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     

技术壁垒与我国对策的思考

合理利用技术壁垒，促进中国经济的发展。要提高创新能力，建立和完善我国的技术保护体系，突破国外的技术壁垒。     

中国书法的民族文化精神

中国书法作为中国文化的核心和精髓，突出地表现在五个方面：一、生生不息的传承精神；二、自由成己的创造精神；三、天人合一的理性精神；四、博大坚毅的人格力量；五、艺术化的人生态度。这些颇具民族性的特点，值得后人认真总结和发扬光大。     

两段绞接式联合收割机输送槽的设计研究

本文介绍一种适宜配置在悬挂式联合收割机上的两段铰接式输送槽设计,这种输送槽后段固定,前段与割台连接并与后输送槽铰接,该铰接点为前悬挂割台的支点,借助液压油缸升降,文中对该铰接点位置的确定,以及两段输送槽铰接处的圆弧过渡板设计进行了分析研究,提供了设计方法.     

混合稀土氯化物催化合成了氧基吡喃的Diels－Alder反应

报道了混合稀土氯化物催化丙烯酸和乙烯基丁醚环加成反应的动力学行为．在温和条件下，反应收率可由非催化反应的５％提高到７４％，反应为二级反应。     

L函数的一类二次均值公式

设整数ｎ≥３，定义二次均值 其中表示对模ｎ的所有偶特征求和．证明了下列公式： 其中 φ（ｎ）为Ｅｕｌｅｒ函数。     

异种钢焊接接头过渡层影响因素的研究

本文根据回归正交优化法原理,用电孤焊试验研究了异种钢焊接接头过渡层的影响因素。通过实测与计算,获得过渡层为零时,最佳匹配的焊接规范参数,为消除过渡层的有害影响探讨了新的途径。     

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     

供需网的一个节点:供应链协同管理与决策

从供应链决策分析、决策支持技术及决策支持方法等几个方面探讨了供应链协同管理决策的理论和方法 .最后以生产商和销售商为例研究了双方共同获利的、协同决策的建模方法 .由此得出 ,为了实现供应链为增值链 ,减少开销 ,双方共同获利 ,协同决策是非常重要的 .     

一种可视化CFD网格生成方法

结合可视化技术 ,提出一种CFD前处理中的网格生成方法。阐述了CFD前处理中存在的问题 ,并针对这些问题构架起专用的绘图平台。详细介绍了网格划分的思路及实现方法 ,并将网格自动生成模块集成在绘图平台中。实践证明 ,这种可视化CFD网格划分平台操作界面方便友好 ,算法简捷正确 ,大大简化了前处理工作的复杂性 ,具有很高的实用价值