Smooth Value Functions for a Class of Nonsmooth Utility Maximization Problems

In this paper we prove that there exists a smooth classical solution to the HJB equation for a large class of constrained problems with utility functions that are not necessarily differentiable or strictly concave. The value function is smooth if admissible controls satisfy an integrability condition or if it is continuous on the closure of its domain. The key idea is to work on the dual control problem and the dual HJB equation. We construct a smooth, strictly convex solution to the dual HJB equation and show that its conjugate function is a smooth, strictly concave solution to the primal HJB equation satisfying the terminal and boundary conditions.Comment: 18 page

Reprocessed emission from warped accretion discs induced by the Bardeen-Petterson effect

The broad Balmer emission-line profiles resulting from the reprocessing of UV/X-ray radiation from a warped accretion disc induced by the Bardeen-Petterson effect are studied. We adopt a thin warped disc geometry and a central ring-like illuminating source in our model. We compute the steady-state shape of the warped disc numerically, and then use it in the calculation of the line profile. We find that, from the outer radius to the inner radius of the disc, the warp is twisted by an angle of $\sim\pi$ before being flattened efficiently into the equatorial plane. The profiles obtained depend weakly on the illuminating source radius in the range from $3r_{g}$ to $10r_g$, but depend strongly on this radius when it approaches the marginally stable orbit of an extreme Kerr black hole. Double- or triplet-peaked line profiles are present in most cases when the illuminating source radius is low. The triplet-peaked line profiles observed from the Sloan Digital Sky Survey may be a {"}signature" of a warped disc.Comment: 8 pages, 6 figures, typos corrected, matches version to appear in MNRA

Weberite Na2_2MM'F7_7 (M,M'=Redox-Active Metal) as Promising Fluoride-Based Sodium-Ion Battery Cathodes

Sodium-ion batteries are a viable alternative to lithium-ion technology due to the plentiful sodium resources. However, certain commercialization challenges, such as low specific energies and poor cycling performance of current Na-ion cathodes, still need to be addressed. To overcome these hurdles, this study explored the potential of a novel class of fluoride-based materials, specifically trigonal-type Na$_2$MM'F$_7$ (M and M' are redox-active metals) belonging to the weberite-type compounds, as promising candidates for Na-ion cathodes. Through a comprehensive assessment utilizing ab initio calculations, twelve prospective compounds were identified, demonstrating high thermodynamic stability, large gravimetric capacities (>170 mAh/g), and low net Na-ion migration barriers (<600 meV). Significantly, ten out of the twelve screened compounds exhibit high specific energies exceeding 580 Wh/kg (approximately equals to the specific energy of LiFePO$_4$), indicating their exceptional electrochemical performance. This study will pave the way for further advancements in fluoride-based electrode materials

A friendly-environmental strategy: application of arbuscular mycorrhizal fungi to ornamental plants for plant growth and garden landscape

The demand for ornamental plants is increasing due to urban greening and rural construction, while the growing environment of plants, especially the soil environment, is deteriorating. Hence, sustainable methods of ornamental plant cultivation need to be developed quickly. The application of arbuscular mycorrhizal fungi (AMF) to ornamental plants can be one of the eco-friendly ways to achieve the objective. Soil AMF establish mycorrhizal symbiosis with roots of ornamental plants, which can develop a marvelous mycorrhizal mycelium network in the rhizosphere to stimulate nutrient and water acquisition of host plants. Numerous researches have proven that AMF improved the quality of ornamental plants, like fruit yield, height, biomass, seed quality, the size and number of flowers, leaf, and root. In addition, mycorrhizal fungi also improve nutrient uptake and endogenous hormone balance of host plants. Another important function of AMF is to regulate the physiological, biochemical, and molecular responses of host plants to adversity, including drought stress, temperature stress, heavy-metal stress, and insect and disease stress. From the perspective of the ecological garden landscape, AMF richness would maintain plant abundance, nutrient and energy balance, and higher productivity in normal and soil environment stress, thus, establishing a friendly-environmental ecosystem. This review also provides the basis to exploit and improve the commercial application of AMF in ornamental plants in the future