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     

论虚概念

把虚概念理解为“所指称的对象不存在于客观世界的概念”，是不对的，将导致把“理想模型”等本该是实概念的概念纳入其中，也把本该是虚概念的如“地心说”中的概念却排除在外。实际上，对象有虚实，属性也有虚实。实对象、实属性，属于实概念；虚对象、实属性，也属于实概念；实对象、虚属性，属于虚概念；虚对象、虚属性，是虚概念。可见，无论对象虚实，只要所指称的对象的主要特征所反映的属性不是客观的概念就是虚概念。     

白兰瓜子叶和幼叶的组织培养

白兰瓜子叶和幼叶作为外植体直接诱导再生芽。讨论了不同种类和不同水平激素的作用、子叶上不同位置组织的差异、子叶与幼叶诱导频率的不同等。试验结果表明:生长素类激素和较高浓度的细胞分裂素激素类使子叶产生白色疏松的愈伤组织,MS+BA0.5ppm+Kt0.5ppm为诱导再生芽的最适培养基;子叶上不同部位组织的诱导率呈显著差异,近生长点端→远生长点端,诱导率由24.4％急剧下降至0％:幼叶的诱导率为27.4％,明显高于子叶9.7％的诱导率;不定芽的生根需极低浓度的Kt(0.01—0.05ppm)或不附加任何激素。     

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.     

拥挤型网络的路流组合及模型求解的搜索法

证明了一个路流组合定理并由此而给出了一个求解网络选址两级模型的算法，讨论了网络的Ｂｒａｅｓｓ性及其在网络规划中的应用前景。     

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     

关于超导机制的一些思考（待续）

对三旋理论应用于超导机制作一些介绍和研究。     

红外光谱法测定乙丙橡胶中丙烯的含量

乙丙胶经热压膜后测定8.65um和13.85um处的吸光度。Ln~(A8.65)/_(A13.85)值与丙烯含量C_3成直线关系,回归方程C_3％=54.43+16.45ln~(A8.65)/_(13.85)。未知样品,A,C,D,F丙烯克分百分数分别为41.0,53.1,40.4和59.7。     

PDE-Based Medial Axis Extraction and Shape Manipulation of Arbitrary Meshes

Shape skeletonization （i.e., medial axis extraction） is powerful in many visual computing applications, such as pattern recognition, object segmentation, registration, and animation. In this paper, the authors expand the use of diffusion equations combined with distance field information to approximate medial axes of arbitrary 3D differential properties. It offers an alternative solids represented by polygonal meshes based on their but natural way for medial axis extraction for commonly used 3D polygonal models. By solving the PDE along time axis, this system can not only quickly extract diffusion-based medial axes of input meshes, but also allow users to visualize the extraction process at each time step. In addition, the proposed model provides users a set of manipulation toolkits to sculpt extracted medial axes, then use diffusion-based techniques to recover corresponding deformed shapes according to the original input datasets. This skeleton-based shape manipulation offers a fast and easy way for animation and deformation of complicated mesh objects.