In this paper, we investigate several families of linear codes over 𝔽_p , where p is an odd prime, defined by the complements of down-sets in 𝔽_p^n . By translating the weight of each codeword into a counting problem on the corresponding down-set, we derive a unified weight formula that provides a direct and efficient method for determining weight distributions. Based on this formula, we construct and analyze several new families of few-weight codes, and determine their parameters, weight distributions, distance optimality, and minimality. We further establish a self-orthogonality criterion for codes arising from single-generator down-sets. As applications, we completely characterize the self-orthogonality of the code families considered in this paper and obtain several infinite families of self-orthogonal few-weight codes that are both minimal and distance-optimal.
We establish calculus rules for the mth-order Studniarski-like derivative introduced in the literature and used in set-valued optimization. New sum and chain rules are derived under additional assumptions, including directional conditions that are weaker than classical regularity requirements. Unlike existing results, which typically yield only one-sided inclusions, our analysis provides sufficient conditions ensuring exact calculus rules in the form of equalities. These results are applied to the sensitivity analysis of solution mappings arising from parametric systems, leading to refined higher-order sensitivity characterizations. Several examples illustrate the applicability of the proposed framework and demonstrate its advantages in situations where other derivative notions fail to exist.
The total variation nonlocal (TV-nonlocal) model integrates several advanced features, including geometric active contours, anisotropic diffusion, regularization, and nonlocal gradients, demonstrating solid mathematical foundations. It achieves remarkable performance and interpretability when processing blurred edge segmentation or enhancement tasks. Although this model has attracted significant research attention, the development of efficient numerical solutions remains unsatisfactory. To enhance the model’s computational efficiency and applicability in image enhancement, this paper proposes a novel parareal method that combines first-order semi-implicit discretization with the split Bregman method, based on fundamental principles of parallel composite methodologies. The rigorous convergence analysis demonstrates that the proposed method exhibits favorable convergence properties under specific conditions. Furthermore, this paper provides a detailed discussion of key implementation aspects of the parallel algorithm. Numerical experiments show that the parareal scheme significantly accelerates the solving process of the TV-nonlocal model while maintaining accuracy, achieving a remarkable improvement in computational efficiency.
The high-wavenumber Helmholtz equation, fundamental to wave propagation phenomena in acoustics, electromagnetics, and seismic imaging, poses significant computational challenges due to its highly oscillatory solutions. While data-driven approaches, particularly physics-informed neural networks (PINNs), have shown promise in solving differential equations, their application to high-frequency regimes remains hindered by severe ill-conditioning induced by high-frequency differential operators in residual loss terms. To address this, we propose a novel shallow intrinsic frequency PINN (SIF-PINN) architecture that explicitly incorporates frequency-domain prior knowledge through an adaptive wavelength parameterization. By rigorously analyzing the spectral properties of neural tangent kernel (NTK) operators, we demonstrate that shallow networks equipped with frequency structure exhibit superior eigenvalue distributions compared to deep architectures, effectively mitigating spectral bias and accelerating convergence in large-wavenumber scenarios. Theoretical justification is established through Herglotz wave function analysis, proving the convergence of SIF-PINNs under appropriate frequency constraints. The various convincing numerical examples are shown to illustrate the effectiveness of the method. This work establishes a paradigm for integrating frequency-aware mechanisms into neural solvers for multiscale wave phenomena.
A set of vertices is called a k-limited dominating set if every vertex of the graph is either contained in the set or adjacent to a vertex of the set, and each vertex of the set has at most k neighbors outside the set. The minimum cardinality among all k-limited dominating sets of G is the k-limited domination number, denoted by γ _k^L(G) . Since Δ (G) -limited domination coincides with the classical domination, we restrict our attention to the nontrivial range 1 ≤ k < Δ (G) , where the degree limitation becomes meaningful and leads to new combinatorial phenomena. In this paper, we initiate the study of this concept by deriving sharp general bounds for γ _k^L(G) and identifying conditions under which these bounds can be further improved. We establish a connection between k-limited domination and (1, t)-domination. In particular, for d-regular graphs we prove that γ _k^L(G)=γ _1,d-k(G) . In the special case k=1 , we show that 1-limited domination is tightly linked to graph packings, yielding the tight bound γ ^L_1(G) ≤ n - ρ (G) . The study reveals several natural open questions and indicates that limited domination provides a rich ground for further research.
