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     

LOGISTICS SCHEDULING： ANALYSIS OFTWO-STAGE PROBLEMS

This paper studies the coordination effects between stages for scheduling problems where decision-making is a two-stage process. Two stages are considered as one system. The system can be a supply chain that links two stages, one stage representing a manufacturer; and the other, a distributor It also can represent a single manufacturer, while each stage represents a different department responsible for a part of operations. A problem that jointly considers both stages in order to achieve ideal overall system performance is defined as a system problem. In practice, at times, it might not be feasible for the two stages to make coordinated decisions due to (i) the lack of channels that allow decision makers at the two stages to cooperate, and/or (ii) the optimal solution to the system problem is too difficult (or costly) to achieve.Two practical approaches are applied to solve a variant of two-stage logistic scheduling problems. The Forward Approach is defined as a solution procedure by which the first sta     

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.     

一个快速的二进制多重精度gcd算法

求两个整数的最大公因子（gcd)的经典的Euclid算法时间复杂度为O（ln^3n)，不适宜于多重精度运算。论文证明了gcd的相关性质，提出了一个基于二进制的、适用于多重精度运算的改进算法，其时间复杂度为O（ln^2n)。     

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．     

高温与液压对快堆座式主容器屈曲的影响

讨论了快中子增殖反应堆在运行工况下，主容器内液态钠的高温和液压的存在对容器的屈曲行为的影响。首先用有限元分析了聚氯乙烯圆柱容器实验模型在室温和高温、无液压和有液压时在模拟水平地震的水平力和弯矩荷载下的屈曲荷载与屈曲模态，高温使容器的弹性模量降低，从而使容器抗屈曲能力降低，而液压的存在使抗屈曲能力增加。计算结果与实验结果吻合得到很好。然后用同一程序计算了中国实验快堆钠池主要容器在室温和高温、无液压和有液压时在模拟水平地震的水平力和弯矩荷载下的静态屈曲荷载与屈曲模态，由于屈曲发生在主容器的裙座，不是在有液体的主圆柱壳上，高温和液压对屈曲荷载与屈曲模态几乎没有影响。     

基于小波包能量特征的滚动轴承故障监测方法

对滚动轴承的故障特点进行分析，根据小波包变换能将信号按任意时频分辨率分解到不同频段的特性，提出了小波包能量特征的概念及算法，并给出两个仿真算例，然后将小波包能量特征与径向基函数网络相结合，提出了一种新的滚动轴承故障诊断方法，数字仿真实验证实了该方法的正确性和有效性。     

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.     

周期信号的孔径抖动及其去除

孔径抖动是指模数转换器采样周期之间出现的相位抖动 ,是由各时钟周期边沿出现时刻的不确定性导致的。当输入信号频率较高时孔径抖动是模数转换器信噪比下降的一个重要原因。对数据采集系统中的孔径抖动进行了仿真实验 ,介绍了反卷积算法 ,并利用反卷积算法对周期信号的孔径抖动进行了去除。仿真结果表明该方法是有效可行的