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    Inverse Spectral Problem for the Diffusion Operator from a Particular Set of Eigenvalues
    Qing Cao,Xiaochuan Xu
    Acta mathematica scientia,Series A    2021, 41 (3): 577-582.  
    Abstract97)   HTML13)    PDF(pc) (316KB)(116)       Save

    In this paper, we study the inverse spectral problem for the diffusion operator on a finite interval with the Robin-Dirichlet boundary conditions, and prove that a particular set of eigenvalues can uniquely determine the diffusion operator, and give the reconstruction algorithm.

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    Properties and Applications of the Core Inverse of an Even-Order Tensor
    Hongxing Wang,Xiaoyan Zhang
    Acta mathematica scientia,Series A    2021, 41 (1): 1-14.  
    Abstract89)   HTML6)    PDF(pc) (326KB)(107)       Save

    Tensor generalized inverse is one of the important contents of tensor theory research. In this paper, based on the research of tensor generalized inverse in recent years, we obtain some properties of the core inverse of tensor with the Einstein product, a tensor partial ordering based on the core inverse and the least-squares solution of $ {{\cal A}} {*}{{\cal X}}={{\cal B}}$ under condition ${{\cal X}}\in{\Bbb {\cal R}}({{\cal A}}) $.

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    Bonnesen-Style Inequalities for Star Bodies
    Zengle Zhang
    Acta mathematica scientia,Series A    2021, 41 (5): 1249-1262.  
    Abstract84)   HTML14)    PDF(pc) (351KB)(161)       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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    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.  
    Abstract66)   HTML8)    PDF(pc) (303KB)(84)       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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    Shape Optimization for p-Torsional Rigidity Problems
    Qihua Ruan
    Acta mathematica scientia,Series A    2021, 41 (6): 1625-1633.  
    Abstract62)   HTML0)    PDF(pc) (301KB)(90)       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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    Second Main Theorem for Algebraic Curves on Compact Riemann Surfaces
    Lizhen Duan,Hongzhe Cao
    Acta mathematica scientia,Series A    2021, 41 (6): 1585-1597.  
    Abstract61)   HTML9)    PDF(pc) (357KB)(109)       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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    A Partial Inverse Problem for the Sturm-Liouville Operator on Quantum Graphs with a Loop
    Shengyu Guan,Dongjie Wu,Sat Murat,Chuanfu Yang
    Acta mathematica scientia,Series A    2021, 41 (2): 289-295.  
    Abstract59)   HTML3)    PDF(pc) (278KB)(65)       Save

    This deals with the Sturm-Liouville operator on the quantum graphs with a loop. Given the potential on a part of edges, we try to recover the remaining potential from the subspectrum. The uniqueness theorem and a constructive algorithm for the solution of this partial inverse problem are provided.

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    Construction of the Planar Bodies with Constant Width
    Deyan Zhang,Botao Duan
    Acta mathematica scientia,Series A    2021, 41 (1): 15-28.  
    Abstract56)   HTML2)    PDF(pc) (490KB)(93)       Save

    Firstly, a class of planar curves "lever wheel" and their arm functions are defined, and the parameter representation of the lever wheel is established in this paper. Secondly, it is shown that the lever wheel is an equivalent characterization of the constant width curve. Finally, it is proven that the Reuleaux polygons are a class of lever wheels with piecewise constant arm functions, and Reuleaux polygons with even edges are constructed.

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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.  
    Abstract46)   HTML3)    PDF(pc) (607KB)(18)       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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    Boundness of Riesz Transforms on Hardy Spaces Associated with Schrödinger Operators on the Heisenberg Group
    Xuan Chen
    Acta mathematica scientia,Series A    2021, 41 (1): 46-62.  
    Abstract44)   HTML5)    PDF(pc) (359KB)(62)       Save

    Let $L=-\Delta_{{\Bbb H}^{n}}+V $ be a Schrödinger operator on the Heisenberg group ${\Bbb H}^{n} $, where $V $ is a nonnegative potential belonging to the reverse Hölder class. By the molecular decomposition of the Hardy space $ H_{L}^{p}({\Bbb H}^{n})$, we obtain the $ H^p_L$-boundedness of the Riesz transform associated with $L $.

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    Stabilities of K- Frames and Tight K- Frames Under the Operator Perturbation
    Dandan Du,Yucan Zhu
    Acta mathematica scientia,Series A    2021, 41 (1): 29-38.  
    Abstract43)   HTML1)    PDF(pc) (311KB)(48)       Save

    In this paper, we discuss the stabilities of K-frames and tight K-frames under the operator perturbation. Firstly, we provide an equivalent characterization of the operator perturbation for a K-frame by using a bounded linear operator $T $ from ${{\cal H}_1} $ to ${{\cal H}_2} $. We also give a simple way to construct new K-frames from two existing Bessel sequences. Meanwhile, we make a discussion on the construction for K-frames from given ones. In the end, we obtain a necessary and sufficient condition to generate tight K-frames from two old Bessel sequences. Our results generalize and improve the remarkable results which had been obtained by Casazza and Christensen.

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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.  
    Abstract41)   HTML1)    PDF(pc) (349KB)(20)       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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    Generalized Transversality Theorem for Fredholm Operator in Global Analysis
    Qiang Li
    Acta mathematica scientia,Series A    2021, 41 (5): 1263-1269.  
    Abstract39)   HTML1)    PDF(pc) (820KB)(18)       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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    Approximation Properties of a New Bernstein-Bézier Operators with Parameters
    Qiulan Qi,Dandan Guo
    Acta mathematica scientia,Series A    2021, 41 (3): 583-594.  
    Abstract39)   HTML2)    PDF(pc) (309KB)(93)       Save

    In this paper, a new generalized Bernstein-Bézier type operators is constructed. The estimates of the moments of these operators are investigated. The rate of convergence in terms of modulus of continuity is given. Then, the equivalent theorem of these operators is studied.

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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.  
    Abstract39)   HTML2)    PDF(pc) (313KB)(44)       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.  
    Abstract38)   HTML1)    PDF(pc) (345KB)(33)       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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    A Generalised Decic Freud-Type Weight
    Dan Wang,Mengkun Zhu,Chen Yang,Xiaoli Wang
    Acta mathematica scientia,Series A    2021, 41 (4): 921-935.  
    Abstract38)   HTML3)    PDF(pc) (559KB)(29)       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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    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.  
    Abstract37)   HTML0)    PDF(pc) (343KB)(33)       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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    Nonlinear Helmholtz Equation Involving Multiple Dirac Masses in $\mathbb{R}^2$
    Yong Ma,Huyuan Chen
    Acta mathematica scientia,Series A    2021, 41 (3): 652-665.  
    Abstract36)   HTML1)    PDF(pc) (405KB)(22)       Save

    Our purpose of this paper is to study weak solutions of nonlinear Helmholtz equation $-\Delta u-u=Q|u{|^{p-2}}u+\sum\limits_{i=1}^N{{k_i}}{\delta_{{A_i}}}$ where $p>1$, $k_i\in$$\mathbb{R}$\{0} with i=1, …, N, Q: $\mathbb{R}^2$→[0, +∞) is a Hölder continuous function and ${\delta_{{A_i}}}$ is the Dirac mass concentrated at ${{A_i}}$.We obtain two solutions of (0.1) if $k=\sum\limits_{i=1}^N{|{k_i}}| < {k^*}$ for some $k^*$>0 when $Q$ decays as $|x|^{α}$ at infinity with $α ≤ $ 0 and $p$>max{2, 3(2+$α$)}. These two sequences of solutions of (0.1) are sign-changing real-valued solutions with isotropic singularity at ${{A_i}}$ by applying Mountain Pass Theorem to an related integral equation. By using the iteration technique, we obtain the decays of solution of (0.1) controlled by $|x|^{-\frac12}$ at infinity when $p>\max\{2, 4(2+\alpha)\}$.

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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.  
    Abstract34)   HTML10)    PDF(pc) (613KB)(132)       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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