The Helmholtzian matrix of a graph G=(V(G),E(G)) is a graph-theoretic analogue of the vector Laplacian (or Helmholtz operator) introduced in S. Li, L. Lu and J. F. Wang, A graph discretization of vector Laplacian, Discrete Appl. Math. 379 (2026), 446–460. Motivated by applications of the graph Helmholtzian to simplicial networks, we investigate its basic spectral properties. The matrix is naturally indexed by the edges of G and is positive semidefinite for every choice of orientations. We characterize when an edge orientation makes it entrywise nonnegative and relate its irreducibility to an associated signed graph with loops. We prove that the Helmholtzian spectrum is independent of the edge and triangle orientations and establish eigenvalue interlacing inequalities for induced subgraphs. We also show that the nonzero Helmholtzian spectrum, with multiplicities, is the multiset union of the nonzero Laplacian spectrum of G and the nonzero spectrum of a shifted adjacency matrix of the signed graph on the triangles of G. These results connect the Helmholtzian spectrum with ordinary Laplacian spectra, signed graphs, and combinatorial Hodge theory.
To address limited accuracy, excessive redundancy, and real-time deployment constraints in current potato surface defect inspection methods, this study developed a lightweight real-time detector based on YOLOv11n, called You Only Look Once for Potato Surface Defects (YOLO-PSD). First, the backbone and neck C3k2 blocks were replaced with C3k2-ConvFormer-CGLU (Convolutional Gated Linear Unit) modules, enhancing multi-scale texture and boundary representation, which yields higher detection accuracy at lower computation. Second, the baseline detection head was substituted with LSCD-LQE (Localization-Quality Estimation–Lightweight Shared Convolutional Detection Head), which employs a shared lightweight convolutional stem, decoupled regression and classification branches, and DGQP (Distribution-Guided Quality Predictor) for localization-quality estimation, suppressing high-score/low-quality boxes while improving separability under overlapping targets. Finally, Wise-IoU was adopted as the bounding-box regression loss; its adaptive focusing mechanism improves localization accuracy at high IoU thresholds and promotes stable late-stage convergence. On the potato-defect benchmark, YOLO-PSD attains 95.1% precision, 89.9% recall, and 95.6% mAP, surpassing the baseline by 4.4, 4.4, and 3.1 %age points, respectively. The computational cost and parameter count reach 5.4 GFLOPs and 2.16 M, making reductions of 15.6% and 16.3% compared with the baseline. In real industrial deployment on an intelligent sorting machine, YOLO-PSD achieves 12.6 ms latency, with online sorting accuracy of up to 94.7% under the tested conveyor-speed conditions, underscoring its practical robustness and readiness for engineering applications. Collectively, this work delivers a high-accuracy, lightweight, and low-latency solution for automated detection and sorting of potato surface defects, handling scalable adoption in advanced agricultural automation.
Postharvest blueberries are highly susceptible to Alternaria alternata (A. alternata) and Fusarium fujikuroi (F. fujikuroi) infections, which induce decay and produce abscisic acid (ABA). This study investigated how exogenous ABA influenced blueberry ferroptosis during fungal pathogenesis. Results showed that ABA treatment was associated with accelerated lesion expansion, with ABA accumulation in adjacent areas but depletion in lesion areas of infected fruit. ABA treatment correlated with chlorophyll degradation in thylakoid membranes and elevated hydrogen peroxide (H2O2), malondialdehyde (MDA), and ferrous ions (Fe2 +) in both areas, while reducing glutathione (GSH) levels. Correlation analysis revealed that ABA primarily modulated the ROS-lipid peroxidation axis rather than iron homeostasis. Notably, ABA treatment was correlated with increased saturated fatty acids and altered fatty acid unsaturation indices, with distinct effects on oxylipins (OAHFAs) and phospholipids (MLCL, PG and PC). ABA treatment was also associated with mitochondrial stress and dysfunction, and exacerbated plasmalemma PC peroxidation in lesion areas, correlating with accelerated decay. These findings suggested that ABA might act as a virulence factor by amplifying ROS-driven ferroptosis, yet exhibit dual roles in modulating membrane lipid dynamics. This research highlighted ABA as a potential therapeutic target for mitigating postharvest blueberry decay.
Two-dimensional materials as saturable absorbers (SA) are advancing ultrafast fiber lasers. Ternary transition metal chalcogenides (TMCs) offer excellent stability and high carrier mobility, but the potential of Nb2GeTe4 in high-repetition-rate mode-locking remains unexplored. In this work, Nb2GeTe4 nanosheets were prepared by liquid-phase exfoliation and deposited onto tapered fibers. When incorporated into an erbium-doped fiber ring cavity, stable conventional soliton mode-locking was achieved. By precisely tuning the polarization state, multiple-order harmonic mode-locking from the 4th-8th and 22nd-26th orders was realized. Moreover, a maximum repetition rate of 319.8 MHz (52nd harmonic) was obtained at increased pump power. The Nb2GeTe4-SA exhibits excellent nonlinear modulation and long-term stability. This work provides a reliable material platform for high-repetition-rate pulsed lasers and soliton nonlinear dynamics, with promise for ultrafast light sources and high-capacity optical communications.
Permutation polynomials over finite fields have widespread applications in cryptography, coding theory and combinatorial design theory. This paper focuses on constructing permutation polynomials of the form x+γTrqq2(h(x)) over finite fields of even characteristic. There are two main contributions. We first propose several classes of such permutation polynomials by partitioning the finite field Fq2 and determining the number of solutions of certain equations. Second, we employ Mesnager's method (Mesnager, 2014 [20]), which utilizes triples of permutation polynomials satisfying a specific property, to construct self-dual bent functions. Moreover, we show that the permutation polynomials constructed in this paper are not quasi-multiplicative equivalent to the known permutation polynomials of the same form.
