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2025, Vol. 45 No. 4   Published date: 28 December 2025
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• Remark on a Symmetric zeta Function

  LI Jiangtao

  2025, 45(4): 1-27. doi: 10.3969/j.issn.1006-8074.2025.04.001

  Abstract ( 421 )   PDF (225KB) ( 39 )   　　

  In this paper we introduce a symmetric zeta function, prove that it can be analytically continued to a meromorphic function on $\mathbb{C}^3$ with only simple poles on certain special hyperplanes, and calculate the multiple residue values at particular singular points. For a divergent multiple series that can be regarded as the value of the symmetric zeta function at the point $(1,1,1)$, we conduct a detailed analysis on its growth behavior and establish a connection with the classical Euler constant.

• Weak Mean Attractor of Stochastic Reaction-Diffusion Equation with Superlinear Multiplicative Noise and Strong Dissipativity Supercritical Nonlinearities

  QI Ailing, JU Xuewei, TONG Xiaoting

  2025, 45(4): 28-49. doi: 10.3969/j.issn.1006-8074.2025.04.002

  Abstract ( 430 )   PDF (240KB) ( 25 )   　　

   This paper investigates global solutions and long-time dynamics for the stochastic reaction-diffusion equation $\mathrm{d}u = \left(\Delta u + f(u) + g(x,t)\right)\mathrm{d}t + \sigma(u)\mathrm{d}W$ on a bounded domain, where the drift term $f(u)$, with polynomial growth rate $\beta$, is strongly dissipative and the diffusion term $\sigma(u)$ has growth rate $\gamma$, satisfying $\beta + 1 > 2\gamma$. Under this condition, we establish the existence, uniqueness, and regularity of solutions in Bochner spaces. Our analysis relies only on weak monotonicity conditions and requires no further growth restrictions on $f$ and $\sig$. Moreover, we prove the existence of a weak mean random attractor for the system. These results offer new insights into the balance mechanism between stochastic perturbations and dissipative effects in superlinear regimes.

• Inverse Problem of the Invariant $D_2(C_2^r)$ of Elementary Abelian 2-Groups

  ZHAO Kaiwen, LUO Caidian

  2025, 45(4): 50-59. doi: 10.3969/j.issn.1006-8074.2025.04.003

  Abstract ( 500 )   PDF (167KB) ( 18 )   　　

  Let $G$ be a finite abelian group and $k$ be a positive integer. The Davenport constant is a central invariant in zero-sum thoery. The invariant $D_k(G)$ generalizes the Davenport constant $D(G)$ and is defined as the maximum length $l$ such that there exists a sequence $B$ of length $l$ over $G$ containing $k$ disjoint non-empty zero-sum subsequences. This paper studies the inverse problem associated with this invariant for the elementary abelian 2-groups $C_2^r$. For $r \in [2,4]$, we characterize the structures of zero-sum sequences of length $ D_2(C_2^r)$ and $D_2(C_2^r) - 1$ in $C_2^r$ that can be decomposed into at most two minimal zero-sum subsequences. For $r \in [2,5]$, we characterize the structures of sequences of length $D_2(C_2^r) - 1$.

• Asymptotic Stability of RLC Systems with Fractional-order and Time Delay

  HAO Yuliang , GUO Yu, LIU Yicheng

  2025, 45(4): 60-72. doi: 10.3969/j.issn.1006-8074.2025.04.004

  Abstract ( 462 )   PDF (918KB) ( 20 )   　　

  This paper presents a systematic study on the modeling and stability analysis of fractional-order cascaded RLC networks with time delays. A generalized model of an $n$-stage cascaded RLC network with time delays is developed using the Caputo fractional derivative. The corresponding fractional-order differential equations are derived for both single-stage ($n=1$) and two-stage ($n=2$) configurations. The transcendental characteristic equation of the system is obtained via Laplace transform. By applying the Matignon stability criterion, asymptotic stability conditions are established for systems with and without time delays. It is shown that stability in the delay-free case depends mainly on the fractional order $\alpha$, whereas in the presence of time delays, stability is independent of $\alpha$ and instead governed by the delay parameter $\tau$. Notably, the critical delay threshold $\tau_{\mathrm{max}}$ for system stability is derived analytically. A detailed numerical study (Table I) further elucidates the effects of key parameters, including the resistance $R$, inductance $L$, capacitance $C$, fractional order $\alpha$, and time delay $\tau$ on the stability behavior.

  This study provides a theoretical basis and practical design guidelines for tuning parameters to ensure stability in fractional-order circuits with time delays.

• New Criteria for Boundedness in Terms of Two Measures for Impulsive Differential Equations

  XIA Zhinan

  2025, 45(4): 73-86. doi: 10.3969/j.issn.1006-8074.2025.04.005

  Abstract ( 442 )   PDF (180KB) ( 22 )   　　

  In this paper, we utilize the theory of Kurzweil-Henstock integrals to investigate new criteria

  for boundedness in terms of two measures for generalized ordinary differential equations.

  As applications, we establish criteria for $(h_{0}, h)$-uniform boundedness

  and $(h_{0}, h)$-uniform ultimate boundedness in terms of two measures for impulsive differential equations.

• A Modified Hybrid Three-term Conjugate Gradient Projection Method for Constrained Nonlinear Monotone Equations

  CHENG Mengfan, WANG Qi, WANG Haijun, LIU Jia

  2025, 45(4): 87-106. doi: 10.3969/j.issn.1006-8074.2025.04.006

  Abstract ( 413 )   PDF (627KB) ( 16 )   　　

  This paper presents a modified hybrid three-term conjugate gradient projection method (MHTTCGPM) for solving large-scale nonlinear monotone equations with convex set constraints. The method incorporates an adaptive line search technique, ensuring that the search direction satisfies the sufficient descent property. Without requiring Lipschitz continuity, the global convergence of the proposed method is rigorously established. Numerical results demonstrate the effectiveness and reliability of the new algorithm.

• Single-Image-Based Precise Camera Calibration for Autofocus Lenses: A Novel Flexible Approach

  YI Xuejun, GUI Minhui, XIE Zhantong

  2025, 45(4): 107-125. doi: 10.3969/j.issn.1006-8074.2025.04.007

  Abstract ( 394 )   PDF (2194KB) ( 23 )   　　

  In various imaging applications such as autonomous vehicles and drones, autofocus lenses are indispensable for capturing clear images. However, conventional camera calibration methods typically rely either on processing multiple images at a fixed focal length or on detecting multi-plane markers in a single image and then applying multi-image calibration models. This paper proposes a flexible and accurate calibration approach that extracts subpixel saddle points from a single image containing three non-coplanar calibration boards. To compute accurate homography matrices for the three boards, outliers are removed by eliminating chessboard points that deviated from the fitted grid lines according to their row and column positions. Initial estimates of the intrinsic parameters and the poses of the three planar chessboards are obtained using the three homography matrices in combination with Zhang's calibration method.

  During parameter refinement, a multi-objective optimization function is constructed, incorporating three error terms:

  (1) Reprojection error of the inlier grid points;

  (2) Mechanism-driven error derived from the relationship between homography matrices and camera parameters;

  (3) Cross-planar linearity constraint error, which preserves the pre-imaging collinearity of any five points across different planes after projection.

  
  

  For weight selection in the optimization process, confidence intervals of the detected grid points are analyzed by horizontally rotating the reprojection lines to reduce bias introduced by line slope. The optimal weights are determined by minimizing the number of points whose confidence intervals does not intersect the reprojected lines. When multiple candidates yield similar reprojection performance, the parameter set with the smallest reprojection error is selected as the final result. This method efficiently estimates both intrinsic and extrinsic camera parameters. Simulations and real-world experiments validate the high precision and effectiveness of the proposed approach. Our technique is straightforward, practical, and holds significant theoretical and practical value for rapid and reliable camera calibration.