For a graph F, let h_F(n,q) be the minimum number of copies of F in an n-vertex graph with ex(n,F)+q edges, where ex(n,F) is the maximum number of edges in an n-vertex F-free graph. Let c(n,F) be the minimum number of copies obtained by adding one edge to an extremal F-free graph. Mubayi's supersaturation conjecture predicts, under a stability hypothesis, that h_F(n,q)≥ q c(n,F). Ma and Yuan recently constructed stable graph counterexamples for every fixed q≥4; they asked whether the one-edge equality h_F(n,1)=c(n,F) might still hold for every graph F containing a cycle. We give a negative answer to their question. For each integer t≥6, let H_t be obtained from the t-vertex path by replacing each edge with a 3t-page book, using disjoint page vertices for different path edges. Then h_H_t(n,1)<c(n,H_t) for infinitely many values of n. Moreover, by taking t large, the ratio h_H_t(n,1)/c(n,H_t) can be made arbitrarily small along infinitely many values of n.
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 $\arccos(1/3)$.
Flame spread under lateral airflow exhibits distinct flame behaviors and heat transfer mechanisms compared to concurrent and opposed flow configurations, yet this scenario remains insufficiently understood. In this work, flame spread behaviors under lateral flow for different pool widths were systematically investigated. Results show that the increasing flow speed causes a transition from pulsating to uniform flame spread, accompanied by the emergence and evolution of flame streaks regardless of pool width. Flame spread rate displays a non-monotonic dependence on the lateral flow speed, leading to three distinct stages based on the evolution of dominant mechanisms of heat transfer. Heat transfer measurements reveal that subsurface convective heat transfer in liquid phase dominates in narrow pool, whereas gas-phase heat feedback becomes significant in wide pool. Accordingly, a model of subsurface flow velocity for narrow pool is established incorporating sidewall viscosity effects. Moreover, flame convective heat transfer is enhanced while radiation is suppressed with enhanced lateral flow, resulting in convective dominance in the gas-phase heat feedback. Additionally, an equation of convective heat flux is proposed using the lateral heating length and mix-flow theory. Finally, a theoretical model for predicting flame spread rate is proposed in narrow and wide pools respectively based on the dominant heat transfer mechanisms.
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.
