RGB–infrared (IR) fusion is an effective way to improve object detection robustness for automotive perception under low-light and adverse-weather conditions. Yet, practical multi-modal detectors still face three issues: imperfect cross-modal alignment, inefficient long-range interaction, and unstable query initialization when modalities exhibit inconsistent evidence. This paper presents CMAFNet, a deployment-oriented cross-modal alignment and fusion network with three key designs. (1) A Dynamic Receptive Backbone (DRB) extracts multi-scale features with adaptive receptive fields for both modalities. (2) A Channel-Split Mamba Block (CSM-Block) models long-range cross-modal dependencies using selective state-space modeling with linear complexity in token length, enabling an efficient accuracy–latency trade-off. (3) A Global Multi-modal Interaction Network (GMIN) performs fine-grained alignment and adaptive fusion via dual-branch cross-attention guided by global average/max pooling. In addition, an uncertainty-minimal query selection strategy and a separable dynamic decoder further enhance detection stability and efficiency. Experiments on M3FD and FLIR-Aligned show that CMAFNet achieves 83.9% mAP50 and 84.2% mAP50, respectively, while maintaining competitive inference efficiency, supporting real-time automotive deployment on compute-constrained platforms.

Binary Waring decomposition seeks to express a homogeneous binary form as a minimal sum of powers of linear forms. In the binary setting, Sylvester’s theorem gives a classical algebraic route for rank determination and parameter recovery through structured Hankel/catalecticant matrices. Although this procedure is exact and interpretable in ideal arithmetic, practical rank identification may become unstable when the input coefficients are contaminated by noise or when the underlying roots are close to degenerate configurations. This paper develops a data-driven rank inference framework coupled with certified Sylvester reconstruction for robust binary Waring decomposition. The proposed method first converts the coefficient sequence into a Hankel-aware graph that captures recurrence-induced dependencies among polynomial coefficients. A graph neural network is then used to infer plausible rank candidates from this structured representation. Instead of accepting a single prediction directly, the framework performs explicit Sylvester reconstruction and algebraic residual verification for candidate ranks. To further improve decision reliability, a lightweight meta-verification module integrates reconstruction residuals, model confidence scores, and stability-related indicators to select the most credible rank. Experiments on large-scale synthetic binary forms demonstrate that the proposed approach improves rank identification accuracy and verified reconstruction success under low-to-moderate noise, while maintaining the transparency and auditability of classical symbolic–numeric computation. These results suggest that data-driven rank inference can serve as an effective front-end for algebraically certified reconstruction, especially in numerically ambiguous regimes where fixed threshold-based Sylvester implementations are fragile.

Small underwater objects are often distinguished by weak contours and textures that multiscale aggregation can suppress. We propose DCP-UDet to preserve these local features as contextual information passes through the detector. Spatially embedded cross-channel attention extracts shared context, and wavelet–context refinement modulates local responses using directional detail. Gated scale fusion then controls the coarse features added to a fine-resolution residual path. During training, normalized scale-and-visibility weighting emphasizes difficult targets, while paired-view consistency regularizes predictions under photometric degradation. On the study-specific DUO split, DCP-UDet achieves 65.2% AP and 44.6% small-object AP, exceeding DEIM-D-FINE-S by 1.4 and 2.4 percentage points, respectively. The model has 14.8 million parameters and requires 36.2 GFLOPs, with a batch-one latency of 8.4 ms on an RTX A6000. Component and objective ablations show gains from both selective propagation and difficulty-normalized supervision. Under severe image corruption, the detector retains 79.8% of its clean-image AP. The results support refining coarse context before selectively adding it to fine features for underwater detection.

Binary Waring decomposition seeks to express a homogeneous binary form as a minimal sum of powers of linear forms. In the binary setting, Sylvester’s theorem gives a classical algebraic route for rank determination and parameter recovery through structured Hankel/catalecticant matrices. Although this procedure is exact and interpretable in ideal arithmetic, practical rank identification may become unstable when the input coefficients are contaminated by noise or when the underlying roots are close to degenerate configurations. This paper develops a data-driven rank inference framework coupled with certified Sylvester reconstruction for robust binary Waring decomposition. The proposed method first converts the coefficient sequence into a Hankel-aware graph that captures recurrence-induced dependencies among polynomial coefficients. A graph neural network is then used to infer plausible rank candidates from this structured representation. Instead of accepting a single prediction directly, the framework performs explicit Sylvester reconstruction and algebraic residual verification for candidate ranks. To further improve decision reliability, a lightweight meta-verification module integrates reconstruction residuals, model confidence scores, and stability-related indicators to select the most credible rank. Experiments on large-scale synthetic binary forms show that the proposed meta-guided variant improves rank identification and verified reconstruction success relative to the one-shot hybrid solver under low-to-moderate noise while maintaining the transparency and auditability of classical symbolic–numeric computation. Additional stress tests indicate that performance can degrade under shifted sampling regimes; so, the method should be interpreted as a robust decision layer within the modeled problem class rather than as unconstrained real-world validation.

We study two-tone factors in edge-connected regular graphs and claw-free cubic graphs under red-blue vertex colorings. A two-tone factor is a spanning subgraph in which vertices of different colors are assigned different allowed degree sets. For edge-connected regular graphs, we prove an existence theorem for two-tone $\left(\right\{k\},\{k,k+2\left\}\right)$-factors under arbitrary red-blue colorings and describe a constructive factor-theoretic route to matching-type formulations. For the claw-free cubic setting, we show that the natural arbitrary-coloring extension for $\left(\right\{0,1\},\{2,3\left\}\right)$-factors is false in general. Nevertheless, we prove a restricted positive result when the graph has a triangle decomposition and the coloring is constant on each triangle, and we give a mask-consistency characterization for triangle-inflated cubic graphs. The computational section is therefore framed as an implementation and proof-audit study: it certifies recovered factors in the regular case, cross-checks claw-free cubic stress tests by independent MILP and mask-CSP formulations, and explains why only provable restrictions are retained as theorems. From the perspective of symmetry, the contrast identifies a boundary between symmetry-stable and symmetry-fragile colored degree constraints.