Xiaoyu Jiang

This paper proposes an innovative omnidirectional antenna space radiation shaped resistor network model to analyze potential distribution characteristics in complex resistor networks and apply it to path planning. Through mathematical modeling based on Kirchhoff's laws and the recursive transformation method, combined with the discrete sine transform of the seventh kind and Chebyshev polynomials of the first kind, we derive precise formulas for node potentials and equivalent resistances. We further develop a novel path planning algorithm that leverages the natural decay properties of potentials, enhanced by directional deviation penalties and a backtracking mechanism. Comparative analyses with classical path planning algorithms demonstrate that the proposed method holds significant potential, particularly in dynamic environments. Finally, a fast algorithm for potential calculation is introduced, achieving a four- to five-fold improvement in computational efficiency over traditional approaches. These advances deepen research on resistor networks and provide strong support for applications in complex systems, autonomous driving, and wireless communications.

In recent years, the exploration and application of resistance networks have expanded significantly, and solving the equivalent resistance between two poin

Investigating spectral properties and operator-space distance measurements, this research focused on tridiagonal Toeplitz matrices under perturbed Dirichlet boundary conditions (hereafter referred to as PDDT Toeplitz matrices). Explicit analytical expressions for eigenvalues and their associated eigenvectors were derived. These expressions emphasized their critical role in characterizing stability under perturbation conditions. Building on the structural features of PDDT Toeplitz matrices, we developed closed-form solutions to quantify normality distance and departure from normality. Additionally, these solutions analyzed $ \varepsilon $-pseudospectra and eigenvalue sensitivity at both local and collective scales. By evaluating eigenvalue sensitivity through these parameters, our framework further enabled the evaluation of spectral sensitivity within PDDT Toeplitz environments. Through rigorous analysis, it has been shown that a dramatic increase in eigenvalue sensitivity depended exclusively on the magnitude ratio of lower to upper diagonal entries, demonstrating remarkable independence from diagonal terms or the complex phases of non-diagonal elements. The degree to which a matrix deviated from normality could be effectively characterized by examining the absolute values of its sub-diagonal and super-diagonal elements. To conclude, we explored an inverse eigenvalue problem embedded within a constrained optimization framework, which produced trapezoidal PDDT Toeplitz matrices as final optimal computational solutions.