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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.  
    Abstract83)   HTML6)    PDF(pc) (326KB)(93)       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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    The Existence of the Measure Solution for the Non-Isentropic Chaplygin Gas
    Yufeng Chen,Tingting Chen,Zhen Wang
    Acta mathematica scientia,Series A    2020, 40 (4): 833-841.  
    Abstract77)   HTML4)    PDF(pc) (339KB)(74)       Save

    In this paper, we consider the Riemann problem of one-dimensional non-isentropic Chaplygin gas dynamics. In the case that the pressure and internal energy are general, we construct the classical solution by the characteristic analysis under a sufficient and necessary condition on the Riemann data. As the density ρ concentrates, the δ shock waves exist. According to the theory of Radon measure, the general Rankine-Hugoniot is induced. Combining with entropy condition, we obtain the measure solution for this problem. This extends the result for the isentropic Chaplygin gas.

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    Classical Solutions for a Kind of New Kirchhoff-Type Problems Without Boundary Constraint
    Yue Wang,Hongmin Suo,Wei Wei
    Acta mathematica scientia,Series A    2020, 40 (4): 857-868.  
    Abstract70)   HTML2)    PDF(pc) (391KB)(48)       Save

    The existence of classical solutions for a class of new Kirchhoff-type problems with unadulterated exponential item at right are considered on boundary cuboid in this article, and all results on the theoretical basis are based on constructors of functions. We show that the exact expressions of classical solutions with all exponents except the minus one by using the techniques of analysis associated with it. At the same time, we give some examples to explaining and verifying our conclusion.

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    Multiple Solutions for Nonlinear Equations Related to Kirchhoff Type Equations
    Wencui Liang,Zhengjie Zhang
    Acta mathematica scientia,Series A    2020, 40 (4): 842-849.  
    Abstract49)   HTML3)    PDF(pc) (302KB)(73)       Save

    In this paper, we will discuss the following Kirchhoff equation
    $\left\{\begin{array}{ll}-\left(a+b\int_{\mathbb{R}^{3}}{|\nabla u{{|}^{2}}}\right)\triangle u+u=\left(1+\varepsilon g(x)\right) u^{p}, x\in\mathbb{R}^{3}, \\u\in H^{1}\left(\mathbb{R}^{3}\right), \end{array}\right. $
    where $\varepsilon$, $a$, $b$ are positive constants, $1< p<5,g(x)\in L^{\infty}\left(\mathbb{R}^{3}\right)$. When $g(x)$ satisfy some conditions, we use perturbation method prove that there exists a $\varepsilon_{0}$, if $0<\varepsilon <\varepsilon_{0}$ there are many solutions for above problem.

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    Behaviour of Meromorphic Solutions of Complex Functional-Differential Equations
    Manli Liu,Lingyun Gao
    Acta mathematica scientia,Series A    2020, 40 (5): 1121-1131.  
    Abstract48)   HTML1)    PDF(pc) (359KB)(70)       Save

    The aim of this paper is twofold. Firstly, we consider the existence of solutions to a type of complex functional-differential equations

    (w')nw(n)=awn+1(g)+bw+d

    in complex variables. We obtain g is linear when w is a transcendental meromorphic function and a≠0, b, d are constants. In addition, due to the different properties between equations and system of equations, it is meaningful to research systems of equations, this paper is also concerned with a type of system of functional equations, properties of meromorphic solutions are obtained under some proper conditions. Examples are constructed to show that our results are accurate.

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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.  
    Abstract47)   HTML2)    PDF(pc) (490KB)(80)       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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    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.  
    Abstract45)   HTML2)    PDF(pc) (278KB)(55)       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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    Multiple Pertubations to a Quasilinear Schrödinger Equation
    Xiaoli Han
    Acta mathematica scientia,Series A    2020, 40 (4): 869-881.  
    Abstract41)   HTML3)    PDF(pc) (292KB)(37)       Save

    In this paper, we will deal with the Cauchy problem of a class of quasilinear Schrödinger equations. The main goal is to obtain some sufficient conditions on the blow up in finite time and global existence of the solution.

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    On Approximation by Bernstein-Durrmeyer-Type Operators in Movable Compact Disks
    Zhaojun Pang,Dansheng Yu,Ping Zhou
    Acta mathematica scientia,Series A    2020, 40 (3): 545-555.  
    Abstract38)   HTML0)    PDF(pc) (294KB)(72)       Save

    To approximate analytic functions in movable compact disks, we introduce a new kind of Bernstein-Durrmeyer-Type polynomials. The order of simultaneous approximation rate of the new polynomials in the movable compact disks is given.

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    The Approximation and Growth of Entire Function Represented by Laplace-Stieltjes Transform with Infinite Order
    Hongyan Xu,Sanyang Liu
    Acta mathematica scientia,Series A    2020, 40 (3): 556-568.  
    Abstract36)   HTML0)    PDF(pc) (368KB)(35)       Save

    The main purpose of this article is to investigate the growth and approximation of Laplace-Stieltjes transform with irregular growth converges in the whole plane, by introducing the concept of the double lower q-type. We obtain some relation theorems concerning the double lower q-type, the error, An* and λn, which are extension and improvement of the previous theorems given by Luo-Kong, Singhal-Srivastava.

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    The Problem of the Radii of a Harmonic Linear Differential Operator
    Zhenyong Hu,Qihan Wang,Boyong Long
    Acta mathematica scientia,Series A    2020, 40 (5): 1163-1174.  
    Abstract34)   HTML0)    PDF(pc) (309KB)(28)       Save

    For harmonic mappings $ f_{i}(z)=h_{i}(z)+\overline{g_{i}(z)}$($ i=1, 2$) defined in the unit disk satisfying the given coefficient conditions, we consider the radii of full convexity and full starlikeness of order $\alpha $ for the convex combination $ (1-t)L^{\epsilon}_{f_{1}}+tL^{\epsilon}_{f_{2}}$, where $ L^{\epsilon}_{f_{i}}=z\frac{\partial f_{i}}{\partial z}-\epsilon\overline{z}\frac{\partial f_{i}}{\partial\overline{z}}(|\epsilon|=1)$ denotes the differential operator of $ f_{i}$. In addition, we obtain the radii of fully convex and full starlikeness of order $\alpha $ for convolution of harmonic mappings under the differential operator. All results are sharp.

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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.  
    Abstract34)   HTML5)    PDF(pc) (359KB)(50)       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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    Existence of Ground States for a Class of Modified Gross-Pitaevskii Equations
    Xiaomeng Huang,Yimin Zhang
    Acta mathematica scientia,Series A    2020, 40 (4): 850-856.  
    Abstract34)   HTML1)    PDF(pc) (340KB)(35)       Save

    In this paper, using scaling technique and some rearrangement inequalities, existence and classification of ground states for a class of Modified Gross-Pitaevskii equations with respect to the nonlinear exponent p. If $2< p < 2+\frac{4}{N}$ , for any $c>0$ , there is at least a minimizer for this problem. If $p=2+\frac{4}{N}$ and $c\leq\|\phi\|_2$ or $c>\left(\frac{3}{2}\right)^{\frac{N}{4}}\|\phi\|_2$ (the definition of $\|\phi\|_2$ see section 1) or $p>2+\frac{4}{N}$ , there is no minimizer for this problem. But it is unclear if $p=2+\frac{4}{N}$ and $\|\phi\|_2<c<\left(\frac{3}{2}\right)^{\frac{N}{4}}\|\phi\|_2$.

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    An Inverse Initial Value Problem for Degenerate Parabolic Equations
    Liu Yang,Zuicha Deng
    Acta mathematica scientia,Series A    2020, 40 (4): 891-903.  
    Abstract33)   HTML0)    PDF(pc) (2054KB)(33)       Save

    This paper investigates an inverse problem of reconstructing the initial value in a degenerate parabolic equation. Problems of this type have important applications in several fields of applied science. The key to numerically solve such problem is to construct highorder difference schemes for corresponding forward problem. However, the dumping point method which is widely-used for numerically solving classical heat conduction equations cannot be applied to degenerate parabolic equations, because the principal coefficients are zero on degenerate boundaries. In this paper, a new but quite simple technique is proposed to construct a difference scheme of second order accuracy, and the stability and convergence of the scheme are proved. In order to accelerate the convergence rate, the conjugate gradient method is adopted to obtain numerical solutions of the inverse problem. Numerical verification on the efficiency and accuracy of the proposed algorithm is also performed.

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    Global Regularity for the 3D Liquid Crystal Equations with Fractional Diffusion
    Qiang Li
    Acta mathematica scientia,Series A    2020, 40 (4): 918-924.  
    Abstract33)   HTML1)    PDF(pc) (280KB)(21)       Save

    In this paper, the focus is the global regularity of three-dimensional liquid crystal equations with fractional dissipations $(-\Delta)^{\alpha}u$ and $(-\Delta)^{\beta}d$. The objective is to establish the global regularity of the fractional liquid crystal equations with the minimal amount of dissipations. And it is obtained that the equations have a global classical solution with sufficiently smooth data if $\alpha\geq\frac{5}{4}$ and $\beta\geq\frac{5}{4}$.

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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.  
    Abstract32)   HTML1)    PDF(pc) (311KB)(43)       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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    Disjointness of Generalized Frames
    Wei Zhang,Yunzhang Li
    Acta mathematica scientia,Series A    2020, 40 (3): 589-596.  
    Abstract31)   HTML0)    PDF(pc) (304KB)(17)       Save

    The notion of disjointness of frames in Hilbert spaces was firstly introduced by Han and Larson, it is closely related with super frames in Hilbert spaces, and plays an important role in construction of super frames and frames. G-frames is a generalization of frames in Hilbert spaces. In this paper, we establish characterization of disjointness of g-frames, strong disjointness of g-frames and weak disjointness of g-frames in terms of super g-frames; With the results obtained, we give different proof method of the known theorem; We obtain the relation between strongly disjoint and weakly disjoint of dual g-frames; Finally, we use strong disjointness of g-frames to construct (super) dual g-frames, which cover the results obtained by other authors.

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    Existence and Uniqueness of Positive Solutions to an Unstirred Chemostat with Toxins
    Haixia Li
    Acta mathematica scientia,Series A    2020, 40 (5): 1175-1185.  
    Abstract29)   HTML0)    PDF(pc) (392KB)(38)       Save

    A food chain model in the unstirred chemostat with toxins is studied. The stability of the trivial solution and semi-trivial solution is analyzed by means of the stability theory, and a priori estimate of positive solution is given by the maximum principle and the super and sub-solution method. Then, by using the fixed point index theory, the sufficient conditions for the existence of positive solutions are achieved. Finally, the effect of the toxins on the dynamic behavior is discussed by virtue of the perturbation theory and bifurcation theory, and the stability and uniqueness of positive solutions are obtained. The results show that the species can coexist when the growth rates of the microorganisms u and v are larger in the presence of the toxins. Furthermore, if the effect of the toxins is sufficiently large, the system has unique stable positive solution when the growth rate of the microorganism v belongs to a certain range.

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    The Non-Existence of Solutions of a Certain type of Nonlinear Complex Differential-Difference Equations
    Shuqing Lin,Junfan Chen
    Acta mathematica scientia,Series A    2021, 41 (1): 69-80.  
    Abstract28)   HTML2)    PDF(pc) (334KB)(44)       Save

    In this paper, we study transcendental entire solutions of a certain type of complex differential-difference equations where $P(z) $ and $Q(z) $ are non-zero polynomials, $\alpha(z) $ is polynomial, $ m$ and $n $ are positive integers, $ \eta\in{\Bbb C}\setminus\{0\}$. Several sufficient conditions on the non-existence of transcendental entire solutions of such equations are supplied.

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    The Novel (2+1)-Dimensional Supersymmetric Integrable Equations
    Fang Chen,Zeyu Sun,Minru Chen,Zhaowen Yan
    Acta mathematica scientia,Series A    2020, 40 (5): 1132-1141.  
    Abstract28)   HTML1)    PDF(pc) (294KB)(21)       Save

    Base on the super Lie algebra osp(2/2), we construct the (2+1)-dimensional supersymmetric integrable equations by means of two approaches. One of the technique is in terms of homogeneous spaces of super-Lie algebra, and in the other one, extending the dimension of the system has been used. Moreover, we derive the Bäcklund transformations for the (2+1)-dimensional supersymmetric integrable equations.

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