Newton's Easy Quadratures ``Omitted for the Sake of Brevity'

In the 1687 Principia, Newton gave a solution to the direct problem (given the orbit and center of force, find the central force) for a conic-section with a focal center of force (answer: a reciprocal square force) and for a spiral orbit with a polar center of force (answer: a reciprocal cube force). He did not, however, give solutions for the two corresponding inverse problems (given the force and center of force, find the orbit). He gave a cryptic solution to the inverse problem of a reciprocal cube force, but offered no solution for the reciprocal square force. Some take this omission as an indication that Newton could not solve the reciprocal square, for, they ask, why else would he not select this important problem? Others claim that ``it is child's play' for him, as evidenced by his 1671 catalogue of quadratures (tables of integrals). The answer to that question is obscured for all who attempt to work through Newton's published solution of the reciprocal cube force because it is done in the synthetic geometric style of the 1687 Principia rather than in the analytic algebraic style that Newton employed until 1671. In response to a request from David Gregory in 1694, however, Newton produced an analytic version of the body of the proof, but one which still had a geometric conclusion. Newton's charge is to find both ``the orbit' and ``the time in orbit.' In the determination of the dependence of the time on orbital position, t(r), Newton evaluated an integral of the form ∫dx/x <sup>n</sup> to calculate a finite algebraic equation for the area swept out as a function of the radius, but he did not write out the analytic expression for time t = t(r), even though he knew that the time t is proportional to that area. In the determination of the orbit, θ (r), Newton obtained an integral of the form ∫dx/√(1−x<sup>2</sup>) for the area that is proportional to the angle θ, an integral he had shown in his 1669 On Analysis by Infinite Equations to be equal to the arcsin(x). Since the solution must therefore contain a transcendental function, he knew that a finite algebraic solution for θ=θ(r) did not exist for ``the orbit' as it had for ``the time in orbit.' In contrast to these two solutions for the inverse cube force, however, it is not possible in the inverse square solution to generate a finite algebraic expression for either ``the orbit' or ``the time in orbit.' In fact, in Lemma 28, Newton offers a demonstration that the area of an ellipse cannot be given by a finite equation. I claim that the limitation of Lemma 28 forces Newton to reject the inverse square force as an example and to choose instead the reciprocal cube force as his example in Proposition 41. (Received August 14, 2002) Published online March 26, 2003 Communicated by G. Smith     

STABILITY CRITERIA FOR A CLASS OF UNCERTAINSYSTEMS WITH TIME—DELAY

Some stability criteria are obtained for a class of uncertain systems with time-delay usingLyapunov functional and analytic techniques. It is easy to check the criteria by making use of theboundedness of the uncertainties.     

Technical Notes ：STUDIES ON THE VALUE OF SECONDARY MARKET TO THE SUPPLY CHAIN

This paper investigates the impact of a secondary market, where retailers can buy and sell excessinventories, on the supply chain. We develop a two-period model with a single manufacturer and tworetailers. At the beginning of the first period the retailers order and receive products from themanufacturer, but at the beginning of the second period, they can trade surplus products betweenthemselves in the secondary market. We investigate the impact of the correlated dependence ofretailers' demand on both the quantity effect and the allocation effect under the secondary market.Lastly,we study potential strategies for the manufacturer to increase sales with the existence of thesecondary market.     

体育专业函授生自学能力培养浅议

首先分析了体育专业函授生的一般特点，并提出了构成体育专业函授生自学能力结构的四大要素，然后着重阐述了体育专业函授生自学能力培养的四项基本措施．     

Non-Minimum Phase Nonlinear System Predictive Control Based on Local Recurrent Neural Networks

Abstract: After a recursive multi-step-ahead predictor for nonlinear systems based on local recurrent neural networks isintroduced, an intelligent PID controller is adopted to correct the errors including identified model errors and accumulatederrors produced in the recursive process. Characterized by predictive control, this method can achieve a good controlaccuracy and has good robustness. A simulation study shows that this control algorithm is very effective.     

FROM MANUFACTURING SCHEDULING TO SUPPLY CHAIN COORDINATION：THE CONTROL OF COMPLEXITY AND UNCERTAINTY

With time-based competition and rapid technology advancements, effective manufacturingscheduling and supply chain coordination are critical to quickly respond to changing marketconditions. These problems, however, are difficult in view of inherent complexity and variousuncertainties involved. Based on a series of results by the authors, decomposition and coordination byusing Lagrangian relaxation is identified in this paper as an effective way to control complexity anduncertainty.A manufacturing scheduling problem is first formulated within the job shop context withuncertain order arrivals, processing times, due dates, and part priorities as a separable optimizationproblem. A solution methodology that combines Lagrangian relaxation, stochastic dynamicprogramming, and heuristics is developed. Method improvements to effectively solve large problemsare also highlighted. To extend manufacturing scheduling within a factory to coordinate autonomicmembers across chains of suppliers, a decentralized supply chai     

反应条件及添加惰性物质对三聚甲醛合成反应的影响

研究了三聚甲醛合成过程进料浓度、催化剂用量等对副产物（特别是甲酸）生成行为的影响。结果表明，该体系副产物的生成强烈地依赖于反应时间和催化剂硫酸的用量，而与原料浓度的波动无关。在反应－精馏过程的反应器中添加惰性高沸有机物，可有效地缩短反应物的停留时间，提高三聚甲酸的选择性，大大降低快速反应－精馏过程对反应器的设计要求。     

实验选课系统的设计及实现

在实行学分制的情况下,配合校级中心实验室的管理模式和实验室管理软件进行设计和开发的实验选课系统,既适应了学分制的要求,完善了实验室的管理机制,也提高了管理水平和工作效率。     

酵母原生质体形成中的电镜观察

用蜗牛酶处理对数期啤酒酵母和异常汉逊酵母细胞,可以有效地从中分离到原生质体透射电镜观察证明了细胞中周质体的存在,揭示了酵母原生质体的释放机理和蜗牛酶的作用位点。     

胆固醇与人体健康

本文论述了人体内胆固醇的分布、主要生理功能、血胆固醇含量与动脉粥样硬化等疾病的关系,同时还介绍了降低血胆固醇含量的途径和措施.