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.     

东白令海三种经济鲽类捕捞群体结构与生长特性比较

对东白令海水域双线鲽(Lepidopsetta bilineata)、太平洋拟庸鲽(Hippoglosso-ides elassodon)和阿拉斯加鲽(Pleuronectes quadrituberculata)等三种经济鲽类的捕捞群体结构和生长特性进行比较。得出结论是:(1)捕捞群体结构复杂;(2)渐近体长和渐近体重较大、寿命长、K值小;(3)雌鱼比雄鱼个体大、生长快、寿命长;(4)三种鲽类的捕捞群体结构和生长特性有明显差异。     

A Survey of the Spatio-Temporal Data Model

Traditional GIS(Geographic Information System)mostly can only describe the transient state data and does not have the ability to deal with the temporal dynamic data. When the data changes, new data will take the place of the old data, namely the transient state change into another, and the old data will disappear. Therefore, it is unable to make an analysis of the updating changes of the data and predict the development trend of the future. In this case, TGIS (Temporal Geographic Information System) emerges and expands the time dimension on the basis of traditional GIS. Spatio-temporal data model is the key to TGIS. Spatio-temporal data modeling is not only related to dynamic expression of spatio-temporal objects, but also gives an important support for spatio-temporal analysis and reasoning. This paper summarized the theories and applications status at home and abroad of spatio-temporal data model in detail, illustrated family tree of spatio-temporal data model for the nearly fifty years, discussed improvement and application status of Base State with Amendments Model, Event-based Spatio-temporal Data Model, Object-oriented Data Model and other spatio-temporal data models, and raised the existing problems of spatio-temporal data model. The current existing problems mainly includes: ①There are a lot of spatio-temporal data models put forward, but some of them only focus on semantic design and neglect the verified; ②Most of existing spatio-temporal data model are for vector data, only the Event-based Spatio-temporal Data Model is raster data structure; ③At present, the expression the time-space information of geographic entity is relative separated with spatio-temporal data model; ④Spatio-temporal data model is mainly used in cadastral management, land use and forestry data updating, less application in other fields. In the end, the future development direction was put forward of spatio-temporal data model. In the time of big data and “Internet plus”, it is necessary to explore the big data spatio-temporal data model that supports multiple data formats.     

Conservation,contraception and controversy: Supporting human rights to enable sustainable fisheries in Madagascar

Environmental NGOs are increasingly called upon to respect human rights when undertaking conservation programs. Evaluating a family planning program running alongside marine management measures in Madagascar, we find that family planning services provided by an environmental NGO can support women’s reproductive rights. Family planning services allow the option of smaller families, and give more time to work, increased income and better health. These benefits do not translate into increased support for, or participation in, marine management, however, and women who are able to work more are typically fishing more. We identify patriarchal norms as a key factor preventing the family planning programme from manifesting in improved resource stewardship, limiting opportunities for women to participate fully in resource management meetings and diversify their livelihood outside traditional tasks, including fishing. We propose that a successful human rights-based approach must be more comprehensive, targeting multiple rights and challenging existing institutions and power structures.     

《颜氏家训》总括范围副词研究

《颜氏家训》是研究魏晋南北朝汉语的重要语料。全面统计分析其中总括范围副词,分为新、旧两种,并逐一分析其语法功能、语法位置、与其它副词叠用情况以及语义指向等,以管窥魏晋南北朝时期总括范围副词的使用情况,这对于系统研究汉语总括范围副词的发展演变具有重要的参考价值。     

Exploring the ‘ethical everyday’: An ethnography of the ethics of family consumption

Using ethnographic research with six families in the North West of England (2007–2009), this paper opens up the ‘black box’ of everyday ethical consumption by adding colour and form to these everyday experiences. While recent geographical literature has recognised the ethical considerations that are implicit in everyday consumption practices, there is a noticeable void of research that explores and fleshes out the everyday ethical actions of individuals and families as consumers. By exploring the everyday ethics of money, waste and health choices in family consumption practices, this paper makes the case for recognising the ethical nature of everyday practices and choices of consumption. It is argued that rather than consumers subscribing to a given set of ethics in consumption, there are multiple ways of recognising consumption as an ethically-embedded process.     

Involvement Geography: The School Partnership Program In The Junior High School

Abstract

This article presents a family program that extends the materials and meeting times of the Michigan Model Family Geography Challenge, including suggestions on creating a successful family night program. The article focuses on developing positive school-community relationships by inviting local community members to participate in fun academic lessons. In addition to watching the news together nightly, families meet four times during an eight-week period to engage in hands-on geography related activities. One lesson is introduced during each session and families work together to complete assigned tasks. Following the activity, families join together to eat and socialize. The program enhances geographic knowledge (including current events), positive family interactions, and school outreach.     

Characteristic frequencies for earthquake families and their tectonic implications: Evidence from earthquake swarms in the kanto district,Japan

Through the analyses of waveform and spectrum for two swarm sequences in the Kanto District, including the results from 15 earthquake swarsm obtained previously, the mechanism of the faulting process in earthquake swarms is clarified in more detail. Earthquakes occurring in short time intervals consist mainly of events with similar waveforms andS-P times. These are called “earthquake families,” and many families are observed during a swarm sequence (70–80 percent); for example, 15 families were observed in the 1983 Izu Peninsula earthquake swarm. The source spectra of earthquake families share the same corner frequency, even though their low-frequency levels may differ by a factor as great as 1000, and the value of the corner frequency depends on the size of the largest event in the family. Local variations of corner frequencies within a factor of 25 are found among the earthquake swarms in the Kanto District. These observations suggest the existence of a characteristic fault length depending on the swarm area, and its length may be responsible for the size of the largest event in the family. The characteristic fault length is about 100 m for Ashio, about 400 m for the Izu Peninsula and about 2.5 km for areas off the Chiba Prefecture, and the magnitudes of the largest events expected from these fault lengths are about 2.5, 4, and 6, respectively.     

大菱鲆（Scophthalmus maximus）大规模家系选育优良家系的生长发育规律

运用Logistic、Gompertz和von Bertalanffy3种生长模型分别对大菱鲆的5个优良家系E♂1×E♀2、F♂4×N♀3、F♂2×E♀4、E♂2×F♀1、F♂1×F♀4及对照组群体的生长曲线进行了拟合和比较分析。结果表明,家系E♂1×E♀2、F♂2×E♀4和E♂2×F♀1的最优拟合模型为Gompert...     

Evolution of the General Solution of the Restricted Problem Covering Symmetric and Escape Solutions

The work presented in paper I (Papadakis, K.E., Goudas, C.L.: Astrophys. Space Sci. (2006)) is expanded here to cover the evolution of the approximate general solution of the restricted problem covering symmetric and escape solutions for values of μ in the interval [0, 0.5]. The work is purely numerical, although the available rich theoretical background permits the assertions that most of the theoretical issues related to the numerical treatment of the problem are known. The prime objective of this work is to apply the ‘Last Geometric Theorem of Poincaré’ (Birkhoff, G.D.: Trans. Amer. Math. Soc. 14, 14 (1913); Poincaré, H.: Rend. Cir. Mat. Palermo 33, 375 (1912)) and compute dense sets of axisymmetric periodic family curves covering the initial conditions space of bounded motions for a discrete set of values of the basic parameter μ spread along the entire interval of permissible values. The results obtained for each value of μ, tested for completeness, constitute an approximation of the general solution of the problem related to symmetric motions. The approximate general solution of the same problem related to asymmetric solutions, also computable by application of the same theorem (Poincaré-Birkhoff) is left for a future paper. A secondary objective is identification-computation of the compact space of escape motions of the problem also for selected values of the mass parameter μ. We first present the approximate general solution for the integrable case μ = 0 and then the approximate solution for the nonintegrable case μ = 10⁻³. We then proceed to presenting the approximate general solutions for the cases μ = 0.1, 0.2, 0.3, 0.4, and 0.5, in all cases building them in four phases, namely, presenting for each value of μ, first all family curves of symmetric periodic solutions that re-enter after 1 oscillation, then adding to it successively, the family curves that re-enter after 2 to 10 oscillations, after 11 to 30 oscillations, after 31 to 50 oscillations and, finally, after 51 to 100 oscillations. We identify in these solutions, considered as functions of the mass parameter μ, and at μ = 0 two failures of continuity, namely: 1. Integrals of motion, exempting the energy one, cease to exist for any infinitesimal positive value of μ. 2. Appearance of a split into two separate sub-domains in the originally (for μ = 0) unique space of bounded motions. The computed approximations of the general solution for all values of μ appear to fulfill the ‘completeness’ criterion inside properly selected sub-domains of the domain of bounded motions in the (x, C) plane, which means that these sub-domains are filled countably densely by periodic family curves, which form a laminar flow-line pattern. The family curves in this pattern may, or may not, be intersected by a ‘basic’ family curve segment of order from 1 up to 3. The isolated points generating asymptotic solutions resemble ‘sink’ points toward which dense sets of periodic family curves spiral. The points in the compact domain in the (x, C) plane resting outside the domain of bounded motions (μ = 0), including the gap between the two large sub-domains (μ > 0) created by the aforementioned split, generate escape motions. The gap between the two large sub-domains of bounded motions grows wider for growing μ. Also, a number of compact gaps that generate escape motions exist within the body of the two sub-domains of bounded motions. The approximate general solutions computed include symmetric, heteroclinic, asymptotic, collision and escape solutions, thus constituting one component of the full approximate general solution of the problem, the second and final component being that of asymmetric solutions.