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2023, Vol. 43 No. 2   Published date: 28 June 2023
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• Progress in Spatio­temporal Dynamics of Vegetation Systems

  Zhang Hongtao, Sun Guiquan

  2023, 43(2): 1-15. doi: 10.3969/j.issn.1006­8074.2023.02.001

  Abstract ( 1262 )   PDF (509KB) ( 403 )   　　

  Vegetation pattern is one of the typical characteristics of ecosystems in arid and semi­arid areas, which can qualitatively describe the spatial distribution structure of vegetation, and can be used as an early indicator of ecosystem improvement and degradation. This paper devotes to summarize the bifurcation phenomena in vegetation system to reveal the formation mechanism of vegetation pattern and provide warning signals of desertification. Firstly, through the Hopf bifurcation theory, the conditions of spatial homogeneous Hopf bifurcation in vegetation model are qualitatively analyzed, and the phenomenon of interannual periodic fluctuation of vegetation biomass is explained. Secondly, the existing vegetation models are analyzed by the Turing bifurcation theory, the regular distribution of

  vegetation in space and the formation mechanism of pattern are revealed, and the types of these patterns are refined by applying the multiple scale analysis method, and the parameter threshold of the system undergoing pattern phase transition is found. Finally, when the Hopf bifurcation and Turing bifurcation occur at the same time, the system will undergo a Turing-­Hopf bifurcation. By means of the normal form theory of reaction­diffusion equation, the normal form of the Turing­-Hopf bifurcation is derived, and the amplitude equation is obtained by the cylindrical coordinate  transformation to analyze its dynamic behavior, and then more complex spatiotemporal patterns of vegetation are revealed.

• Neural Solution Operator of PDEs Based on Function­valued Reproducing Kernel Hilbert Space

  Bao Kaijun, Liu Ziyuan, Wang Haifeng, Qian Xu, Song Songhe

  2023, 43(2): 16-31. doi: 10.3969/j.issn.1006­8074.2023.02.002

  Abstract ( 1246 )   PDF (555KB) ( 213 )   　　

  By learning the mappings between infinite dimensional function spaces using carefully designed neural networks, the operator learning methodology–neural operator has exhibited significantly more efficient than traditional methods in solving complex problems such as differential equations. Toward this end, we incorporate the function­valued reproducing kernel Hilbert spaces (function­valued RKHS) and propose a novel neural operator–reproducing kernel neural operator (RKNO). Motivated by the recently successful operator learning

  methodology–deep operator network (DeepONet), RKNO is formulated by generalizing the Hilbert­Schmidt integral operator and the representer theorem. Numerical experiments on the Advection, KdV, Burgers’, and Poisson equations show that the RKNO allows for an expressive and efficient architecture, in contrast to DeepONet and other models. Futhermore, the RKNO possesses the property of discretization­independence, which can find the solution of a high­resolution input after learning from low­resolution data.

• Some New Classes of n­cycle Permutations over Finite Fields

  Zhang Zhilin, Yuan Pingzhi

  2023, 43(2): 32-47. doi: 10.3969/j.issn.1006­8074.2023.02.003

  Abstract ( 1195 )   PDF (176KB) ( 149 )   　　

  This paper presents some new classes of $n$-cycle permutations over finite fields. At first, we present a concise criterion for Dickson polynomials over finite fields being $n$-cycle permutations.

  Then, we give a necessary and sufficient condition for linearized polynomials over finite fields being involutions.

  Finally, some interesting new classes of $n$-cycle permutations are demonstrated by considering polynomials of different forms.

• T-CS Inverse and Partial Order of Tensors under T-Product

  2023, 43(2): 48-67. doi: 10.3969/j.issn.1006-8074.2023.02.004

  Abstract ( 1772 )   PDF (215KB) ( 150 )   　　

  The generalized inverse and partial order of tensor are important components of tensor theory.

  In this paper, we introduce the T-CS inverse of third-order tensor,

  obtain some characterizations and properties of it under the T-product,

  and apply it to

  introduce a new binary relation:

  the ${\textcircled{S}}$ order, which is equivalent to the T-star order under the set of i-EP tensors.

  Based on the ${\textcircled{S}}$ order, we further introduce the T-CS partial order and give some characterizations of it.

• Existence and Multiple Solutions of $p(x)$-Laplace Equation with two Singular Terms

  Hu Xincun, Chen Haibo

  2023, 43(2): 68-81. doi: 10.3969/j.issn.1006-8074.2023.02.005

  Abstract ( 1004 )   PDF (190KB) ( 206 )   　　

  In this paper, we study the existence and multiplicity of positive solutions for the following double singular problem with $p(x)$-Laplace operator

  \begin{equation*}

  \left\{

  \begin{array}{ll}

  -\Delta _{p(x)}u+V(x)|u|^{p(x)-2}u=\mu\frac{|u|^{s(x)-2}u}{|x|^{s(x)}}+\lambda h(x)u^{-\gamma(x)}&\quad \text{in}\quad \Omega,\\

  u=0&\quad \text{on}\quad \partial\Omega.

  \end{array}%

  \right.

  \end{equation*}

  Due to the presence of singular term $u^{-\gamma(x)}$ and singular potential $|x|^{-s(x)}$ in the equation, it is more difficult to deal with the existence of positive solutions. By using the decomposition of Nehari manifold and some refined estimates, we show that there admits at least two positive solutions for the double singular problem.

• Maps Preserving Generalized Entropy of Convex Combination of Density Operators on Infinite Dimensional Hilbert Spaces

  Jia Fengyang, Zhang Yanfang, He Kan

  2023, 43(2): 82-91. doi: 10.3969/j.issn.1006-8074.2023.02.006

  Abstract ( 1067 )   PDF (161KB) ( 161 )   　　

   Let $\mathcal H$ be an infinite dimensional complex Hilbert space, and $S(\mathcal H)$ the set of density operators on $\mathcal H$, i.e., the set of all positive and trace one bounded linear operators on $\mathcal H$. In this paper, we give a characterization of the maps preserving generalized entropy of convex combination of density operators on $S(\mathcal H)$.

• The Inverse Eigenvalue Problem for a Class of Quaternion Conjugate Symplectic Tensors

  Bai Rui, Huang Jingpin

  2023, 43(2): 92-106. doi: 10.3969/j.issn.1006-8074.2023.02.007

  Abstract ( 938 )   PDF (228KB) ( 179 )   　　

  This paper studies the inverse eigenvalue problem for a class of quaternion conjugate symplectic tensors under the Einstein product. Firstly, the properties and characteristic structures of conjugate symplectic tensors are obtained by using the transformation operator of quaternion tensors. Secondly, for the given ${I_1}{I_2} \cdots {I_N}$ characteristic pairs of quaternion tensors, a quaternion self-conjugated symplectic tensor $\mathcal{S}$ is found to include all the given characteristic pairs. As an application,

  we give a necessary and sufficient condition for the existence of conjugate symplectic tensor solutions and the expression of solutions to the quaternion tensor equation $\mathcal{B}\ast_{N}\mathcal{S} = \mathcal{D}$. The feasibility of the proposed method is showed with numerical examples.

• Precompact Sets in Variable Exponent Bounded Variation Spaces

  Si Yanan, Xu Jingshi

  2023, 43(2): 107-121. doi: 10.3969/j.issn.1006­8074.2023.02.008

  Abstract ( 1169 )   PDF (168KB) ( 148 )   　　

  By equivariated sets, we consider sufficient conditions for precompact sets in variable exponent Banach space-valued bounded Wiener variation spaces and bounded Riesz variation spaces in univariate and bivariate cases, respectively.

• Regulation Strategies for Competitive Environment

  Zhang Jingchuan, Liu Lu

  2023, 43(2): 122-130. doi: 10.3969/j.issn.1006-8074.2023.02.009

  Abstract ( 1004 )   PDF (210KB) ( 246 )   　　

  Regulating multiple companies in a competitive environment is much more complex than regulating a single company in a non-competitive environment. The number of companies not following regulations significantly impacts whether other companies stick to the rules. Using an effective regulation strategy is particularly important. The inspection mechanism is one of the most widely used mechanisms. This paper analyzes and compares the two fundamental inspection mechanisms for two competing companies: thorough checking and uniform checking. Under a thorough checking strategy, the regulator either checks all companies or does not check at all. Under a uniform checking strategy, the regulator decides to check each company independently. As usual, we adopt a game model to find the equilibrium under two inspection strategies and compare two inspection strategies from the perspective of the regulator. The result indicates that the more intensive the competition of companies is, the more suitable for the regulator to adopt the thorough checking strategy.