Symmetry

Printed circuit board (PCB) defect detection is essential for ensuring manufacturing quality and product reliability in electronics production. PCB designs often exhibit inherent symmetry in circuit layouts and periodic patterns, which defects disrupt by introducing asymmetries detectable in spatial and frequency domains. Contemporary methods predominantly emphasize either global context modeling or local detail preservation, yet fail to synergistically leverage their complementary characteristics. Existing approaches predominantly constrain operations to the spatial domain, neglecting complementary frequency-domain representations that encode periodic structures and spectral anomalies. We present PCB-FS, a unified frequency–spatial learning framework that augments YOLOv8 with three synergistic components: (i) Dual-Domain Convolution (DD-Conv) for adaptive spatial frequency feature extraction, (ii) Global–Local Axial Attention (GLA-Attention) for context-aware defect modeling, and (iii) Cross-Stage Partial Dynamic Shifted Large Kernel Convolution (C2f-DSLKConv) for efficient large-receptive-field fusion. Our DD-Conv module adaptively fuses spatial and frequency-domain representations, while the GLA-Attention mechanism enhances global context modeling without sacrificing local detail preservation. The C2f-DSLKConv module further enables efficient large-receptive-field spatial modeling and hierarchical feature aggregation. Our method exploits symmetry-breaking artifacts in PCB layouts to detect structural anomalies, achieving superior defect localization accuracy. A comprehensive evaluation on the enhanced PCB surface defect dataset (DsPCBSD+) demonstrates that PCB-FS achieves 86.2% mAP@50 and 52.4% mAP@50-95, establishing new state-of-the-art performance with significant margins over existing methods. Integrating frequency and spatial domain features significantly enhances PCB defect detection reliability and efficiency in practical applications.

In this paper, we study a generalization of Padovan numbers. We define a generating function and matrix generators for the generalized Padovan sequence. Moreover, using graph methods and a special family of generalized Padovan sequences, we derive a multinomial formula for generalized Padovan numbers. We also prove some identities that generalize known formulae for the classical Padovan numbers.

The modernization of compressed air systems represents both a strategic challenge and an opportunity to achieve a balanced symmetry, understood as the equilibrium among energy efficiency, industrial optimization, and operational sustainability. This study combines the experimental validation of a centralized compressed air system operating under real industrial conditions with a bibliometric analysis that contextualizes the work within global research trends in energy efficiency and industrial optimization. The system, implemented at the Oleohidráulica Company in Cienfuegos, Cuba, consists of two BOGE C 22-2 screw compressors and a newly upgraded distribution network. The analysis involved calculating pressure drops using the methodology proposed by Atlas Copco and verifying the results in situ through measurements at the most distant point of the network. The obtained pressure drop of 0.059 bar, below the international threshold of 0.1 bar, confirms the adequacy and reliability of the design. Moreover, the discussion highlights future perspectives for improvement, where integrating a hybrid approach that combines computational fluid dynamics (CFD) simulations with experimental validation could enhance the accuracy of flow and pressure predictions and facilitate the optimization of the pipeline network design. Overall, the study demonstrates that while the current system complies with international standards, achieving symmetry as an operational balance among efficiency, reliability, and sustainability remains an ongoing process, guiding future optimization efforts.