Acta Mathematica Scientia (Series B)
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ISSN 0252-9602
CN  42-1227/O
25 December 2020, Volume 40 Issue 6 Previous Issue   
Articles
CONTINUITY PROPERTIES FOR BORN-JORDAN OPERATORS WITH SYMBOLS IN HÖRMANDER CLASSES AND MODULATION SPACES
Maurice de GOSSON, Joachim TOFT
Acta mathematica scientia,Series B. 2020, 40 (6):  1603-1626.  DOI: 10.1007/s10473-020-0601-z
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We show that the Weyl symbol of a Born-Jordan operator is in the same class as the Born-Jordan symbol, when Hörmander symbols and certain types of modulation spaces are used as symbol classes. We use these properties to carry over continuity, nuclearity and Schatten-von Neumann properties to the Born-Jordan calculus.
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ASYMPTOTIC STABILITY OF A BOUNDARY LAYER AND RAREFACTION WAVE FOR THE OUTFLOW PROBLEM OF THE HEAT-CONDUCTIVE IDEAL GAS WITHOUT VISCOSITY
Lili FAN, Meichen HOU
Acta mathematica scientia,Series B. 2020, 40 (6):  1627-1652.  DOI: 10.1007/s10473-020-0602-y
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This article is devoted to studying the initial-boundary value problem for an ideal polytropic model of non-viscous and compressible gas. We focus our attention on the outflow problem when the flow velocity on the boundary is negative and give a rigorous proof of the asymptotic stability of both the degenerate boundary layer and its superposition with the 3-rarefaction wave under some smallness conditions. New weighted energy estimates are introduced, and the trace of the density and velocity on the boundary are handled by some subtle analysis. The decay properties of the boundary layer and the smooth rarefaction wave also play an important role.
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ON REFINEMENT OF THE COEFFICIENT INEQUALITIES FOR A SUBCLASS OF QUASI-CONVEX MAPPINGS IN SEVERAL COMPLEX VARIABLES
Qinghua XU, Yuanping LAI
Acta mathematica scientia,Series B. 2020, 40 (6):  1653-1665.  DOI: 10.1007/s10473-020-0603-x
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Let $\mathcal{K}$ be the familiar class of normalized convex functions in the unit disk. In [14], Keogh and Merkes proved that for a function $f(z)=z+\sum\limits_{k=2}^\infty a_kz^k$ in the class $\mathcal{K}$, \begin{align*} |a_3-\lambda a_2^2|\leq \max \left\{\frac{1}{3}, |\lambda-1|\right\},\ \ \lambda \in \mathbb{C}. \end{align*} The above estimate is sharp for each $\lambda$.
In this article, we establish the corresponding inequality for a normalized convex function $f$ on $\mathbb{U}$ such that $z=0$ is a zero of order $k+1$ of $f(z)-z$, and then we extend this result to higher dimensions. These results generalize some known results.
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EXISTENCE OF SOLUTIONS FOR THE FRACTIONAL (p, q)-LAPLACIAN PROBLEMS INVOLVING A CRITICAL SOBOLEV EXPONENT
Fanfan CHEN, Yang YANG
Acta mathematica scientia,Series B. 2020, 40 (6):  1666-1678.  DOI: 10.1007/s10473-020-0604-9
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In this article, we study the following fractional $(p,q)$-Laplacian equations involving the critical Sobolev exponent: \[ (P_{\mu, \lambda}) \begin{cases} (-\Delta)_{p}^{s_{1}}u+(-\Delta)_{q}^{s_{2}}u=\mu |u|^{q-2}u +\lambda|u|^{p-2}u + |u|^{p_{s_{1}}^{*}-2}u, & \text{in $\Omega$,} \\ u=0, & \text{in $\mathbb{R}^{N} \setminus \Omega$}, \end{cases} \] where $\Omega \subset \mathbb{R}^{N}$ is a smooth and bounded domain, $\lambda,\ \mu >0, \ 0 < s_{2} < s_{1} < 1,\ 1 < q < p < \frac{N}{s_{1}} $. We establish the existence of a non-negative nontrivial weak solution to $(P_{\mu, \lambda})$ by using the Mountain Pass Theorem. The lack of compactness associated with problems involving critical Sobolev exponents is overcome by working with certain asymptotic estimates for minimizers.
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RADIALLY SYMMETRIC SOLUTIONS FOR QUASILINEAR ELLIPTIC EQUATIONS INVOLVING NONHOMOGENEOUS OPERATORS IN AN ORLICZ-SOBOLEV SPACE SETTING
Jae-Myoung KIM, Yun-Ho KIM, Jongrak LEE
Acta mathematica scientia,Series B. 2020, 40 (6):  1679-1699.  DOI: 10.1007/s10473-020-0605-8
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We investigate the following elliptic equations: $$ \begin{cases} -M\Bigl(\int_{\mathbb{R}^N}\phi(|\nabla u|^2){\rm d}x\Bigr)\text{div}(\phi^{\prime}(|\nabla u|^2)\nabla u) +|u|^{\alpha-2}u=\lambda h(x,u), \\[2mm] u(x) \rightarrow 0, \quad \text{as} \ |x| \rightarrow \infty, \end{cases} \quad \text{ in } \ \ \mathbb{R}^N, $$ where $N \geq 2$, $1 < p < q < N$, $\alpha < q$, $1 < \alpha \leq p^*q^{\prime}/p^{\prime}$ with $p^*=\frac{Np}{N-p}$, $\phi(t)$ behaves like $t^{q/2}$ for small $t$ and $t^{p/2}$ for large $t$, and $p^{\prime}$ and $q^{\prime}$ are the conjugate exponents of $p$ and $q$, respectively. We study the existence of nontrivial radially symmetric solutions for the problem above by applying the mountain pass theorem and the fountain theorem. Moreover, taking into account the dual fountain theorem, we show that the problem admits a sequence of small-energy, radially symmetric solutions.
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ON VORTEX ALIGNMENT AND THE BOUNDEDNESS OF THE Lq-NORM OF VORTICITY IN INCOMPRESSIBLE VISCOUS FLUIDS
Siran LI
Acta mathematica scientia,Series B. 2020, 40 (6):  1700-1708.  DOI: 10.1007/s10473-020-0606-7
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We show that the spatial $L^q$-norm ($q>5/3$) of the vorticity of an incompressible viscous fluid in $\mathbb{R}^3$ remains bounded uniformly in time, provided that the direction of vorticity is Hölder continuous in space, and that the space-time $L^q$-norm of vorticity is finite. The Hölder index depends only on q. This serves as a variant of the classical result by Constantin-Fefferman (Direction of vorticity and the problem of global regularity for the Navier-Stokes equations, Indiana Univ. J. Math. 42 (1993), 775-789), and the related work by Grujić-Ruzmaikina (Interpolation between algebraic and geometric conditions for smoothness of the vorticity in the 3D NSE, Indiana Univ. J. Math. 53 (2004), 1073-1080).
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A SUBCLASS OF QUASI-CONVEX MAPPINGS ON A REINHARDT DOMAIN IN $\mathbb{C}^n$
Xiaosong LIU
Acta mathematica scientia,Series B. 2020, 40 (6):  1709-1722.  DOI: 10.1007/s10473-020-0607-6
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Let $D_{p_1,p_2,\cdots,p_n}=\{z\in \mathbb{C}^n: \sum\limits_{l=1}^n|z_l|^{p_l}<1\}, p_l> 1, l=1,2,\cdots,n$. In this article, we first establish the sharp estimates of the main coefficients for a subclass of quasi-convex mappings (including quasi-convex mappings of type $\mathbb{A}$ and quasi-convex mappings of type $\mathbb{B}$) on $D_{p_1,p_2,\cdots,p_n}$ under some weak additional assumptions. Meanwhile, we also establish the sharp distortion theorems for the above mappings. The results that we obtain reduce to the corresponding classical results in one dimension.
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A GLOBALLY CONVERGENT QP-FREE ALGORITHM FOR INEQUALITY CONSTRAINED MINIMAX OPTIMIZATION
Jinbao JIAN, Guodong MA
Acta mathematica scientia,Series B. 2020, 40 (6):  1723-1738.  DOI: 10.1007/s10473-020-0608-5
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Although QP-free algorithms have good theoretical convergence and are effective in practice, their applications to minimax optimization have not yet been investigated. In this article, on the basis of the stationary conditions, without the exponential smooth function or constrained smooth transformation, we propose a QP-free algorithm for the nonlinear minimax optimization with inequality constraints. By means of a new and much tighter working set, we develop a new technique for constructing the sub-matrix in the lower right corner of the coefficient matrix. At each iteration, to obtain the search direction, two reduced systems of linear equations with the same coefficient are solved. Under mild conditions, the proposed algorithm is globally convergent. Finally, some preliminary numerical experiments are reported, and these show that the algorithm is promising.
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SPECTRA OF COMPOSITION GROUPS ON THE WEIGHTED DIRICHLET SPACE OF THE UPPER HALF-PLANE
M. O. AGWANG, J. O. BONYO
Acta mathematica scientia,Series B. 2020, 40 (6):  1739-1752.  DOI: 10.1007/s10473-020-0609-4
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We prove that the group of weighted composition operators induced by continuous automorphism groups of the upper half plane $\mathbb{U}$ is strongly continuous on the weighted Dirichlet space of $\mathbb{U}$, $\mathcal{D}_{α}$($\mathbb{U}$). Further, we investigate when they are isometries on $\mathcal{D}_{α}$($\mathbb{U}$). In each case, we determine the semigroup properties while in the case that the induced composition group is an isometry, we apply similarity theory to determine the spectral properties of the group.
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GENERALIZED ROPER-SUFFRIDGE OPERATOR FOR $\epsilon$ STARLIKE AND BOUNDARY STARLIKE MAPPINGS
Jie WANG, Jianfei WANG
Acta mathematica scientia,Series B. 2020, 40 (6):  1753-1764.  DOI: 10.1007/s10473-020-0610-y
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This article is devoted to a deep study of the Roper-Suffridge extension operator with special geometric properties. First, we prove that the Roper-Suffridge extension operator preserves $\epsilon$ starlikeness on the open unit ball of a complex Banach space $\mathbb{C}\times X$, where $X$ is a complex Banach space. This result includes many known results. Secondly, by introducing a new class of almost boundary starlike mappings of order $\alpha$ on the unit ball $B^n$ of ${\mathbb{C}}^{n}$, we prove that the Roper-Suffridge extension operator preserves almost boundary starlikeness of order $\alpha$ on $B^n$. Finally, we propose some problems.
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STRONG EQUIVALENCES OF APPROXIMATION NUMBERS AND TRACTABILITY OF WEIGHTED ANISOTROPIC SOBOLEV EMBEDDINGS
Jidong HAO, Heping WANG
Acta mathematica scientia,Series B. 2020, 40 (6):  1765-1782.  DOI: 10.1007/s10473-020-0611-x
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In this article, we study multivariate approximation defined over weighted anisotropic Sobolev spaces which depend on two sequences a=$\{a_j\}_{j\geq1}$ and b=$\{b_j\}_{j\geq1}$ of positive numbers. We obtain strong equivalences of the approximation numbers, and necessary and sufficient conditions on a, b to achieve various notions of tractability of the weighted anisotropic Sobolev embeddings.
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THE CAUCHY PROBLEM FOR THE TWO LAYER VISCOUS SHALLOW WATER EQUATIONS
Pengcheng MU, Qiangchang JU
Acta mathematica scientia,Series B. 2020, 40 (6):  1783-1807.  DOI: 10.1007/s10473-020-0612-9
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In this paper, the Cauchy problem for the two layer viscous shallow water equations is investigated with third-order surface-tension terms and a low regularity assumption on the initial data. The global existence and uniqueness of the strong solution in a hybrid Besov space are proved by using the Littlewood-Paley decomposition and Friedrichs' regularization method.
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THE EXISTENCE OF A NONTRIVIAL WEAK SOLUTION TO A DOUBLE CRITICAL PROBLEM INVOLVING A FRACTIONAL LAPLACIAN IN $\mathbb{R}^N$ WITH A HARDY TERM
Gongbao LI, Tao YANG
Acta mathematica scientia,Series B. 2020, 40 (6):  1808-1830.  DOI: 10.1007/s10473-020-0613-8
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In this paper, we consider the existence of nontrivial weak solutions to a double critical problem involving a fractional Laplacian with a Hardy term: \begin{equation} \label{eq0.1} (-\Delta)^{s}u-{\gamma} {\frac{u}{|x|^{2s}}}= {\frac{{|u|}^{ {2^{*}_{s}}(\beta)-2}u}{|x|^{\beta}}}+ \big [ I_{\mu}* F_{\alpha}(\cdot,u) \big](x)f_{\alpha}(x,u), \ \ u \in {\dot{H}}^s(\mathbb{R}^n), (0.1)\end{equation} where $s \in(0,1)$, $0\leq \alpha,\beta < 2s < n$, $\mu \in (0,n)$, $\gamma < \gamma_{H}$, $I_{\mu}(x)=|x|^{-\mu}$, $F_{\alpha}(x,u)=\frac{ {|u(x)|}^{ {2^{\#}_{\mu} }(\alpha)} }{ {|x|}^{ {\delta_{\mu} (\alpha)} } }$, $f_{\alpha}(x,u)=\frac{ {|u(x)|}^{{ 2^{\#}_{\mu} }(\alpha)-2}u(x) }{ {|x|}^{ {\delta_{\mu} (\alpha)} } }$, $2^{\#}_{\mu} (\alpha)=(1-\frac{\mu}{2n})\cdot 2^{*}_{s} (\alpha)$, $\delta_{\mu} (\alpha)=(1-\frac{\mu}{2n})\alpha$, ${2^{*}_{s}}(\alpha)=\frac{2(n-\alpha)}{n-2s}$ and $\gamma_{H}=4^s\frac{\Gamma^2(\frac{n+2s}{4})} {\Gamma^2(\frac{n-2s}{4})}$. We show that problem (0.1) admits at least a weak solution under some conditions.
To prove the main result, we develop some useful tools based on a weighted Morrey space. To be precise, we discover the embeddings \begin{equation} \label{eq0.2} {\dot{H}}^s(\mathbb{R}^n) \hookrightarrow {L}^{2^*_{s}(\alpha)}(\mathbb{R}^n,|y|^{-\alpha}) \hookrightarrow L^{p,\frac{n-2s}{2}p+pr}(\mathbb{R}^n,|y|^{-pr}), (0.2)\end{equation} where $s \in (0,1)$, $0 < \alpha < 2s < n$, $p\in[1,2^*_{s}(\alpha))$ and $r=\frac{\alpha}{ 2^*_{s}(\alpha) }$. We also establish an improved Sobolev inequality, \begin{equation} \label{eq0.3} \Big( \int_{ \mathbb{R}^n } \frac{ |u(y)|^{ 2^*_{s}(\alpha)} } { |y|^{\alpha} }{\rm d}y \Big)^{ \frac{1}{ 2^*_{s} (\alpha) }} \leq C ||u||_{{\dot{H}}^s(\mathbb{R}^n)}^{\theta} ||u||^{1-\theta}_{ L^{p,\frac{n-2s}{2}p+pr}(\mathbb{R}^n,|y|^{-pr}) },~~~~\forall u \in {\dot{H}}^s(\mathbb{R}^n), (0.3)\end{equation} where $s \in (0,1)$, $0 < \alpha < 2s < n$, $p\in[1,2^*_{s}(\alpha))$, $r=\frac{\alpha}{ 2^*_{s}(\alpha) }$, $0 < \max \{ \frac{2}{2^*_{s}(\alpha)}, \frac{2^*_{s}-1}{2^*_{s}(\alpha)} \} < \theta < 1$, ${2^{*}_{s}}=\frac{2n}{n-2s}$ and $C=C(n,s,\alpha) > 0$ is a constant. Inequality (0.3) is a more general form of Theorem 1 in Palatucci, Pisante [1].
By using the mountain pass lemma along with (0.2) and (0.3), we obtain a nontrivial weak solution to problem (0.1) in a direct way. It is worth pointing out that (0.2) and 0.3) could be applied to simplify the proof of the existence results in [2] and [3].
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EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR A COUPLED SYSTEM OF KIRCHHOFF TYPE EQUATIONS
Yaghoub JALILIAN
Acta mathematica scientia,Series B. 2020, 40 (6):  1831-1848.  DOI: 10.1007/s10473-020-0614-7
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In this paper, we study the coupled system of Kirchhoff type equations \begin{equation*} \left\{ \begin{array}{ll} -\bigg(a+b\int_{\mathbb{R}^3}{|\nabla u|^{2}{\rm d}x}\bigg)\Delta u+ u = \frac{2\alpha}{\alpha+\beta}|u|^{\alpha-2}u|v|^{\beta}, & x\in \mathbb{R}^3, \\[3mm] -\bigg(a+b\int_{\mathbb{R}^3}{|\nabla v|^{2}{\rm d}x}\bigg)\Delta v+ v = \frac{2\beta}{\alpha+\beta}|u|^{\alpha}|v|^{\beta-2}v, & x\in \mathbb{R}^3, \\[2mm] u,v\in H^{1}(\mathbb{R}^3), \end{array} \right. \end{equation*} where $a,b > 0$, $ \alpha, \beta > 1$ and $3 < \alpha+\beta < 6$. We prove the existence of a ground state solution for the above problem in which the nonlinearity is not 4-superlinear at infinity. Also, using a discreetness property of Palais-Smale sequences and the Krasnoselkii genus method, we obtain the existence of infinitely many geometrically distinct solutions in the case when $ \alpha, \beta \geq 2$ and $4\leq\alpha+\beta < 6$.
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GLOBAL OPTIMIZATION OF THE DIFFERENCE OF TWO INCREASING PLUS-CONVEX-ALONG-RAYS FUNCTIONS
H. SHAHRIARIPOUR, H. MOHEBI
Acta mathematica scientia,Series B. 2020, 40 (6):  1849-1873.  DOI: 10.1007/s10473-020-0615-6
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The theory of increasing and convex-along-rays (ICAR) functions defined on a convex cone in a real locally convex topological vector space X was already well developed. In this paper, we first examine abstract convexity of increasing plus-convex-along-rays (IPCAR) functions defined on a real normed linear space X. We also study, for this class of functions, some concepts of abstract convexity, such as support sets and subdifferentials. Finally, as an application, we characterize the maximal elements of the support set of strictly IPCAR functions and give optimality conditions for the global minimum of the difference between two IPCAR functions.
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PARAMETRIC REPRESENTATIONS OF QUASICONFORMAL MAPPINGS
Zhenlian LIN, Qingtian SHI
Acta mathematica scientia,Series B. 2020, 40 (6):  1874-1882.  DOI: 10.1007/s10473-020-0616-5
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In this article, we first give two simple examples to illustrate that two types of parametric representation of the family of $\Sigma_{K}^{0}$ have some gaps. Then we also find that the area derivative formula (1.6), which is used to estimate the area distortion of $\Sigma_{K}^{0}$, cannot be derived from [6], but that formula still holds for $\Sigma_{K}^{0}$ through our amendatory parametric representation for the one obtained by Eremenko and Hamilton.
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DYNAMIC FOR A STOCHASTIC MULTI-GROUP AIDS MODEL WITH SATURATED INCIDENCE RATE
Qixing HAN, Daqing JIANG
Acta mathematica scientia,Series B. 2020, 40 (6):  1883-1896.  DOI: 10.1007/s10473-020-0617-4
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In this paper, a stochastic multi-group AIDS model with saturated incidence rate is studied. We prove that the system is persistent in the mean under some parametric restrictions. We also obtain the sufficient condition for the existence of the ergodic stationary distribution of the system by constructing a suitable Lyapunov function. Our results indicate that the existence of ergodic stationary distribution does not rely on the interior equilibrium of the corresponding deterministic system, which greatly improves upon previous results.
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ON THE COMPLETE 2-DIMENSIONAL λ-TRANSLATORS WITH A SECOND FUNDAMENTAL FORM OF CONSTANT LENGTH
Xingxiao LI, Ruina QIAO, Yangyang LIU
Acta mathematica scientia,Series B. 2020, 40 (6):  1897-1914.  DOI: 10.1007/s10473-020-0618-3
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In this article we study the two-dimensional complete $\lambda$-translators immersed in the Euclidean space $\mathbb{R}^3$ and Minkovski space $\mathbb{R}^3_1$. We obtain two classification theorems: one for two-dimensional complete $\lambda$-translators $x:M^2\to\mathbb{R}^3$ and one for two-dimensional complete space-like $\lambda$-translators $x:M^2\to\mathbb{R}^3_1$, with a second fundamental form of constant length.
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VAR AND CTE BASED OPTIMAL REINSURANCE FROM A REINSURER'S PERSPECTIVE
Tao TAN, Tao CHEN, Lijun WU, Yuhong SHENG, Yijun HU
Acta mathematica scientia,Series B. 2020, 40 (6):  1915-1927.  DOI: 10.1007/s10473-020-0619-2
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In this article, we study optimal reinsurance design. By employing the increasing convex functions as the admissible ceded loss functions and the distortion premium principle, we study and obtain the optimal reinsurance treaty by minimizing the VaR (value at risk) of the reinsurer's total risk exposure. When the distortion premium principle is specified to be the expectation premium principle, we also obtain the optimal reinsurance treaty by minimizing the CTE (conditional tail expectation) of the reinsurer's total risk exposure. The present study can be considered as a complement of that of Cai et al.[5].
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THE DECAY ESTIMATES FOR MAGNETOHYDRODYNAMIC EQUATIONS WITH COULOMB FORCE
Wenxuan ZHENG, Zhong TAN
Acta mathematica scientia,Series B. 2020, 40 (6):  1928-1940.  DOI: 10.1007/s10473-020-0620-9
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In this article we consider the compressible viscous magnetohydrodynamic equations with Coulomb force. By spectral analysis and energy methods, we obtain the optimal time decay estimate of the solution. We show that the global classical solution converges to its equilibrium state at the same decay rate as the solution of the linearized equations.
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ASYMPTOTICS OF THE CROSS-VARIATION OF YOUNG INTEGRALS WITH RESPECT TO A GENERAL SELF-SIMILAR GAUSSIAN PROCESS
Soukaina DOUISSI, Khalifa ES-SEBAIY, Soufiane MOUSSATEN
Acta mathematica scientia,Series B. 2020, 40 (6):  1941-1960.  DOI: 10.1007/s10473-020-0621-8
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We show in this work that the limit in law of the cross-variation of processes having the form of Young integral with respect to a general self-similar centered Gaussian process of order β ∈ (1/2, 3/4] is normal according to the values of β. We apply our results to two self-similar Gaussian processes:the subfractional Brownian motion and the bifractional Brownian motion.
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EXISTENCE AND UNIQUENESS OF THE POSITIVE STEADY STATE SOLUTION FOR A LOTKA-VOLTERRA PREDATOR-PREY MODEL WITH A CROWDING TERM
Xianzhong ZENG, Lingyu LIU, Weiyuan XIE
Acta mathematica scientia,Series B. 2020, 40 (6):  1961-1980.  DOI: 10.1007/s10473-020-0622-7
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This paper deals with a Lotka-Volterra predator-prey model with a crowding term in the predator equation. We obtain a critical value $\lambda_1^D(\Omega_0)$, and demonstrate that the existence of the predator in $\overline{\Omega}_0$ only depends on the relationship of the growth rate $\mu$ of the predator and $\lambda_1^D(\Omega_0)$, not on the prey. Furthermore, when $\mu<\lambda_1^D(\Omega_0)$, we obtain the existence and uniqueness of its positive steady state solution, while when $\mu\geq \lambda_1^D(\Omega_0)$, the predator and the prey cannot coexist in $\overline{\Omega}_0$. Our results show that the coexistence of the prey and the predator is sensitive to the size of the crowding region $\overline{\Omega}_0$, which is different from that of the classical Lotka-Volterra predator-prey model.
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ON THE NUCLEARITY OF COMPLETELY 1-SUMMING MAPPING SPACES
Zhe DONG, Yafei ZHAO
Acta mathematica scientia,Series B. 2020, 40 (6):  1981-1988.  DOI: 10.1007/s10473-020-0623-6
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In this paper, we investigate the $\lambda$-nuclearity in the system of completely 1-summing mapping spaces $(\Pi_{1}(\cdot, \cdot), \pi_{1})$. In Section 2, we obtain that $\mathbb{C}$ is the unique operator space that is nuclear in the system $(\Pi_{1}(\cdot, \cdot), \pi_{1})$. We generalize some results in Section 2 to $\lambda$-nuclearity in Section 3.
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ITÔ DIFFERENTIAL REPRESENTATION OF SINGULAR STOCHASTIC VOLTERRA INTEGRAL EQUATIONS
Nguyen Tien DUNG
Acta mathematica scientia,Series B. 2020, 40 (6):  1989-2000.  DOI: 10.1007/s10473-020-0624-5
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In this paper we obtain an Itô differential representation for a class of singular stochastic Volterra integral equations. As an application, we investigate the rate of convergence in the small time central limit theorem for the solution.
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