Mathematics

Small objects in underwater and traffic-sign images often have weak boundaries and low contrast. Downsampling and multi-scale fusion can further weaken the features needed to detect them. We propose ContextFlow-YOLO, a YOLOv8n-based detector designed to preserve context during feature extraction and reuse it across scales. Its context-preserving backbone combines local features with surrounding and global information during downsampling. The global-guided scale-sequence neck distributes this context to features at different levels. A nonlinear context-attention fusion unit then refines the responses used for prediction. ContextFlow-YOLO obtains mAP@0.5 scores of 0.855, 0.838, and 0.826 on URPC2019, URPC2020, and TT100K, respectively. On TT100K, the model uses 6.11 M parameters and runs at 134 FPS. Heatmap and receptive-field visualizations show more concentrated target responses and broader feature coverage than YOLOv8n.

Physical exercise knowledge is usually distributed in flat catalogs, where muscles, equipment and training attributes are stored as independent columns and the dependencies that organize the domain are never made explicit. This work presents a heterogeneous multi-relational Fitness Knowledge Graph built from a public-domain exercise dataset and refined through expert-guided curation: 922 entities and 8598 typed assertions across nine semantic relations, each validated against an explicit domain and range schema before the graph is written. Its organization was assessed through relational representation learning, comparing seven models over thirty independent runs under the same protocol: free node embeddings, DistMult, TransE, ComplEx, a homogeneous Graph Convolutional Network, the same encoder combined with a relation-aware decoder, and a Relational Graph Convolutional Network. Models without propagation reconstruct observed edges more accurately, while propagation yields a markedly larger separation between exercise types, improving similarity scores from 0.652 for DistMult to between 0.810 and 0.968, significant under a paired Wilcoxon test with Holm correction. The advantage does not extend to every criterion: a translational decoder without propagation reaches a Silhouette score indistinguishable from that of the Relational Graph Convolutional Network with less than a third of its parameters. A control experiment removing the two relations derived from the source category field shows that this organization is carried by them, since every type-based metric collapses and the Silhouette score becomes negative for all seven models. The information does not leave the graph, as a parameter-free similarity over the seven remaining relations still recovers exercise type at 0.828 against 0.479 for random neighborhoods, placing the limitation in the training objective rather than in the relational design. Edge reconstruction and semantic organization therefore measure different properties of a representation, and the relational composition of the graph determines which property the representation space reflects.

Nonmonotone transformations of fuzzy numbers arise in loss functions, dispersion measures, and higher-order risk criteria, but inverse-credibility laws apply directly only to monotone maps. Existing methods typically either compute transformed alpha-cuts without a credibilistic scalar functional or use endpoint formulas whose validity depends on monotonicity. We develop a calculus that preserves interior extrema before integration. For any continuous transformation of a fuzzy number whose transformed output is integrable, the expected value admits an extrema-based alpha-cut representation that requires neither regularity nor an inverse credibility distribution. For continuous nondecreasing radial losses, we derive the exact risk, prove a sharp factor-two bound for the conservative endpoint proxy, characterize both equality regimes, and localize the proxy excess to center-containing cuts. We also establish Lipschitz perturbation and certified midpoint-quadrature bounds. At the decision level, proxy constraints are conservative and proxy minimization has a factor-two guarantee. A reproducible experiment further shows that endpoint substitution can reverse the optimizer and contract the feasible set. The framework is further specialized to higher-order credibilistic moments: for regular fuzzy numbers, signed odd moments retain inverse-distribution formulas, whereas absolute and even moments are obtained from the extrema-based formulation without requiring regularity. Triangular and Gaussian closed forms, standardized coefficients, a scale-free bias function, and deterministic audits demonstrate the resulting error, stability, computation, and decision guarantees.