\;|\leq\varepsilon.$$ The mapping $g$ is said to be a phase isometry provided that $\varepsilon=0$. In this paper, we show the following universal inequality of $g$: for each $u^*\in w^*$-exp $\|u^*\|B_{X^*}$, there exist a phase function $\sigma_{u^*}: X\rightarrow \{-1,1\}$ and $\varphi$ $\in$ $Y^*$ with $\|\varphi\|= \|u^*\|\equiv \alpha $ satisfying that $$\;\;\;\;\; |\langle u^*,u\rangle-\sigma_{u^*} (u)\langle \varphi, g(u)\rangle |\leq\frac{5}{2}\varepsilon\alpha ,\;\;{\rm for\;all\;}u\in X.$$ In particular, let $X$ be a smooth Banach space. Then we show the following: (1) the universal inequality holds for all $u^*\in X^*$; (2) the constant $\frac{5}{2}$ can be reduced to $\frac{3}{2}$ provided that $Y^\ast$ is strictly convex; (3) the existence of such a $g$ implies the existence of a phase isometry $\Theta:X\rightarrow Y$ such that $\Theta(u)=\lim\limits_{n,\mathcal{U}}\frac{g(nu)}{n}$ provided that $Y^{**}$ has the $w^*$-Kadec-Klee property (for example, $Y$ is both reflexive and locally uniformly convex). Reference | Related Articles | Metrics

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THE EXACT MEROMORPHIC SOLUTIONS OF SOME NONLINEAR DIFFERENTIAL EQUATIONS* Huifang Liu, Zhiqiang Mao Acta mathematica scientia,Series B    2024, 44 (1): 103-114.   DOI: 10.1007/s10473-024-0104-4 Abstract （20）      PDF       Save We find the exact forms of meromorphic solutions of the nonlinear differential equations $f^n+q(z){\rm e}^{Q(z)}f^{(k)}=p_1{\rm e}^{\alpha_1 z}+p_2{\rm e}^{\alpha_2 z}, \quad n\geq3, ~k\geq1,$ where $q, Q$ are nonzero polynomials, $Q\not\equiv Const.$, and $p_1, p_2, \alpha_1, \alpha_2$ are nonzero constants with $\alpha_1\neq\alpha_2$. Compared with previous results on the equation $p(z)f^3+q(z)f''=-\sin \alpha(z)$ with polynomial coefficients, our results show that the coefficient of the term $f^{(k)}$ perturbed by multiplying an exponential function will affect the structure of its solutions. Reference | Related Articles | Metrics

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THE LIMIT CYCLE BIFURCATIONS OF A WHIRLING PENDULUM WITH PIECEWISE SMOOTH PERTURBATIONS Jihua yang Acta mathematica scientia,Series B    2024, 44 (3): 1115-1144.   DOI: 10.1007/s10473-024-0319-4 Abstract （17）            Save This paper deals with the problem of limit cycles for the whirling pendulum equation $\dot{x}=y,\ \dot{y}=\sin x(\cos x-r)$ under piecewise smooth perturbations of polynomials of $\cos x$, $\sin x$ and $y$ of degree $n$ with the switching line $x=0$. The upper bounds of the number of limit cycles in both the oscillatory and the rotary regions are obtained using the Picard-Fuchs equations, which the generating functions of the associated first order Melnikov functions satisfy. Furthermore, the exact bound of a special case is given using the Chebyshev system. At the end, some numerical simulations are given to illustrate the existence of limit cycles. Reference | Related Articles | Metrics

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MATHEMATICAL MODELING AND BIFURCATION ANALYSIS FOR A BIOLOGICAL MECHANISM OF CANCER DRUG RESISTANCE Kangbo Bao, Guizhen LIANG, Tianhai TIAN, Xinan ZHANG Acta mathematica scientia,Series B    2024, 44 (3): 1165-1188.   DOI: 10.1007/s10473-024-0321-x Abstract （17）            Save Drug resistance is one of the most intractable issues in targeted therapy for cancer diseases. It has also been demonstrated to be related to cancer heterogeneity, which promotes the emergence of treatment-refractory cancer cell populations. Focusing on how cancer cells develop resistance during the encounter with targeted drugs and the immune system, we propose a mathematical model for studying the dynamics of drug resistance in a conjoint heterogeneous tumor-immune setting. We analyze the local geometric properties of the equilibria of the model. Numerical simulations show that the selectively targeted removal of sensitive cancer cells may cause the initially heterogeneous population to become a more resistant population. Moreover, the decline of immune recruitment is a stronger determinant of cancer escape from immune surveillance or targeted therapy than the decay in immune predation strength. Sensitivity analysis of model parameters provides insight into the roles of the immune system combined with targeted therapy in determining treatment outcomes. Reference | Related Articles | Metrics

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THE ASYMPTOTIC BEHAVIOR AND OSCILLATION FOR A CLASS OF THIRD-ORDER NONLINEAR DELAY DYNAMIC EQUATIONS Xianyong Huang, Xunhuan Deng, Qiru Wang Acta mathematica scientia,Series B    2024, 44 (3): 925-946.   DOI: 10.1007/s10473-024-0309-6 Abstract （17）            Save In this paper, we consider a class of third-order nonlinear delay dynamic equations. First, we establish a Kiguradze-type lemma and some useful estimates. Second, we give a sufficient and necessary condition for the existence of eventually positive solutions having upper bounds and tending to zero. Third, we obtain new oscillation criteria by employing the P ötzsche chain rule. Then, using the generalized Riccati transformation technique and averaging method, we establish the Philos-type oscillation criteria. Surprisingly, the integral value of the Philos-type oscillation criteria, which guarantees that all unbounded solutions oscillate, is greater than $\theta_{4}(t_1,T)$. The results of Theorem 3.5 and Remark 3.6 are novel. Finally, we offer four examples to illustrate our results. Reference | Related Articles | Metrics

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THE GLOBAL EXISTENCE OF STRONG SOLUTIONS TO THERMOMECHANICAL CUCKER-SMALE-STOKES QUATIONS IN THE WHOLE DOMAIN Weiyuan Zou Acta mathematica scientia,Series B    2024, 44 (3): 887-908.   DOI: 10.1007/s10473-024-0307-8 Abstract （17）            Save We study the global existence and uniqueness of a strong solution to the kinetic thermomechanical Cucker-Smale (for short, TCS) model coupled with Stokes equations in the whole space. The coupled system consists of the kinetic TCS equation for a particle ensemble and the Stokes equations for a fluid via a drag force. In this paper, we present a complete analysis of the existence of global-in-time strong solutions to the coupled model without any smallness restrictions on the initial data. Reference | Related Articles | Metrics

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THE BOUNDEDNESS OF OPERATORS ON WEIGHTED MULTI-PARAMETER LOCAL HARDY SPACES* Wei Ding, Yan Tang, Yueping Zhu Acta mathematica scientia,Series B    2024, 44 (1): 386-404.   DOI: 10.1007/s10473-024-0121-3 Abstract （17）      PDF       Save Though atomic decomposition is a very useful tool for studying the boundedness on Hardy spaces for some sublinear operators, untill now, the boundedness of operators on weighted Hardy spaces in a multi-parameter setting has been established only by almost orthogonality estimates. In this paper, we mainly establish the boundedness on weighted multi-parameter local Hardy spaces via atomic decomposition. Reference | Related Articles | Metrics