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    The General Inverse Bonnesen-Style Inequalities in $\mathbb{R}^n$
    Xu Dong,Yan Zhang,Chunna Zeng,Xingxing Wang
    Acta mathematica scientia,Series A    2022, 42 (3): 641-650.  
    Abstract124)   HTML9)    PDF(pc) (337KB)(122)       Save

    The isoperimetric problem plays an important role in integral geometry. In this paper we mainly investigate the inverse form of the isoperimetric inequality, i.e. the general inverse Bonnesen-type inequalities. The upper bounds of several new general isoperimetric genus are obtained. Futhermore, as corollaries, we get a series of classical inverse Bonnesen-type inequalities in the plane. Finally, the best estimate between the results of three upper bounds is given.

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    Bonnesen-Style Inequalities for Star Bodies
    Zengle Zhang
    Acta mathematica scientia,Series A    2021, 41 (5): 1249-1262.  
    Abstract102)   HTML14)    PDF(pc) (351KB)(213)       Save

    Motivated by works of Lutwak and Petty[25-26, 37], a new star body ${\cal G}K$ associated with a given convex body $K$ is constructed. The isoperimetric inequality for ${\cal G}K$ and the reverse Bonnesen-style inequalities for K are established.

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    Second Main Theorem for Algebraic Curves on Compact Riemann Surfaces
    Lizhen Duan,Hongzhe Cao
    Acta mathematica scientia,Series A    2021, 41 (6): 1585-1597.  
    Abstract101)   HTML10)    PDF(pc) (357KB)(189)       Save

    In this paper, we first establish some second main theorems for algebraic curves from a compact Riemann surface into a complex projective subvariety of the complex projective space, which is ramified over hypersurfaces in subgeneral position. Then we use it to study the ramification for the generalized Gauss map of complete regular minimal surfaces in $\mathbb{R}^{m}$ with finite total curvature.

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    Shape Optimization for p-Torsional Rigidity Problems
    Qihua Ruan
    Acta mathematica scientia,Series A    2021, 41 (6): 1625-1633.  
    Abstract82)   HTML0)    PDF(pc) (301KB)(122)       Save

    In this paper, we construct a shape functional for p-torsional rigidity problems and prove that the optimal shape of this shape functional is a ball. Using a method of the shape derivative, we give an alternative proof of the overdetermined problem for p-torsional rigidity.

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    Ranks of Quantum States with Prescribed Reduced States in an Infinite Dimensional Quantum System
    Shuyuan Yang,Kan He
    Acta mathematica scientia,Series A    2022, 42 (1): 1-8.  
    Abstract75)   HTML6)    PDF(pc) (869KB)(110)       Save

    Suppose that $ H$ and $K $ are two infinite dimensional quantum systems (i.e. infinite dimensional complex Hilbert space). Let $\rho $ be a quantum state on $ H\otimes K$ with two reduced states $tr_H(\rho) $ and $ tr_K(\rho)$. Then all possible ranks of $\rho $ are determined in this paper.

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    Meromorphic Solutions of Finite Order to the Equation $f^{n}+f^{m}(z+c)=e^{Az+B}$
    Minfeng Chen,Zongsheng Gao,Zhibo Huang
    Acta mathematica scientia,Series A    2021, 41 (4): 913-920.  
    Abstract73)   HTML8)    PDF(pc) (303KB)(97)       Save

    In this paper, we study the meromorphic solutions of finite order to the difference equations $f^{n}(z)+f^{m}(z+c)=e^{Az+B}$ $(c\neq 0)$ over the complex plane ${\mathbb C}$ for integers $n, m$, and $A, B, c\in {\mathbb C}$.

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    Approximate Optimality Conditions and Mixed Type Duality for a Class of Non-Convex Optimization Problems
    Jiaolang Wang,Donghui Fang
    Acta mathematica scientia,Series A    2022, 42 (3): 651-660.  
    Abstract64)   HTML4)    PDF(pc) (298KB)(67)       Save

    By using the properties of the Fréchet subdifferentials, we first introduce a new constraint qualification and then establish some approximate optimality conditions for the non-convex constrained optimization problem with objective function and/or constraint function being α-convex function. Moreover, some results for the weak duality, strong duality and converse-like duality theorems between this non-convex optimization problem and its mixed type dual problem are also given.

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    Classification of Calabi Hypersurfaces in ${\mathbb R}^5$ with Parallel Fubini-Pick Form
    Ruiwei Xu,Miaoxin Lei
    Acta mathematica scientia,Series A    2022, 42 (2): 321-337.  
    Abstract61)   HTML2)    PDF(pc) (391KB)(83)       Save

    The classifications of locally strongly convex equiaffine hypersurfaces (centroaffine hypersurfaces) with parallel Fubini-Pick form with respect to the Levi-Civita connection of the Blaschke-Berwald metric (centroaffine metric) have been completed in the last decades. In [20], the authors studied Calabi hypersurfaces with parallel Fubini-Pick form with respect to the Levi-Civita connection of the Calabi metric and classified 2 and 3-dimensional cases. In this paper, we extend such calssification results to 4-dimensional Calabi hypersurfaces in the affine space ${\mathbb R}^5$.

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    The Two-Dimensional Steady Chaplygin Gas Flows Passing a Straight Wedge
    Jia Jia
    Acta mathematica scientia,Series A    2021, 41 (5): 1270-1282.  
    Abstract58)   HTML1)    PDF(pc) (349KB)(33)       Save

    The purpose of this paper is to investigate the two-dimensional steady supersonic chaplygin gas flows passing a straight wedge. By the definition of Radon measure solution, the accurate expressions are obtained for all cases where the Mach number is greater than 1. It is quite different from the polytropic gas, for the chaplygin gas flows passing problems, there exists a Mach number $ M^{\ast}_{0} $, when the Mach number of incoming flows is greater than or equal to $ M^{\ast}_{0} $, the quality will be concentrated on the surface of the straight wedge. At this time, there are not piecewise smooth solutions in the Lebesgue sense. The limit analysis is used to prove that the limit obtained by Lebesgue integral is consistent with the solution obtained in the sence of Radon measure solution.

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    Normal Family Theorems for Meromorphic Functions with Discrete Values of One Leaf
    Xiaojing Guo,Fujie Chai,Daochun Sun
    Acta mathematica scientia,Series A    2021, 41 (6): 1598-1605.  
    Abstract56)   HTML3)    PDF(pc) (313KB)(103)       Save

    In this paper, the normal theorems of meromorphic functions involving discrete values are studied by using the theory of Ahlfors covering surfaces. Firstly, the discrete values with one leaf of meromorphic functions are defined, then the inequalities about islands are investigated and two precise inequalities about islands are obtained. Finally, the inequalities are used to study the discrete values and the normal family of meromorphic functions, then a normal theorem involving a monophyletic island and a normal theorem involving discrete values of one leaf are obtained. All these theorems promote the famous Ahlfors' five islands theorem and five single valued theorem of Nevanlinna.

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    A p-Laplace Eigenvalue Problem with Coercive Potentials
    Jingran He,Helin Guo,Wenqing Wang
    Acta mathematica scientia,Series A    2021, 41 (5): 1323-1332.  
    Abstract56)   HTML1)    PDF(pc) (345KB)(41)       Save

    In this paper, we are concerned with the asymptotic behavior of solutions for a p-Laplace eigenvalue problem with coercive potentials. The bottom of the potential (The set of global minimum points of the potential) is an ellipsoid, we prove that the solutions of the problem will blow up at one of the endpoints of the major axis of the ellipsoid as the related parameter tends to a threshold value, and we also give the exact blow-up rate.

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    On the Observer Design for Fractional Singular Linear Systems
    Zaiyong Feng,Ning Chen,Yongpeng Tai,Zhengrong Xiang
    Acta mathematica scientia,Series A    2021, 41 (5): 1529-1544.  
    Abstract54)   HTML3)    PDF(pc) (607KB)(30)       Save

    The paper discussed the Caputo fractional derivatives of impulse function $\delta(t) $, i.e., ${^C_0}{D}_t^\alpha\delta(t) $ and its Laplace transform, the distributional solution of Fractional Singular Linear Systems (FSLS) was consequently obtained. Based on the distributional solution of the system, the asymptotic stability theorem for FSLS was given, and the existence theorem of state observer for FSLS was proved. A simplified design method of full state observer for FSLS was investigated and summarized by pole assignment only for the slow subsystem. Finally, a state observer as an example was designed to verify the effectiveness of the proposed method.

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    On a $p$-Adic Integral Operator Induced by a Homogeneous Kernel and Its Applications
    Jianjun Jin,Shuan Tang,Xiaogao Feng
    Acta mathematica scientia,Series A    2021, 41 (4): 968-977.  
    Abstract52)   HTML0)    PDF(pc) (343KB)(38)       Save

    In this paper, we introduce and study a $p$-adic integral operator induced by a homogeneous kernel of degree $-λ$ and obtain its sharp norm estimates. As applications, we establish some new $p$-adic inequalities with the best constant factors and their equivalent forms, which extend some known results in the literature.

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    Commutators of Weighted Composition Operators on Hardy Space of the Unit Ball
    Ning Xu,Zehua Zhou,Ying Ding
    Acta mathematica scientia,Series A    2021, 41 (6): 1606-1615.  
    Abstract48)   HTML0)    PDF(pc) (330KB)(61)       Save

    In this paper, we study commutators of weighted composition operators with linear fractional non-automorphisms on Hardy space of the unit ball. First, we obtain the formula of commutators of weighted composition operators. Then, we characterize compactness of commutators according to two special situations of linear fractional maps. Finally, we obtain that commutators are compact when linear fractional maps are parabolic and commutators are not compact when linear fractional maps are hyperbolic.

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    Generalized Transversality Theorem for Fredholm Operator in Global Analysis
    Qiang Li
    Acta mathematica scientia,Series A    2021, 41 (5): 1263-1269.  
    Abstract47)   HTML1)    PDF(pc) (820KB)(26)       Save

    Generalized transversality theorem for $ C^r $ mapping $ F(u, s):M\times S\rightarrow N $ is established in infinite dimensional Banach manifolds $ M, S, N $. If the mapping $ F(u, s) $ is generalized transversal to a single point set $ \{\hat{\theta}\} $, and $ f_s(u)=F(u, s) $ is a Fredholm operator in the sense of parameter s, then there exists a residual set $ \Sigma\subset S, $ such that $ f_s(u) $ are generalized transversal to $ \{\hat{\theta}\} $, for all $ s\in \Sigma. $

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    A Generalised Decic Freud-Type Weight
    Dan Wang,Mengkun Zhu,Chen Yang,Xiaoli Wang
    Acta mathematica scientia,Series A    2021, 41 (4): 921-935.  
    Abstract46)   HTML3)    PDF(pc) (559KB)(35)       Save

    In this paper, the authors focus on a generalised decic Freud-type weight function and study the properties of the orthogonal polynomials with respect to this weight. The difference-differential equations of their associated recurrence coefficients are derived; meanwhile, the authors also find the asymptotic behavior of recurrence coefficients via above mentioned equations. What's more, the authors discuss the Hankel determinant in regard to this weight as $n\rightarrow\infty$, and calculate the smallest eigenvalues of large Hankel matrices generated by this weight when $\alpha=t=0$.

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    A Class of Differential Operators with Eigenparameter Dependent Boundary Conditions
    Kang Sun,Yunlan Gao
    Acta mathematica scientia,Series A    2022, 42 (3): 661-670.  
    Abstract45)   HTML3)    PDF(pc) (355KB)(46)       Save

    In this paper, A class of third-order differential operators with transition conditions and two boundary conditions containing spectral parameters is studied, and the analytical method is used to do two aspects of work. First, by constructing a new space and a new operator, the eigenvalues of the problem and the operator are connected so that the eigenvalues of the original problem are consistent with the eigenvalues of the new operator. Second, the properties of the eigenvalues of the original problem are studied, and the conclusion that the spectrum of the original problem has only point spectrum is given.

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    Blow-Up Properties of Solutions to a Class of Parabolic Type Kirchhoff Equations
    Hui Yang,Yuzhu Han
    Acta mathematica scientia,Series A    2021, 41 (5): 1333-1346.  
    Abstract43)   HTML3)    PDF(pc) (405KB)(54)       Save

    In this paper, blow-up properties of solutions to an initial-boundary value problem for a parabolic type Kirchhoff equation are studied. The main results contain two parts. In the first part, we consider this problem with a general diffusion coefficient $M(\|\nabla u\|_2^2)$ and general nonlinearity $f(u)$. A new finite time blow-up criterion is established, and the upper and lower bounds for the blow-up time are also derived. In the second part, we deal with the case that $M(\|\nabla u\|_2^2)=a+b\|\nabla u\|_2^2$ and $f(u)=|u|^{q-1}u$, which was considered in[Computers and Mathematics with Applications, 2018, 75:3283-3297] with $q>3$, where global existence and finite time blow-up of solutions were obtained for subcritical, critical and supercritical initial energy. Their results are complemented in this paper in the sense that $q=3$ will be shown to be critical for the existence of finite time blow-up solutions to this problem.

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    Bifurcation of Limit Cycles from a Liénard System of Degree 4
    Hongying Zhu,Minzhi Wei,Sumin Yang,Caoqing Jiang
    Acta mathematica scientia,Series A    2021, 41 (4): 936-953.  
    Abstract43)   HTML10)    PDF(pc) (613KB)(165)       Save

    In this paper, we study the number of limit cycles by Poincaré bifurcation for some Liénard system of degree 4. We prove that the system can bifurcate at most 6 limit cycles from the periodic annulus, by the tools of regular chain theory in polynomial algebra and Chebyshev criteria, at least 3 limit cycles by asymptotic expansions of the related Abelian integral (first order Melnikov functions).

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    Global Stability of an Epidemic Model with Quarantine and Incomplete Treatment
    Lixiang Feng,Defen Wang
    Acta mathematica scientia,Series A    2021, 41 (4): 1235-1248.  
    Abstract43)   HTML2)    PDF(pc) (855KB)(124)       Save

    An epidemic model with quarantine and incomplete treatment is constructed. The model allows for the susceptibles to the unconscious and conscious susceptible compartment. We establish that the global dynamics are completely detremined by the basic reproduction number $R_{0}$. If $R_{0}≤1$, then the disease free equilibrium is globally asymptotically stable. If $R_{0}>1$, the endemic equilibrium is globally asymptotically stable. Some numerical simulations are also carried out to confirm the analytical results.

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