Restricted Three-Body Problem: An Approximation of its General Solution – Part One – the Manifold of Symmetric Periodic Solutions

This paper gives the results of a programme attempting to exploit ‘la seule bréche’ (Poincaré, 1892, p. 82) of non-integrable systems, namely to develop an approximate general solution for the three out of its four component-solutions of the planar restricted three-body problem. This is accomplished by computing a large number of families of ‘solutions précieuses’ (periodic solutions) covering densely the space of initial conditions of this problem. More specifically, we calculated numerically and only for μ = 0.4, all families of symmetric periodic solutions (1st component of the general solution) existing in the domain D:(x ₀ ∊ [−2,2],C ∊ [−2,5]) of the (x ₀, C) space and consisting of symmetric solutions re-entering after 1 up to 50 revolutions (see graph in Fig. 4). Then we tested the parts of the domain D that is void of such families and established that they belong to the category of escape motions (2nd component of the general solution). The approximation of the 3rd component (asymmetric solutions) we shall present in a future publication. The 4th component of the general solution of the problem, namely the one consisting of the bounded non-periodic solutions, is considered as approximated by those of the 1st or the 2nd component on account of the `Last Geometric Theorem of Poincaré' (Birkhoff, 1913). The results obtained provoked interest to repeat the same work inside the larger closed domain D:(x ₀ ∊ [−6,2], C ∊ [−5,5]) and the results are presented in Fig. 15. A test run of the programme developed led to reproduction of the results presented by Hénon (1965) with better accuracy and many additional families not included in the sited paper. Pointer directions construed from the main body of results led to the definition of useful concepts of the basic family of order n, n = 1, 2,… and the completeness criterion of the solution inside a compact sub-domain of the (x ₀, C) space. The same results inspired the ‘partition theorem’, which conjectures the possibility of partitioning an initial conditions domain D into a finite set of sub-domains D i that fulfill the completeness criterion and allow complete approximation of the general solution of this problem by computing a relatively small number of family curves. The numerical results of this project include a large number of families that were computed in detail covering their natural termination, the morphology, and stability of their member solutions. Zooming into sub-domains of D permitted clear presentation of the families of symmetric solutions contained in them. Such zooming was made for various values of the parameter N, which defines the re-entrance revolutions number, which was selected to be from 50 to 500. The areas generating escape solutions have being investigated. In Appendix A we present families of symmetric solutions terminating at asymptotic solutions, and in Appendix B the morphology of large period symmetric solutions though examples of orbits that re-enter after from 8 to 500 revolutions. The paper concludes that approximations of the general solution of the planar restricted problem is possible and presents such approximations, only for some sub-domains that fulfill the completeness criterion, on the basis of sufficiently large number of families.     

Geoacoustic Inversion via Genetic Algorithm and Its Application to Manganese Sediment Identification

An acoustic inversion method using a wide-band signal and two near field receivers is proposed and applied to multiple layered seabed models including a manganese sediment. The inversion problem can be formulated into a probabilistic model comprised of signals, a forward model, and additive noise. The forward model simulates wide-band signals, such as chirp signals, and is chosen to be the source-waveletconvolution plane wave modeling method. The wavelet matching technique, using weighted least-squares fitting, estimates the sediment sound-speed and thickness on which determination of the possible numerical ranges for a priori uniform distribution is based. The genetic algorithm is applied to a global optimization problem to find a maximum a posteriori solution for determined a priori search space. Here the object function is defined by an L 2 norm of the difference between measured and modeled signals. Not only the marginal pdf but also its statistics are calculated by numerical evaluation of integrals using the samples selected during importance sampling process of the genetic algorithm.     

不同浓度Li2SO4水溶液结构的X射线衍射研究

用θ～ 2θ型粉末衍射仪反射法精确测量了不同浓度的Li2 SO4水溶液的衍射数据 ,通过数据处理给出了溶液的结构函数和径向分布函数 ,由几何结构模型的最小二乘法精修 ,得到了溶液中阳离子和阴离子第一、二水合层的结构参数。     

溶液结构的X射线衍射研究──4.用粉末衍射仪精确测定溶液结构的新方法

本文叙述了用θ—2θ型X射线衍射仪精确测定电解质溶液结构的新的实验技术，设计制作了具有恒温功能的超厚液体样品池，并建立了样品池窗口强度的校正方法。建立在Johansson和Yamaguchi工作基础上，发展了液体X射线衍射数据处理和结构参数精细化的计算机程序，获得了非常令人满意的实验结果。由θ－2θ型X射线衍射仪精确测定的径向分布函数与θ－θ型衍射自由表面散射比较，表明DRF分辩率有所提高。     

双线性插值近似网格的栅格数据投影变换

在栅格数据投影变换常用算法的基础上,介绍一种新的基于双线性内插近似网格算法的栅格数据投影变换,并详细描述其算法过程,认真分析其特点,并证明该算法在大范围栅格数据投影变换中有显著优点。     

大气中一类非线性方程的非线性波解及周期的近似计算法

大气中许多模式可化为Ｆ＇＇＝ＡＦ／Ｆ＋Ｂ（ＡＢ≠０）一类非线性常微分方程。利用常微分方程定性理论说明了该方程存在非线性波解，讨论了其周期的计算问题，并求得了波解及其周期的近似表达式。     

Whole mantle SH velocity model constrained by waveform inversion based on three-dimensional Born kernels

A whole mantle SH velocity model is obtained by using a unique data set and techniques. Body and surface waveforms including major and multi-orbit phases are used as a data set and are inverted by using 3-D Born kernels. The resultant model, SH18CE, reveals the different natures of the two major upwelling systems: the strong low velocity anomalies beneath Africa extend for more than 1000 km from the core–mantle boundary (CMB), whereas those beneath the Pacific are restricted to 300–400 km from the CMB. The results also show the variable natures of stagnant slabs on the 670 discontinuity around Japan: the depths of the strongest high velocity anomalies within the stagnant slabs are different region by region, which is consistent with the detailed delay time tomography model in this area.     

疏松砂岩稠油油藏出砂冷采机理及配套技术研究

稠油出砂冷采技术对地层原油含有溶解气的各类疏松砂岩稠油油藏具有较广泛的适应性，它通过大量砂子的产出和泡沫油的形成，获得较高的原油产量，河南油田于1996年在国内率先开展稠油出砂冷采技术调研及可行性研究，出砂冷采机理及配套工艺技术研究，并于1997年6月进入现场先导试验，矿场试验已展示出明显出砂冷采特征的G4906井，日产油量20t-30t，是常规开采和蒸汽吞吐的10倍和4倍以上，开采成本比蒸汽吞吐降低47%，出砂冷采工业化应用配套技术的形成和应用，将使国内及河南油田稠油资源利用程度进一步提高。     

Effects of Deicing-Salts on Heavy Metal Mobility Zum Einfluß von Streusalzen auf die Beweglichkeit von Schwermetallen

Heavy metal input into roadside soils and effects of deicing salts are often discussed as separate environmental problems. However, a relationship between salt input and heavy metal mobility has already been detected in laboratory tests. During a 2-year field study, the soil solution of a roadside soil was continuously sampled. The results of soil solution analysis not only showed a strong rise in deicing-salt related sodium, calcium and chloride concentrations, but also high zinc and cadmium values in winter and spring. Additional laboratory tests confirmed the mobilizing effects of salt solution which contained NaCl and CaCl₂. Cation exchange and the formation of chlorocomplexes are discussed as the responsible mechanisms.     

Prediction of limestone solution rates from rainfall and runoff data: Some comments

Recent work by Láng (1977) on solutional erosion rates in limestone drainage basins in relation to rainfall contains serious errors and has a questionable logical basis. Although much sounder, Smith and Atkinson's (1976) work on erosion rates in relation to runoff also requires modification as it greatly overestimates erosion rates in two areas of New Zealand. Greater resolution could probably be obtained by relating autogenic runoff to the rate of solution by autogenic waters and by computing separate relationships for areas where solution takes place under open and closed system equilibrium conditions. These factors should be taken into account in the experimental design of future studies.