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.     

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     

BioStarM-140s芯片基因信息查询系统的构建及应用

应用生物信息学数据库查询的方法,对B ioS tarM-140s小鼠基因表达芯片上的14 112个基因进行相应注释信息的查询和所对应蛋白质功能的分类.首先参照人类综合型芯片上基因的功能分类,设定了14个蛋白质类别,然后采用V isual C 编程语言,构建了基因信息查询系统.应用该查询系统,进行了网络生物学数据库信息资源的自动直接查询,共搜索到有注释信息的基因12 070个;进一步按照所设定的14个蛋白质类别对12 070个基因进行分类查询,搜索到了与这14个分类相符合的、具有详细蛋白质功能注释信息的基因共1 606个.     

基于IP多播的机房信息管理系统的设计与实现

提出了一种基于IP多播技术的解决方案并加以实现,满足了对上机学生按系或班级分组进行管理与各类统计的需要,实现了传统C/S结构所能实现的全部功能,解决了传统机房信息管理系统使用带来的问题。     

Specific binding of antigenic peptides to cell-associated MHC class I molecules

T lymphocytes recognize antigen in the form of peptides that associate with specific alleles of class I or class II major histocompatibility (MHC) molecules. By contrast with the clear MHC allele-specific binding of peptides to purified class II molecules purified solubilized class I molecules either bind relatively poorly or show degenerate specificity. Using photo-affinity labelling, we demonstrate here the specific interaction of peptides with cell-associated MHC class I molecules and show that this involves metabolically active processes.     

Characterization of the rough endoplasmic reticulum ribosome-binding activity.

The rough endoplasmic reticulum membranes of mammalian cells contain specific ribosome-binding sites. A purification to apparent homogeneity of a negatively charged protein (ERp180) of relative molecular mass 180,000 (180 K) was reported which was proposed to function as a rough endoplasmic reticulum ribosome receptor. We report here that ribosome-binding site activity quantitatively solubilized from rough endoplasmic reticulum membranes does not cofractionate with ERp180. By contrast, ribosome-binding site activity fractionates as a much smaller, positively charged protein.