Symmetry

There are several graphs naturally associated with rings. The unitary Cayley graph of a ring R is the graph with vertex set R, where two elements are adjacent if and only if $x-y$ is a unit of R. We show that the unitary Cayley graph $C_{T_n\left(\mathbb{F}\right)}$ of the ring $T_n\left(\mathbb{F}\right)$ of all upper triangular matrices over a finite field $\mathbb{F}$ is isomorphic to a semistrong product of a complete graph and the antipodal graph of a Hamming graph. In particular, when , the graph $C_{T_n\left(\mathbb{F}\right)}$ has a highly symmetric structure: it is the union of $2^{n-1}$ complete bipartite graphs. Moreover, we prove that the clique number and the chromatic number of $C_{T_n\left(\mathbb{F}\right)}$ are both equal to $\left|\mathbb{F}\right|$, and we establish tight upper and lower bounds for the domination number of $C_{T_n\left(\mathbb{F}\right)}$.

Aiming to solve the bottlenecks of the traditional Grey Wolf Optimizer (GWO) in UAV three-dimensional path planning—including uneven initial population distribution, slow convergence speed, and proneness to local optima—this paper proposes an improved algorithm (CPS-GWO) that integrates the Kent chaotic map with Particle Swarm Optimization (PSO) to mitigate these limitations. To enhance the diversity of the initial population, the Kent chaotic map is employed, as ergodicity ensures the symmetric distribution of the initial population, expanding search coverage; meanwhile, a nonlinear adaptive strategy is adopted to dynamically adjust the control parameter a, enabling flexible search behaviour. Furthermore, the grey wolf position update rule is optimized by incorporating the inertia weight and social learning mechanism of PSO, which strengthens the algorithm’s ability to balance exploration and exploitation. Additionally, a multi-objective comprehensive cost function is constructed, encompassing path length, collision penalty, height constraints, and path smoothness, to fully align with the practical demands of UAV path planning. To validate the performance of CPS-GWO, a three-dimensional urban simulation environment is established on the MATLAB platform. Comparative experiments with different population sizes are conducted, with the traditional GWO as the benchmark. The results demonstrate that, compared with the original GWO, (1) the average fitness of CPS-GWO is significantly reduced by 31.30–38.53%; (2) the path length is shortened by 15.62–22.12%; (3) path smoothness is improved by 43.44–51.52%; and (4) the fitness variance is only 9.58–12.16% of that of the traditional GWO, indicating notably enhanced robustness. Consequently, the proposed CPS-GWO effectively balances global exploration and local exploitation capabilities, thereby providing a novel technical solution for efficient path planning in UAV logistics and distribution under complex urban environments, which holds important engineering application value.

This paper establishes a comprehensive analytical framework for a new transformation of probability measures, denoted by , which unifies the classical t- and $T_a$-transformations in free probability. We derive the functional equation characterizing through the Cauchy–Stieltjes transform and explicitly show how it specializes to known deformations when $a=0$ or $t=1$. Within the setting of Cauchy-Stieltjes kernel families, we prove structural symmetry and invariance properties of the transformation, demonstrating in particular that both the free Meixner family and the free analog of the Letac-Mora class remain invariant under . Furthermore, we obtain several new limiting theorems that uncover symmetric relationships among fundamental free distributions, including the semicircular, Marchenko–Pastur, and free binomial laws.