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     

The behavior of the homozygous and heterozygous sub-types of rats which are genetically-selected for diabetes insipidus: A comparison with Long Evans and Wistar stocks

Several aspects of spontaneous and conditioned behavior (food and water intake, locomotion and emotionality, passive and active avoidance acquisition and retention) of standard (albino and pigmented) rats, and rats heterozygous (HEDI) and homozygous (HODI) for diabetes insipidus, are reviewed. As would be expected, HODI rats have been repeatedly found to consume far more fluid than either HEDI or control rats. Pigmented rats appear to be more active than albinos. HODI rats exhibit less marked emotional responses than do control rats, among which the pigmented ones exhibit the highest emotionality. Light aversion is more evident in albino than in pigmented rats. No differences are found among HEDI, HODI and normal Long Evans rats. It is quite difficult to provide a clear-cut statement concerning inter-strain differences in passive avoidance behavior, possibly because of the variety of techniques employed. In any case, HODI rats do not perform worse than normal controls do. In one-way active avoidance paradigms, pigmented rats perform better than albinos, and the performance of HODI rats does not differ from that of controls. In two-way avoidance paradigms, albinos appear to outperform pigmented rats. Once again, there are no obvious differences between HODI and control animals.In addition to indicating that HODI rats may actually be less emotional than the other groups of rats reviewed here, the studies described once again fail to confirm the previously alleged functions of vasopressin in memory consolidation.     

Sound communication inNezara viridula (L.) (Heteroptera: Pentatomidae): Further evidence that signal transmission is substrate-borne

We investigated the sound communication associated with mating inNezara viridula (L.) to clarify conflicting reports on the medium through which acoustic signals are borne. Recordings made from virgin pairs ofN. viridula during precopulatory and compulatory behaviour have revealed that signals transmitted as vibrations through the substrate are a more likely means of communication than signals transmitted by a radiation of sound pressure waves through the air (at least in Australian populations). We present evidence (additional to that already in the literature) that air-borne acoustics are an unlikely mechanism of sexual communication in this species. Specifically, the recorded pulse train patterns showed no consistency or repeatability in relation to different stages of mating behaviour. In contrast, frequency spectrograms of substrate-borne signals reveal repeatable patterns that do correspond with the stage of mating behaviour.     

A latent class probit model for analyzing pick any/N data

A latent class probit model is developed in which it is assumed that the binary data of a particular subject follow a finite mixture of multivariate Bermoulli distributions. An EM algorithm for fitting the model is described and a Monte Carlo procedure for testing the number of latent classes that is required for adequately describing the data is discussed. In the final section, an application of the latent class probit model to some intended purchase data for residential telecommunication devices is reported. Geert De Soete is supported as “Bevoegdverklaard Navorser” of the Belgian “Nationaal Fonds voor Wetenschappelijk Onderzoek.”     

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。     

乙烷—丙烷混合裂解模型及产率预计

本文研究了乙烷和丙烷的混合裂解,并与乙烷、丙烷单独裂解进行了比较。解决了微分方程组的刚性问题,提出了混合裂解反应动力学模型。明确了三种选择性的概念,即;真实选择性、迭加选择性和总选择性,从而澄清了国际上对于选择性有正偏差和负偏差的不同观点。为乙烯生产提供了有效利用乙烷和丙烷的分析评价方法。     

关于M(x)=o(x)的一个初等证明

本文的主要结果为:设μ(n)是M?bius函数,x>0为实数,若M(x)=■,则M(x)=o(x),x→∞.完成了该定理的初等证明.     

现代技术中的微量级热现象

近年来，在制造具有新颖结构的微型器件和材料（例如微电子器件、微机电系统和各种先进材料）方面取得了巨大的进展。这种发展使得热科学领域中的热现象。从发生在大量级结构中转移到发生在级结构中。对于更好地理解和控制能量传递而言，这种转移既提出了挑战又提供了机遇。本文从微尺寸、微结构和微时间三方面说明了上述微量级热现象。