Chinese Quarterly Journal of Mathematics

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30 June 2026, Volume 41 Issue 2

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Dynamical Mechanisms and Energy Conversion in Rayleigh-Bénard Convection

WANG He-yuan, HE Xiang-hong, YANG Guang

2026, 41(2): 111-127.

doi:10.13371/j.cnki.chin.q.j.m.2026.02.001

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Numerous studies have addressed the stability of fluid between two thermally conducting plates (frequently abbreviated as Rayleigh-B´enard convection). The Lorenz system serves as the classical model of Rayleigh-B´enard convection problems and provides a paradigm for the laminar-to-turbulent transition. In this paper we study the dynamical mechanism and energy conversion of the Lorenz equation. The Lorenz chaotic system is transformed into a Kolmogorov-type system, which is decomposed into four types of torques: inertial torque, internal torque, dissipation torque and external torque. By combining different torques, the key factors for the generation of chaos in the Lorenz
system – the mathematical model corresponding to the Rayleigh-B´enard convection problem – have been studied. We further investigate the conversion among Hamiltonian, kinetic and potential energies, as well as the correlation between the energies and the Reynolds number. It is concluded that the combination of all four torques is necessary
to produce chaos, and the system can produce chaos only when the dissipative torques match the driving (external) torques. Any combination of three types of torques cannot produce chaos. The external torque, driven by heat from the bottom plate, supplies energy and that leads to the production of roll vortices and chaos. Moreover, we introduce
the Casimir function to analyze the system dynamics, and use its derivative to formulate the energy conversion. The bound of the chaotic attractor is obtained by the Casimir function and Lagrange multiplier. It is found that the Casimir function reflects the energy conversion and the distance between the orbit and the equilibria.

Inequalities for Lower Order Eigenvalues of Fourth-Order Elliptic System of Differential Equations

WANG Lin-lin, SUN He-jun, XIAO Meng-ge

2026, 41(2): 128-138.

doi:10.13371/j.cnki.chin.q.j.m.2026.02.002

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In this paper, we investigate the Dirichlet eigenvalue problem of fourth-order elliptic system of differential equations on an n-dimensional Euclidean space as follows
\begin{equation*}
\left\{\begin{aligned}
&A\Delta^2\boldsymbol{u} = -\Gamma\Delta \boldsymbol{u}, && \text{in} \quad \Omega, \\
&\boldsymbol{u} = \frac{\partial \boldsymbol{u}}{\partial \boldsymbol{\nu}} = \boldsymbol{0}, && \text{on} \quad \partial \Omega,
\end{aligned}\right.
\end{equation*}
where A is a symmetric coefficient matrix and ν is the outward unit normal vector field of ∂Ω. We derive some inequalities for lower order eigenvalues of this problem. Our results cover some previous results for the buckling problem.

Rota-Baxter Operators on pre-Lie Coalgebras

CHENG Yong-sheng, CUI Ya-xin, DING Wen-jing, WU Lin-li

2026, 41(2): 139-146.

doi:10.13371/j.cnki.chin.q.j.m.2026.02.003

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In the present paper, we mainly study Rota-Baxter operators (of weight zero) on pre-Lie coalgebras. In particular, the invertible Rota-Baxter operators on pre-Lie coalgebras is well studied, and some fundamental results of the invertible Rota-Baxter relations and derivations on pre-Lie coalgebras are given. Furthermore, using the Rota-Baxter operators on pre-Lie coalgebras, some new pre-Lie coalgebraic structures are obtained.

Ground State Normalized Solution for Nonhomogeneous Schrödinger-Poisson-Slater Equations with Potential

QIU Shuang-shuang, XU Li-ping

2026, 41(2): 147-161.

doi:10.13371/j.cnki.chin.q.j.m.2026.02.004

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In this paper, we primarily investigate the existence of ground state normalized solutions to the following nonhomogeneous Schrödinger-Poisson-Slater equation......

Higher-Order Expansions of Powered Order Statistics of Maxwell Sequences

HUANG Jian-wen, LIU Xin-ling, JIA Jin-ping, WANG Run-ke

2026, 41(2): 162-173.

doi:10.13371/j.cnki.chin.q.j.m.2026.02.005

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Under the optimal norming constants, this paper studies the higher-order expansions of the distributions and densities of the powered order statistics of Maxwell sequence. As auxiliary results, the corresponding convergence rates are obtained. The results show that the convergence rates of distributions and densities of normalized power order statistics are related to power index in principle. Finally, we compared the accuracy of each approximations with its true values through numerical experiments.

2-Edge Connected Triangle-Free Supereulerian Graphs

LV Sheng-mei, MA Xing-zhong, ZHOU Dan

2026, 41(2): 174-183.

doi:10.13371/j.cnki.chin.q.j.m.2026.02.006

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In 2-connected Hamiltonian claw-free graphs involving degree sum of adjacent vertices [Discuss. Math. Graph Theory 40(2020) 85-106 [24]], Tian and Xiong proved that 2-edge connected triangle-free graph of order at most 7 is either supereulerian or one of G₁. Furthermore, in Edge degree conditions for dominating and spanning closed trails
[Discuss. Math. Graph Theory 44(2024) 363-381 [23]], Tian, Broersma and Xiong proved that 2-edge connected graph of order at most 8 is either supereulerian or one of G₁∪G₂. Motivated by the advance above, we considered 2-edge connected graph G of order at most 9, and show that G is either supereulerian or one of G₁∪G₂∪G₃. Although only one vertex is added, forbidden graphs are more complex and harder to derived.

New Bounds of the Infinity Norm of the Inverse of GS-SDD Matrices with Their Applications

DUAN Wei-ting, WEN De-kun, HUI Min, WANG Ya-qiang

2026, 41(2): 184-196.

doi:10.13371/j.cnki.chin.q.j.m.2026.02.007

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In this paper, we present a new bound of the infinity norm of inverse of GS-SDD matrices. Meanwhile, a new error bound for the linear complementarity problem (LCP) of GS-SDD matrices is given, which depends only on the entries of the involved matrices. In addition, some numerical examples are given to illustrate the corresponding results.

Fault Diagnosis Based on BA-SVM Trapezoidal Region Pole Classification

DU Zi-wei, YAO Bo, WANG Fu-zhong

2026, 41(2): 197-206.

doi:10.13371/j.cnki.chin.q.j.m.2026.02.008

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For a class of linear constant systems, the problem of continuous gain type fault diagnosis and reliable control of a single component of an actuator is investigated based on the trapezoidal region. Firstly, in order to solve the problem that the pole information of closed-loop system is difficult to observe, a design scheme of full-dimensional state observer is given to realize the real-time observation of pole information and form a pole classification database for system failure. Secondly, according to the characteristics that the poles are located in different areas when different channels have faults, support vector machine is applied to design a pole classifier to diagnose faults in the system and
achieve accurate and reliable control of the system based on the fault diagnosis results. Then, in order to solve the problem of difficulty in selecting parameters for support vector machines, the bat algorithm (BA) is proposed to achieve automatic parameter optimization, which has the advantages of strong robustness and easy to combine with
other methods.

New Weighted Estimates for Commutators of θ-Type Calderόn-Zygmund Operator

XUE Feng-yu

2026, 41(2): 207-220.

doi:10.13371/j.cnki.chin.q.j.m.2026.02.009

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In this paper, we investigate the boundedness of commutators generated by the θ-Calderόn-Zygmund operator with symbol function belonging to two kinds of weighted Campanato spaces. Specifically, we show that the commutators of the θ-type Calderόn-Zygmund operator are bounded from weighted Lebesgue spaces to weighted Campanato spaces when symbol function belongs to weighted Lipschitz spaces; we also show that the commutators are bounded from weighted Morrey spaces to weighted Campanato spaces when symbol function belongs to a certain weighted Campanato space.