Optimal error estimates of a locally one-dimensional method for the multidimensional heat equation

For the multidimensional heat equation in a parallelepiped, optimal error estimates inL ₂(Q) are derived. The error is of the order of +¦h¦² for any right-hand sidef L ₂(Q) and any initial function ; for appropriate classes of less regularf andu ₀, the error is of the order of ((+¦h¦² ), 1/2<1.Translated fromMatematicheskie Zametki, Vol. 60, No. 2, pp. 185–197, August, 1996.     

A note on smoothed estimating functions

The kernel estimate of regression function in likelihood based models has been studied in Staniswalis (1989,J. Amer. Statist. Assoc.,84, 276–283). The notion of optimal estimation for the nonparametric kernel estimation of semimartingale intensity (t) is proposed. The goal is to arrive at a nonparametric estimate of ₀=(t ₀) for a fixed pointt ₀ [0, 1]. We consider the estimator that is a solution of the smoothed optimal estimating equation is the optimal estimating function as in Thavaneswaran and Thompson (1986,J. Appl. Probab.,23, 409–417).     

A discrete method of optimal control based upon the cell state space concept

A discrete method of optimal control is proposed in this paper. The continuum state space of a system is discretized into a cell state space, and the cost function is discretized in a similar manner. Assuming intervalwise constant controls and using a finite set of admissible control levels (u) and a finite set of admissible time intervals (), the motion of the system under all possible interval controls (u, ) can then be expressed in terms of a family of cell-to-cell mappings. The proposed method extracts the optimal control results from these mappings by a systematic search, culminating in the construction of a discrete optimal control table.The possibility of expressing the optimal control results in the form of a control table seems to give this method a means to make systems real-time controllable.Dedicated to G. LeitmannThe material is based upon work supported by the National Science Foundation under Grant No. MEA-82-17471. The author is also indebted to Professor G. Leitmann for his many helpful comments.     

Optimal dynamic advertising policies for hereditary processes

In this paper, the Nerlove-Arrow model of optimal dynamic advertising policies is generalized by assuming a general probability distribution of the forgetting time, rather than the exponential one. A control problem with integrodifferential equations of motion is defined for which the transitory and steady-state properties of the optimal advertising policy are examined. The effects of assumptions like IHR-distributions and DHR-distributions, the existence of an upper bound for the forgetting time, etc., are explained. It is shown that there are two (in the case of an exponential distribution even three) different current-value adjoint functions associated with the problem, and relations between the two (three) are established. Also provided is a sensitivity analysis.Thanks are due to G. Feichtinger and S. Jorgensen for useful discussions.     

2003～2011年中国科普法规政策与化学普及书籍关系研究

2002年中国颁布科普法后,2003至2011年出版的化学普及书籍种数总体呈上升趋势。出版地和出版社总体上较为分散,局部集中度高。主题上基础知识类化学普及书籍最多,科学文化类次之,实用知识类较少,化学前沿类最少。涉及科普的一般法律、法规和政策对化学普及书籍的创作影响甚微。只有与国家约束性社会发展指标相关,发生涉及公民生命安全的重大社会事件,针对特定人群的具体实施政策出台,科普法规和政策才对化学普及书籍的出版起到推动作用。     

The existence of sensitive optimal policies in two multi-dimensional queueing models

Recently Dekker and Hordijk [3,4] introduced conditions for the existence of deterministic Blackwell optimal policies in denumerable Markov decision chains with unbounded rewards. These conditions include- uniform geometric recurrence.The-uniform geometric recurrence property also implies the existence of average optimal policies, a solution to the average optimality equation with explicit formula's and convergence of the value iteration algorithm for average rewards. For this reason, the verification of-uniform geometric convergence is also useful in cases where average and-discounted rewards are considered.On the other hand,-uniform geometric recurrence is a heavy condition on the Markov decision chain structure for negative dynamic programming problems. The verification of-uniform geometric recurrence for the Markov chain induced by some deterministic policy together with results by Sennott [14] yields the existence of a deterministic policy that minimizes the expected average cost for non-negative immediate cost functions.In this paper-uniform geometric recurrence will be proved for two queueing models: theK competing queues and the two centre open Jackson network with control of the service rates.The research of the author is supported by the Netherlands Organization for Scientific Research N.W.O.     

Numerical solution of minimax problems of optimal control,part 1

This paper contains general transformation techniques useful to convert minimax problems of optimal control into the Mayer-Bolza problem of the calculus of variations [Problem (P)]. We consider two types of minimax problems: minimax problems of Type (Q), in which the minimax function depends on the state and does not depend on the control; and minimax problems of Type (R), in which the minimax function depends on both the state and the control. Both Problem (Q) and Problem (R) can be reduced to Problem (P).For Problem (Q), we exploit the analogy with a bounded-state problem in combination with a transformation of the Jacobson type. This requires the proper augmentation of the state vectorx(t), the control vectoru(t), and the parameter vector , as well as the proper augmentation of the constraining relations. As a result of the transformation, the unknown minimax value of the performance index becomes a component of the parameter vector being optimized.For Problem (R), we exploit the analogy with a bounded-control problem in combination with a transformation of the Valentine type. This requires the proper augmentation of the control vectoru(t) and the parameter vector , as well as the proper augmentation of the constraining relations. As a result of the transformation, the unknown minimax value of the performance index becomes a component of the parameter vector being optimized.In a subsequent paper (Part 2), the transformation techniques presented here are employed in conjunction with the sequential gradient-restoration algorithm for solving optimal control problems on a digital computer; both the single-subarc approach and the multiple-subarc approach are discussed.This research was supported by the National Science Foundation, Grant No. ENG-79-18667, and by Wright-Patterson Air Force Base, Contract No. F33615-80-C3000. This paper is a condensation of the investigations reported in Refs. 1–7. The authors are indebted to E. M. Coker and E. M. Sims for analytical and computational assistance.     

Facility location in the presence of forbidden regions,I: Formulation and the case of Euclidean distance with one forbidden circle

A constrained form of the Weber problem is formulated in which no path is permitted to enter a prespecified forbidden region $\text{R}$ of the plane. Using the calculus of variations the shortest path between two points $\text{x}\text{,}\text{y}\text{?}\text{R}$ which does not intersect $\text{R}$ is determined. If $\text{d(}\text{x}\text{,}\text{y}\text{)}$ is unconstrained distance, we denote the shortes distance along a feasible path by $\text{d}\text{(}\text{x}\text{y}\text{)}$. The constrained Weber problem is, then: given points $\text{x}{}_{\mathrm{j}}\text{?}\text{R}$ and positive weights w_j, j = 1,2,…,n, find a point $\text{x}\text{?}\text{R}$ such that

$\text{f(}\text{x}\text{)=}\text{Σ}\text{n}\text{j=1}\text{d(}\text{x}\text{,}\text{x}{}_{\mathrm{j}}\text{)}$

is a minimum.An algorithm is formulated for the solution of this problem when $\text{d(}\text{x}\text{,}\text{y}\text{)}$ is Euclidean distance and $\text{R}$ is a single circular region. Numerical results are presented.     

Is Bitcoin’s Carbon Footprint Persistent? Multifractal Evidence and Policy Implications

The Bitcoin mining process is energy intensive, which can hamper the much-desired ecological balance. Given that the persistence of high levels of energy consumption of Bitcoin could have permanent policy implications, we examine the presence of long memory in the daily data of the Bitcoin Energy Consumption Index (BECI) (BECI upper bound, BECI lower bound, and BECI average) covering the period 25 February 2017 to 25 January 2022. Employing fractionally integrated GARCH (FIGARCH) and multifractal detrended fluctuation analysis (MFDFA) models to estimate the order of fractional integrating parameter and compute the Hurst exponent, which measures long memory, this study shows that distant series observations are strongly autocorrelated and long memory exists in most cases, although mean-reversion is observed at the first difference of the data series. Such evidence for the profound presence of long memory suggests the suitability of applying permanent policies regarding the use of alternate energy for mining; otherwise, transitory policy would quickly become obsolete. We also suggest the replacement of ‘proof-of-work’ with ‘proof-of-space’ or ‘proof-of-stake’, although with a trade-off (possible security breach) to reduce the carbon footprint, the implementation of direct tax on mining volume, or the mandatory use of carbon credits to restrict the environmental damage.     

Optimal Energy-Storage Configuration for Microgrids Based on SOH Estimation and Deep Q-Network

Energy storage is an important adjustment method to improve the economy and reliability of a power system. Due to the complexity of the coupling relationship of elements such as the power source, load, and energy storage in the microgrid, there are problems of insufficient performance in terms of economic operation and efficient dispatching. In view of this, this paper proposes an energy storage configuration optimization model based on reinforcement learning and battery state of health assessment. Firstly, a quantitative assessment of battery health life loss based on deep learning was performed. Secondly, on the basis of considering comprehensive energy complementarity, a two-layer optimal configuration model was designed to optimize the capacity configuration and dispatch operation. Finally, the feasibility of the proposed method in microgrid energy storage planning and operation was verified by experimentation. By integrating reinforcement learning and traditional optimization methods, the proposed method did not rely on the accurate prediction of the power supply and load and can make decisions based only on the real-time information of the microgrid. In this paper, the advantages and disadvantages of the proposed method and existing methods were analyzed, and the results show that the proposed method can effectively improve the performance of dynamic planning for energy storage in microgrids.