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

Employee business at various levels of a hierarchy for organisations completing case work

This article describes a model for examining the contribution of supervisors to an organization by considering the case work they complete as a production system. The average delay in case work is referred to as the service level. At a given service level, the minimization of total wages within one hour can be studied as a cost function. With this cost function, wage spending on handled-case time and idle time can be formulated. The ratio between the handled-case time and idle time of all employees at the kth level within 1 hour is defined as the‘ busy index’ at the kth level. From the optimal hierarchical structure, we find the following two properties: (1) Given any two levels i and j, the ratio between the idle times of ith and jth levels is independent not only of the service level but also the rate of arriving cases; and (2) At each level, the busy indices are proportional to the square root of each level’s wage rates. This implies that the busy indices decrease with the hierarchical level. Ultimately, when the wage rates at all levels are equal, the increment also becomes equal