Inspired by Borcherds' questions [2, 17.10], Guerzhoy constructed a new type of Hecke operators T(p), called the multiplicative Hecke operators, which acts on the space of meromorphic modular forms on the full modular group SL2(Z). In [17], this result was extended in two directions: to higher levels and to T(n) with a positive integer n. In this paper, building on the results in [17], we further generalize the result in another direction by considering alternative infinite product expansions of meromorphic modular forms. As an application, we demonstrate how multiplicative Hecke operators relate both the divisor of modular forms and traces of singular moduli. Additionally, we prove the existence of a modular form with nonintegral coefficients whose poles or zeros are only supported at the cusps and which is not a multiplicative Hecke eigenform. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
Motivated by the demographic effects, we propose and analyze a novel opinion dynamics model with density-dependent interactions and a nonlinear communication function reminiscent of the celebrated Hegselmann-Krause (H-K) model. The model also incorporates heterogeneity through intrinsic opinion tendencies, leading to richer dynamics. In the homogeneous case, where all agents share the same intrinsic tendency, the relative dissimilarities between opinions converge to zero and a single cluster forms. In contrast, under heterogeneous tendencies, we prove that multiple clusters emerge together with velocity alignment within each cluster, provided suitable structural conditions on the system parameters. These theoretical findings are further supported by numerical simulations, which offer qualitative insights beyond the analytical results.
Ammonia (NH3) co-firing offers a pragmatic pathway for decarbonizing coal-fired power plants; however, the impact of coal rank on the flame structure and emission characteristics in swirl-stabilized combustion environments remains under-investigated. This study evaluated the co-firing characteristics of bituminous and subbituminous coals in an 80 kWth pilot-scale down-fired furnace under a constant NH3 injection velocity, which enabled a stable combustion atmosphere even at high co-firing ratios of up to 40%. Internal temperature distributions were corrected using a suction-pyrometer to ensure data integrity and eliminate radiation-induced measurement errors. The results revealed that combustion behaviors and optimal emission control strategies diverged significantly depending on the coal rank. Bituminous coal maintained a stable high-temperature zone (1150-1200 degrees C) near the burner. Under a 0.9 burner air ratio, it significantly reduced NO levels to 172.7 ppm at a 40% co-firing ratio, which was facilitated by its moderate volatile release and high char-surface reactivity. By contrast, the high-temperature zone for sub-bituminous coal exhibited a downstream shift owing to rapid devolatilization and intensified oxygen competition between NH3 and coal volatiles. For sub-bituminous coal, the optimal burner air ratio (1.0) limited NO emissions to 201.4 ppm at a 40% co-firing ratio. This study demonstrated that stable combustion can be sustained at high substitution rates by controlling the NH3 injection velocity. By providing mechanistic insights and quantitative guidelines for reaction zone control based on coal rank, the findings serve as a fundamental technical basis for developing and designing large-scale NH3 co-firing systems.
