In this paper, we give a new proof of the Lemmens-Seidel conjecture on the maximum number of equiangular lines with a common angle $\arccos(1/5)$. This conjecture was previously resolved by Cao, Koolen, Lin, and Yu in 2022 through an analysis involving forbidden subgraphs for the smallest Seidel eigenvalue $-5$. Our new proof is based on bounds on eigenvalue multiplicities of graphs with degree no larger than $14$. To control the maximum degree of the graph associated with equiangular lines, we employ a recent inequality of Balla derived by matrix projection techniques. Our strategy also leads to a new proof for the classical result obtained by Lemmens and Seidel in 1973 for the case where the common angle is $\arccos(1/3)$.
Rapid recombination of charge carriers remains a critical bottleneck limiting the efficiency of conventional photocatalytic systems. Piezoelectric assisted photocatalysis has emerged as a promising strategy to address this challenge, yet maximizing the synergistic effect between piezoelectricity and photocatalysis is largely restricted by rational catalyst architecture design. Herein, a series of core-shell BaTiO3@TiO2 (BTO@TO) heterojunctions with tunable TiO2 shell thicknesses were rationally fabricated via a sol-gel assisted controlled deposition method, aiming to boost the synergy between the heterojunction built-in electric field and the piezoelectric field for efficient hydrogen evolution. The TiO2 shell thickness exerts a decisive regulatory effect on the charge transfer kinetics in BTO@TO composites, and an optimized moderate shell thickness is indispensable to strike an excellent balance between the heterojunction built-in electric field and piezoelectric field coupling, thereby maximizing the dual-field synergistic effect. Impressively, the optimized BTO@TO-1 composite with a TiO2 shell thickness of similar to 180 nm exhibits exceptional piezo-photocatalytic performance (2.42 times that of photocatalysis alone), with a synergy factor of 2.4 and a synergy efficiency of 58.5%, far exceeding the sum of its individual photocatalytic and piezocatalytic performances, which confirms a prominent synergistic effect. This work highlights the critical role of optimal shell thickness in balancing heterojunction charge separation and piezoelectric field coupling, offering a promising design strategy for ferroelectric-based core-shell catalysts in synergistic piezo-photocatalytic water splitting.
Time-series forecasting is widely applied in traffic systems, energy management and space telemetry analysis, where reliable long-horizon prediction is essential. In multivariate long-horizon forecasting, performance depends not only on capturing temporal dynamics within each variable but also on modeling stable dependencies across variables. Although existing method has explored modeling global inter-variable correlations within the attention mechanism, it only reconstructs the query. This neglect of reconstructing the key and value leads to inaccurate similarity estimation in the attention computation. To address this issue, we propose the Shared Temporal Reference Transformer (STRFormer). STRFormer introduces a shared temporal reference mechanism that aligns query, key, and value representations within the attention module, establishing a unified temporal coordinate system for cross-variable interactions. This design encourages more coherent channel-wise attention patterns without modifying the overall attention structure. In addition, STRFormer incorporates a Multi-scale Refinement module that jointly models global temporal structure and local dynamics, forming a multi-scale architecture for long-horizon forecasting. Extensive experiments on thirteen public datasets from diverse domains demonstrate consistent improvements over competitive baselines, with particularly strong performance on high-dimensional traffic benchmarks.
Foliar iron delivery requires simultaneous control of ferrous-state stability, nutrient transport, and leaf-surface residence, yet these functions are rarely integrated within a single formulation. Here, we developed a carbon dot-stabilized and coordination-gated nanocarrier formulation (NFCT) by co-confining Fe2+ and carbon dots (CDs) within amino-functionalized mesoporous silica and depositing an external tannic acid-Fe (TA-Fe) coordination network. Electron spin resonance measurements and density functional theory calculations support complementary contributions of the CDs to reactive-oxygen-species attenuation and Fe2+ coordination, whereas the outer metal-phenolic network provides pH-dependent regulation of nutrient transport. Under the tested storage conditions, NFCT retained 93.0% of recoverable Fe2+ after 7 d and showed substantially greater recoverable Fe2+ availability under mildly acidic than near-neutral or alkaline conditions. On model waxy leaf surfaces, NFCT improved droplet spreading and achieved 72% of the pre-rain total-Fe deposit after a 70.7 mm simulated-rainfall challenge. These redox, transport, and interfacial attributes were associated with greater foliar Fe accumulation, chlorophyll index, biomass production, and root-system remodeling in Fe-deficient rice. In a single-site, single-season proof-of-concept field trial, no statistically significant differences were detected between NFCT and commercial EDTA-Fe for the measured agronomic traits under the tested application program. The results define a carbon-dot-centered structure-property-function framework in which internal Fe2+ protection, external coordination-gated transport, and rainfast foliar residence provide complementary delivery functions.
In this paper, we systematically review a series of effective methods for studying the qualitative properties of solutions to fractional equations. Beginning with the pioneering extension method and the method of moving planes in integral forms, we introduce a variety of direct methods, including the direct method of moving planes, the method of moving spheres, blow-up and rescaling techniques, the sliding method, regularity lifting, and approaches for interior and boundary regularity estimates. To elucidate the core ideas behind these methods, we employ simple examples that demonstrate how they can be applied to investigate qualitative properties of solutions. We also provide a comparative discussion of their respective strengths and limitations. It is our hope that this paper will serve as a useful handbook for researchers engaged in the study of fractional equations.
