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2023, Vol. 43 No. 3   Published date: 28 September 2023
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• Gradient Estimate for Solutions of $\Delta v+v^r-v^s= 0$ on a Complete Riemannian Manifold

  Wang Youde, Zhang Aiqi

  2023, 43(3): 1-22. doi: 10.3969/j.issn.1006­8074.2023.03.001

  Abstract ( 908 )   PDF (213KB) ( 136 )   　　

  In this paper we consider the gradient estimates on the positive solutions to the elliptic equation $\Delta v+v^r-v^s= 0,$ defined on a complete Riemannian manifold $(M,\,g)$,

  where $r$ and $s$ are two real constants.

  When$(M,\,g)$ satisfies $Ric \geq -(n-1)\kappa$ (where $n\geq2$ is the dimension of $M$ and $\kappa$ is a nonnegative constant), we employ the Nash-Moser iteration technique to derive a Cheng-Yau type gradient estimate for the positive solutions to the above equation under some suitable geometric and analysis conditions.

  Moreover, it is shown that when the Ricci curvature of $M$ is nonnegative, this elliptic equation does not admit any positive solutions except for $v\equiv 1$ if\ $r<s$ and $1<r<\frac{n+3}{n-1}~\mbox{or}~1<s<\frac{n+3}{n-1}.$

• Variational Calculation and Gap Phenomena of Low Order Curvature Functional of Sub-manifolds

  Liu Jin

  2023, 43(3): 23-60. doi: 10.3969/j.issn.1006-8074.2023.03.002

  Abstract ( 1569 )   PDF (331KB) ( 136 )   　　

  Let $\varphi:M^{n}\to N^{n+p}$ be an $n$-dimensional compact without boundary sub-manifold in a general real ambient manifold. Its three important low order curvatures: the square length $S$ of second fundamental form, the square length $H^{2}$ of mean curvature, and the square length $\rho=S-nH^{2}$ of trace zero second fundamental form, respectively describe the geometric properties of totally geodesic, minimal, and totally umbilical. Let $F: [0,+\infty)\times [0,+\infty)\to \mathbb{R}$ be an abstract smooth bivariate function. In this paper, we construct two functionals ${\mathcal L}_{(I,n,F)}(\varphi)=\int_{M}F(S,H^{2}){\rm{d}}v$ and $ {\mathcal L}_{(II,n,F)}(\varphi)=\int_{M}F(\rho,H^{2}){\rm{d}}v$, which include some well-known functionals as special cases, measure how derivations $\varphi$ from totally geodesic, minimal, or totally umbilical sub-manifolds globally, and have a closed relation to the Willmore conjecture. For these functionals, we obtain the first variational equations, and construct a few examples of critical points in space forms. Moreover, we derive out some integral inequalities, and based on which classify the gap phenomenon.

• The Extremal $p$-spectral Radii of Trees, Unicyclic and Bicyclic Graphs with Given Number of Segments

  Qiu Mairong, He Xiaocong

  2023, 43(3): 61-80. doi: 10.3969/j.issn.1006-8074.2023.03.003

  Abstract ( 816 )   PDF (233KB) ( 150 )   　　

  Let $G$ be a finite and simple graph. A walk $S$ is called a segment of $G$ if the endpoints (not necessarily distinct) of $S$ are of degree 1 or at least 3, and each of the rest vertices is of degree 2 in $G$. In this paper, we determine the graphs that maximize the $p$-spectral radius for $p>1$ among trees, unicyclic and bicyclic graphs with given order and number of segments, respectively.

• A-Weyl's Theorem and the Property  $(WE)$ under Perturbations

  Che Yuhong, Dai Lei

  2023, 43(3): 81-94. doi: 10.3969/j.issn.1006-8074.2023.03.004

  Abstract ( 731 )   PDF (168KB) ( 152 )   　　

  In this paper we investigate the necessary and sufficient conditions such that both a-Weyl's theorem and the

  property $(WE)$ hold for a bounded linear operator, and study the stability of a-Weyl's theorem and

  the property $(WE)$ under quasi-nilpotent

  or compact perturbations. As an application, the relative stability of special operators is studied.

• High-accuracy Randomized Completion Algorithm for the Low Rank Toeplitz Tensor

  Wen Ruiping, Li Wenwei

  2023, 43(3): 95-110. doi: 10.3969/j.issn.1006-8074.2023.03.005

  Abstract ( 751 )   PDF (390KB) ( 235 )   　　

  In this paper, we consider the solution for the completion problem of low rank Toeplitz tensors , and based on the high-accuracy completion algorithm, present a high-accuracy completion algorithm with randomized technique, in which the n-mode tensor is stochastically unfolded and the corresponding singular value decomposition (SVD) is modified in each iteration. Moreover, the convergence of the new algorithm is established. The results of numerical experiments on the Toeplitz tensor and the Toeplitz mean tensor show that the new algorithm is significantly better than the high-accuracy completion algorithm for low rank Toeplitz tensors in terms of computational cost.

• Research on Dynamical Properties of Resilient System under Cyber Attack

  Wu Yichun, Yan Jingjing, Ding Ran, Tang Yilei

  2023, 43(3): 111-129. doi: 10.3969/j.issn.1006-8074.2023.03.006

  Abstract ( 762 )   PDF (460KB) ( 164 )   　　

  In order to improve the continuous support capability of the resilient system under complex cyber attacks, this paper first

  builds a class of cyber attack model based on the principle of virus epidemical dynamics to simulate the complex cyber threat environment in the resilient system,

  and deeply studies the dynamic characteristics of the model. Then, a variant target model of the resilient system under cyber attacks is constructed, and the local stability of the resilient system is analyzed by

  using the center manifold theorem and the bifurcation theory.

  Finally, with the Matlab software, the dynamic epidemical process of cyber attacks and the dynamic variant process of the target of the resilient system are displayed

  in some numerical portraits.