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     

MeV Pb离子在非晶Si中的射程离散──离子束混合和增强扩散的影响

１．０ＭｅＶ２０８Ｐｂ离子在非晶Ｓｉ中的投影射程ＲＰ和射程偏差ΔＲＰ作为注量和温度二者的函数用背散射法进行测定．注量的变化范围为５×１０１３～７×１０１４ｃｍ－２．注入是在室温和ｔ＝－１２０℃下完成的．由由实验所确定的投影射程，射程偏差与注量或温度无关，并且分别等于２９５和７２．２ｎｍ．与ＴＲＩＭ８６的计算结果相比较，发现ＲＰ的偏离为１８％，而ΔＲＰ的偏离为３６％．ＲＰ和ΔＲＰ二者与注量及温度的无关性，排除了所观察到的与ＴＲＩＭ的矛盾是由于注入期间辐射增强扩散或离子束混合效应而引起的解释。     

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.     

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     

Purealidin A,a new cytotoxic bromotyrosine-derived alkaloid from the Okinawan marine spongePsammaplysilla purea

Summary A new bromotyrosine-derived alkaloid with antileukemic activity, purealidin A (5), has been isolated from the Okinawan marine spongePsammaplysilla purea and its chemical structure elucidated on the basis of the spectroscopic data.     

Aeromonas caviae: ecologic adaptation in the intestinal tract of infants coupled to adherence and enterotoxin production as factors in enteropathogenicity

Aeromonas caviae isolated from stools of diarrheic formula-fed infants and environmental sources produce acetic acid when grown in glucose broth, which is bactericidal (suicide phenomenon). A. caviae grows anaerobically in a minimal medium or under permissive conditions such as the intestinal tract of formula-fed infants. These isolates adhered to HEp-2 cells and produced a cytotoxic and a cytotonic enterotoxin which underscore their enteropathogenicity.     

Business Process Reengineering (BPR) Versus Outsourcing—Critical Perspectives

This article presents a case example involving the renaming of an organizational change process from BPR to outsourcing. The paper discusses the important ramifications of such a name change in that the recognition of outsourcing allowed an organizational commitment to counseling and termination advice. The paper discusses two ways of looking at the name change, one from a critical theory perspective and one from a realist perspective. It demonstrates how theory can provide useful yet markedly different interpretations of such organizational events. Critical theory operates from within what can be termed the transitive epistemological dimension, whereas critical realism tends to emphasize the importance of ontological issues. Each has important things to say about the situation and improves our understanding overall.