Axioms

In this study, we investigate the upper- and lower-bound approximations of numerical eigenvalues derived by weak Galerkin spectral element methods on arbitrary convex quadrilateral meshes for the Laplace eigenvalue problem. Firstly, the Piola transformation is employed to construct the approximation space for weak gradients on each convex quadrilateral element, while a one-to-one mapping is used to establish the approximation space for weak functions. Subsequently, based on the weak Galerkin spectral element approximation space defined on convex quadrilateral meshes, a Galerkin approximation scheme is formulated, and its well-posedness is then analyzed. Furthermore, numerical experiments are performed on arbitrary convex quadrilateral meshes of the square and L-shaped domains to explore the upper- and lower-bound approximations of numerical eigenvalues. Numerical findings indicate that the presented method not only obtains optimal orders of convergence with respect to both the mesh size and the polynomial degree, but also provides upper- and lower-bound approximations for the reference eigenvalues by proper choices of polynomial degrees in approximation spaces and parameters of the approximation scheme in both h-version and p-version weak Galerkin spectral element methods. This study offers new perspectives and methodologies for the high-precision numerical solution of eigenvalue problems in elliptic equations.

The Energy-Efficient Hybrid Flow Shop Scheduling Problem poses a significant multi-objective optimization challenge, necessitating the simultaneous minimization of conflicting objectives: Total Tardiness, Total Energy Cost, and Carbon Trading Cost. The Non-dominated Sorting Genetic Algorithm II (NSGA-II) is a classic algorithm in the field of multi-objective optimization. However, this algorithm frequently lacks the adaptive capability required to navigate high-dimensional solution spaces, often trapping the search in local optima, particularly when constrained by practical energy states of heterogeneous machines. To address these complexities, this study proposes a hybrid algorithm, named QGN, integrating Q-learning, the Grey Wolf Optimizer (GWO), and the NSGA-II. Specifically, QGN algorithm integrates NSGA-II for robust diversity maintenance with GWO for high-precision intensification. Unlike static hybrid methods, QGN employs a Q-learning agent as an adaptive controller to dynamically balance global exploration and local refinement, providing a theoretically grounded response to the rugged search landscape created by machine heterogeneity. Comprehensive experimental validation across diverse production scenarios confirms that QGN significantly outperforms baselines, including NSGA-II, Jaya, and Multi-Objective Evolutionary Algorithm based on Decomposition (MOEA/D), as well as the state-of-the-art Q-learning- and GVNS-driven NSGA-II (QVNS) algorithm, in terms of both convergence and diversity. The results indicate that the proposed algorithm yields superior solution dominance, generates a substantially larger set of non-dominated solutions, and maintains a more uniform distribution along the Pareto front.

We present Hyperbolic Symmetric Hypermodular Neural Operators (ONHSH), a novel operator learning framework for solving partial differential equations (PDEs) in curved, anisotropic, and modularly structured domains. The architecture integrates three components: hyperbolic-symmetric activation kernels that adapt to non-Euclidean geometries, modular spectral smoothing informed by arithmetic regularity, and curvature-sensitive kernels based on anisotropic Besov theory. In its theoretical foundation, the Ramanujan–Santos–Sales Hypermodular Operator Theorem establishes minimax-optimal approximation rates and provides a spectral-topological interpretation through noncommutative Chern characters. These contributions unify harmonic analysis, approximation theory, and arithmetic topology into a single operator learning paradigm. In addition to theoretical advances, ONHSH achieves robust empirical results. Numerical experiments on thermal diffusion problems demonstrate superior accuracy and stability compared to Fourier Neural Operators and Geo-FNO. The method consistently resolves high-frequency modes, preserves geometric fidelity in curved domains, and maintains robust convergence in anisotropic regimes. Error decay rates closely match theoretical minimax predictions, while Voronovskaya-type expansions capture the tradeoffs between bias and spectral variance observed in practice. Notably, ONHSH kernels preserve Lorentz invariance, enabling accurate modeling of relativistic PDE dynamics. Overall, ONHSH combines rigorous theoretical guarantees with practical performance improvements, making it a versatile and geometry-adaptable framework for operator learning. By connecting harmonic analysis, spectral geometry, and machine learning, this work advances both the mathematical foundations and the empirical scope of PDE-based modeling in structured, curved, and arithmetically.