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CONSTANT DISTANCE BOUNDARIES OF THE $t$-QUASICIRCLE AND THE KOCH SNOWFLAKE CURVE* Xin Wei, Zhi-Ying Wen Acta mathematica scientia,Series B    2023, 43 (3): 981-993.   DOI: 10.1007/s10473-023-0301-6 Abstract （93）      PDF       Save Let $\Gamma$ be a Jordan curve in the complex plane and let $\Gamma_\lambda$ be the constant distance boundary of $\Gamma$. Vellis and Wu \cite{VW} introduced the notion of a $(\zeta,r_0)$-chordal property which guarantees that, when $\lambda$ is not too large, $\Gamma_\lambda$ is a Jordan curve when $\zeta=1/2$ and $\Gamma_\lambda$ is a quasicircle when $0<\zeta<1/2$. We introduce the $(\zeta,r_0,t)$-chordal property, which generalizes the $(\zeta,r_0)$-chordal property, and we show that under the condition that $\Gamma$ is $(\zeta,r_0,\sqrt t)$-chordal with $0<\zeta < r_0^{1-\sqrt t}/2$, there exists $\varepsilon>0$ such that $\Gamma_\lambda$ is a $t$-quasicircle once $\Gamma_\lambda$ is a Jordan curve when $0<\zeta<\varepsilon$. In the last part of this paper, we provide an example: $\Gamma$ is a kind of Koch snowflake curve which does not have the $(\zeta,r_0)$-chordal property for any $0<\zeta\le 1/2$, however $\Gamma_\lambda$ is a Jordan curve when $\zeta$ is small enough. Meanwhile, $\Gamma$ has the $(\zeta,r_0,\sqrt t)$-chordal property with $0<\zeta < r_0^{1- \sqrt t}/2$ for any $t\in (0,1/4)$. As a corollary of our main theorem, $\Gamma_\lambda$ is a $t$-quasicircle for all $0<t<1/4$ when $\zeta$ is small enough. This means that our $(\zeta,r_0,t)$-chordal property is more general and applicable to more complicated curves. 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 （54）      PDF（pc） （457KB）（22）       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 （52）      PDF（pc） （1402KB）（26）       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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STABILITY CONDITIONS AND THE MIRROR SYMMETRY OF K3 SURFACES IN ATTRACTOR BACKGROUNDS* Wenxuan Lu Acta mathematica scientia,Series B    2023, 43 (3): 1007-1030.   DOI: 10.1007/s10473-023-0303-4 Abstract （43）      PDF       Save We study the space of stability conditions on $K3$ surfaces from the perspective of mirror symmetry. This is done in the attractor backgrounds (moduli). We find certain highly non-generic behaviors of marginal stability walls (a key notion in the study of wall crossings) in the space of stability conditions. These correspond via mirror symmetry to some non-generic behaviors of special Lagrangians in an attractor background. The main results can be understood as a mirror correspondence in a synthesis of the homological mirror conjecture and SYZ mirror conjecture. 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 （43）      PDF（pc） （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. Reference | Related Articles | Metrics

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SINGULAR DOUBLE PHASE EQUATIONS* Zhenhai Liu, Nikolaos S. Papageorgiou Acta mathematica scientia,Series B    2023, 43 (3): 1031-1044.   DOI: 10.1007/s10473-023-0304-3 Abstract （42）      PDF       Save We study a double phase Dirichlet problem with a reaction that has a parametric singular term. Using the Nehari manifold method, we show that for all small values of the parameter, the problem has at least two positive, energy minimizing solutions. Reference | Related Articles | Metrics

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RANDERS SPACES WITH SCALAR FLAG CURVATURE* Jintang LI Acta mathematica scientia,Series B    2023, 43 (3): 994-1006.   DOI: 10.1007/s10473-023-0302-5 Abstract （42）      PDF       Save Let $(M, F)$ be an $n$-dimensional Randers space with scalar flag curvature. In this paper, we will introduce the definition of a weak Einstein manifold. We can prove that if $(M, F)$ is a weak Einstein manifold, then the flag curvature is constant. 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 （40）      PDF（pc） （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. 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 （36）      PDF（pc） （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. 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 （32）      PDF（pc） （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. Reference | Related Articles | Metrics

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BIFURCATION ANALYSIS IN A PREDATOR-PREY MODEL WITH AN ALLEE EFFECT AND A DELAYED MECHANISM* Danyang LI, Hua LIU, Haotian ZHANG, Ming MA, Yong YE, Yumei WEI Acta mathematica scientia,Series B    2023, 43 (3): 1415-1438.   DOI: 10.1007/s10473-023-0324-z Abstract （32）      PDF       Save Regarding delay-induced predator-prey models, much research has been done on delayed destabilization, but whether delays are stabilizing or destabilizing is a subtle issue. In this study, we investigate predator-prey dynamics affected by both delays and the Allee effect. We analyze the consequences of delays in different feedback mechanisms. The existence of a Hopf bifurcation is studied, and we calculate the value of the delay that leads to the Hopf bifurcation. Furthermore, applying the normal form theory and a center manifold theorem, we consider the direction and stability of the Hopf bifurcation. Finally, we present numerical experiments that validate our theoretical analysis. Interestingly, depending on the chosen delay mechanism, we find that delays are not necessarily destabilizing. The Allee effect generally increases the stability of the equilibrium, and when the Allee effect involves a delay term, the stabilization effect is more pronounced. Reference | Related Articles | Metrics

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THE SASA-SATSUMA EQUATION ON A NON-ZERO BACKGROUND: THE INVERSE SCATTERING TRANSFORM AND MULTI-SOLITON SOLUTIONS* Lili WEN, Engui FAN, Yong CHEN Acta mathematica scientia,Series B    2023, 43 (3): 1045-1080.   DOI: 10.1007/s10473-023-0305-2 Abstract （31）      PDF       Save We concentrate on the inverse scattering transformation for the Sasa-Satsuma equation with $3\times 3$ matrix spectrum problem and a nonzero boundary condition. To circumvent the multi-value of eigenvalues, we introduce a suitable two-sheet Riemann surface to map the original spectral parameter $k$ into a single-valued parameter $z$. The analyticity of the Jost eigenfunctions and scattering coefficients of the Lax pair for the Sasa-Satsuma equation are analyzed in detail. According to the analyticity of the eigenfunctions and the scattering coefficients, the $z$-complex plane is divided into four analytic regions of $D_j: \ j=1, 2, 3, 4$. Since the second column of Jost eigenfunctions is analytic in $D_{j}$, but in the upper-half or lower-half plane, we introduce certain auxiliary eigenfunctions which are necessary for deriving the analytic eigenfunctions in $D_{j}$. We find that the eigenfunctions, the scattering coefficients and the auxiliary eigenfunctions all possess three kinds of symmetries; these characterize the distribution of the discrete spectrum. The asymptotic behaviors of eigenfunctions, auxiliary eigenfunctions and scattering coefficients are also systematically derived. Then a matrix Riemann-Hilbert problem with four kinds of jump conditions associated with the problem of nonzero asymptotic boundary conditions is established, from this $N$-soliton solutions are obtained via the corresponding reconstruction formulae. The reflectionless soliton solutions are explicitly given. As an application of the $N$-soliton formula, we present three kinds of single-soliton solutions according to the distribution of discrete spectrum. Reference | Related Articles | Metrics

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PROPERTIES OF SOLUTIONS TO A HARMONIC-MAPPING TYPE EQUATION WITH A DIRICHLET BOUNDARY CONDITION* Bo Chen, Zhengmao Chen, Junhui Xie Acta mathematica scientia,Series B    2023, 43 (3): 1161-1174.   DOI: 10.1007/s10473-023-0310-5 Abstract （29）      PDF       Save In the present paper, we consider the problem $\begin{equation} \left\{\begin{array}{ll}\label{0001} -\Delta u=u^{\beta_1}|\nabla u|^{\beta_2}, &\ \ { in} \ \Omega,\\ u=0,&\ \ { on} \ \partial{\Omega},\\ u>0,&\ \ { in} \ {\Omega},\\ \end{array}\right. \end{equation}$ $ \ \ \ \ \ $ (0.1) where $\beta_1,\beta_2>0$ and $\beta_1+\beta_2<1$, and $\Omega $ is a convex domain in $ \mathbb{R}^{n} $. The existence, uniqueness, regularity and $\frac{2-\beta_{2}}{1-\beta_1-\beta_2}$-concavity of the positive solutions of the problem (0.1) are proven. Reference | Related Articles | Metrics

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DOUBLE INERTIAL PROXIMAL GRADIENT ALGORITHMS FOR CONVEX OPTIMIZATION PROBLEMS AND APPLICATIONS* Kunrada Kankam, Prasit Cholamjiak Acta mathematica scientia,Series B    2023, 43 (3): 1462-1476.   DOI: 10.1007/s10473-023-0326-x Abstract （29）      PDF       Save In this paper, we propose double inertial forward-backward algorithms for solving unconstrained minimization problems and projected double inertial forward-backward algorithms for solving constrained minimization problems. We then prove convergence theorems under mild conditions. Finally, we provide numerical experiments on image restoration problem and image inpainting problem. The numerical results show that the proposed algorithms have more efficient than known algorithms introduced in the literature. Reference | Related Articles | Metrics

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THE EXISTENCE AND LOCAL UNIQUENESS OF MULTI-PEAK SOLUTIONS TO A CLASS OF KIRCHHOFF TYPE EQUATIONS* Leilei CUI, Jiaxing GUO, Gongbao LI Acta mathematica scientia,Series B    2023, 43 (3): 1131-1160.   DOI: 10.1007/s10473-023-0309-y Abstract （28）      PDF       Save In this paper, we study the existence and local uniqueness of multi-peak solutions to the Kirchhoff type equations $\begin{equation*} -\left(\varepsilon^2 a+\varepsilon b\int_{\mathbb R^3}|\nabla u|^2\right)\Delta u +V(x)u =u^{p}, u>0 \text{in} \mathbb{R}^3, \end{equation*}$ which concentrate at non-degenerate critical points of the potential function $V(x)$, where $a,b>0$, $1<p<5$ are constants, and $\varepsilon>0$ is a parameter. Applying the Lyapunov-Schmidt reduction method and a local Pohozaev type identity, we establish the existence and local uniqueness results of multi-peak solutions, which concentrate at $\{a_i\}_{1\leq i\leq k}$, where $\{a_{i}\}_{1\leq i\leq k}$ are non-degenerate critical points of $V(x)$ as $\varepsilon\to 0$. Reference | Related Articles | Metrics

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POSITIVE SOLUTIONS WITH HIGH ENERGY FOR FRACTIONAL SCHRÖDINGER EQUATIONS* Qing Guo, Leiga Zhao Acta mathematica scientia,Series B    2023, 43 (3): 1116-1130.   DOI: 10.1007/s10473-023-0308-z Abstract （28）      PDF       Save In this paper, we study the Schrödinger equations $ (-\Delta)^s u+ V(x)u= a(x)|u|^{p-2}u+b(x)|u|^{q-2}u,\ \ x\in\ {\mathbb{R}}^{N},$ where $0<s<1$, $2<q<p<2^*_s$, $2^*_s$ is the fractional Sobolev critical exponent. Under suitable assumptions on $V$, $a$ and $b$ for which there may be no ground state solution, the existence of positive solutions are obtained via variational methods. 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 （27）      PDF（pc） （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]. Reference | Related Articles | Metrics

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FIXED/PREASSIGNED-TIME SYNCHRONIZATION OF QUATERNION-VALUED NEURAL NETWORKS INVOLVING DELAYS AND DISCONTINUOUS ACTIVATIONS: A DIRECT APPROACH* Wanlu WEI, Cheng HU, Juan YU, Haijun JIANG Acta mathematica scientia,Series B    2023, 43 (3): 1439-1461.   DOI: 10.1007/s10473-023-0325-y Abstract （26）      PDF       Save The fixed-time synchronization and preassigned-time synchronization are investigated for a class of quaternion-valued neural networks with time-varying delays and discontinuous activation functions. Unlike previous efforts that employed separation analysis and the real-valued control design, based on the quaternion-valued signum function and several related properties, a direct analytical method is proposed here and the quaternion-valued controllers are designed in order to discuss the fixed-time synchronization for the relevant quaternion-valued neural networks. In addition, the preassigned-time synchronization is investigated based on a quaternion-valued control design, where the synchronization time is preassigned and the control gains are finite. Compared with existing results, the direct method without separation developed in this article is beneficial in terms of simplifying theoretical analysis, and the proposed quaternion-valued control schemes are simpler and more effective than the traditional design, which adds four real-valued controllers. Finally, two numerical examples are given in order to support the theoretical results. Reference | Related Articles | Metrics

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THE SINGULAR LIMIT OF SECOND-GRADE FLUID EQUATIONS IN A 2D EXTERIOR DOMAIN* Xiaoguang You, Aibin Zang Acta mathematica scientia,Series B    2023, 43 (3): 1333-1346.   DOI: 10.1007/s10473-023-0319-9 Abstract （26）      PDF       Save In this paper, we consider the second-grade fluid equations in a 2D exterior domain satisfying the non-slip boundary conditions. The second-grade fluid model is a well-known non-Newtonian fluid model, with two parameters: $\alpha$, which represents the length-scale, while $\nu > 0$ corresponds to the viscosity. We prove that, as $\nu, \alpha$ tend to zero, the solution of the second-grade fluid equations with suitable initial data converges to the one of Euler equations, provided that $\nu = {o}(\alpha^\frac{4}{3})$. Moreover, the convergent rate is obtained. Reference | Related Articles | Metrics

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ON THE RIGOROUS MATHEMATICAL DERIVATION FOR THE VISCOUS PRIMITIVE EQUATIONS WITH DENSITY STRATIFICATION* Xueke Pu, Wenli Zhou Acta mathematica scientia,Series B    2023, 43 (3): 1081-1104.   DOI: 10.1007/s10473-023-0306-1 Abstract （25）      PDF       Save In this paper, we rigorously derive the governing equations describing the motion of a stable stratified fluid, from the mathematical point of view. In particular, we prove that the scaled Boussinesq equations strongly converge to the viscous primitive equations with density stratification as the aspect ratio goes to zero, and the rate of convergence is of the same order as the aspect ratio. Moreover, in order to obtain this convergence result, we also establish the global well-posedness of strong solutions to the viscous primitive equations with density stratification. Reference | Related Articles | Metrics