We consider the N-body problem given by quasihomogeneous force functions of the form (C_1)/r^a + (C_2)/r^b (C_1, C_2, a, b constants and a, b positive with a less than or equal to b) and address the fundamentals of homographic solutions. Generalizing techniques of the classical N-body problem, we prove necessary and sufficient conditions for a homographic solution to be either homothetic, or relative equilibrium. We further prove an analogue of the Lagrange-Pizzetti theorem based on our techniques. We also study the central configurations for quasihomogeneous force functions and settle the classification and properties of simultaneous and extraneous central configurations. In the last part of the thesis, we combine these findings with the Lagrange-Pizzetti theorem to show the link between homographic solutions and central configurations, to prove the existence of homographic solutions and to give algorithms for their construction. / Graduate

No presente trabalho apresentaremos alguns aspectos do problema Newtoniano de n Corpos. Estudaremos o caso de dois corpos, que tem solução direta, embora não seja possível obter todas as variáveis como função do tempo. No caso n maior ou igual a 3 mostraremos que não existe método para integrar este problema via quadraturas. Podemos tirar apenas algumas informações sobre o caso geral, como a Identidade de Lagrange-Jacobi, o Teorema de Sundman-Weierstrass entre outros. Veremos alguns casos de soluções particulares, que serão chamadas de soluções homográficas. Nestas soluções a forma geométrica da configuração inicial dos corpos é preservada durante o movimento. Veremos condições necessárias sobre as configurações iniciais para que seja possível obter estas soluções. Mostraremos uma relação existente entre estas soluções particulares e os pontos críticos de uma aplicação, que associa a uma configuração a energia total e o momento angular total do sistema. Nestes vários casos, cairemos numa mesma equação algébrica, que será chamada de equação das configurações centrais. Mostraremos, em seguida, que as equações de configurações centrais são equivalentes a um outro conjunto de equações algébricas, que servem também para calcular as chamadas configurações centrais, porém, com estas equações as simetrias do problema ficam mais claras, às vezes. Faremos algumas aplicações diretas destas equações algébricas. Uma subclasse interessante da classe das configurações centrais são as chamadas de equações diferenciais empilhadas, nas quais um subconjunto próprio dos corpos também forma uma configuração central. Nos dois últimos capítulos veremos alguns exemplos de configurações centrais deste tipo, em especial aquelas onde podemos retirar uma massa e ainda ter uma configuração central. / In this work we present some aspects of the Newtonian n--body problem. We study the case of two bodies, which have a straightforward solution, although we can not get all the variables as functions of the time. For n greater or equal to 3 we show that there is no method to integrate this problem by quadratures. We can have just some information about the general case, as the Lagrange-Jacobi\'s Identity the Sundman-Weierstrass\'s theorem and others. We will see some cases of particular solutions, which will be called homographic solutions. In these solutions the geometric shape of initial configuration of the bodies is preserved during the movement. We will see necessary conditions on the initial positions that turn possible to obtain these solutions. We show a relation between these particular solutions and critical points of an application, that associate the total energy and total angular momentum of the system. In these several cases, we will fall in same algebraic equation, which we called of the central configurations equations. We show that the central configurations equations are equivalent to another set of algebraic equations, which are also used to compute the central configurations, but with these equations the symmetries of the problem become clearer. We will make some direct applications these algebraic equations. An interesting subclass of the class of central configurations are called stacked differential equations, in which a proper subset of the bodies form a central configuration too. In the last two chapters we will see some examples of central configurations of this kind, especially those where we can remove a mass and still have a central configuration.

No presente trabalho apresentaremos alguns aspectos do problema Newtoniano de n Corpos. Estudaremos o caso de dois corpos, que tem solução direta, embora não seja possível obter todas as variáveis como função do tempo. No caso n maior ou igual a 3 mostraremos que não existe método para integrar este problema via quadraturas. Podemos tirar apenas algumas informações sobre o caso geral, como a Identidade de Lagrange-Jacobi, o Teorema de Sundman-Weierstrass entre outros. Veremos alguns casos de soluções particulares, que serão chamadas de soluções homográficas. Nestas soluções a forma geométrica da configuração inicial dos corpos é preservada durante o movimento. Veremos condições necessárias sobre as configurações iniciais para que seja possível obter estas soluções. Mostraremos uma relação existente entre estas soluções particulares e os pontos críticos de uma aplicação, que associa a uma configuração a energia total e o momento angular total do sistema. Nestes vários casos, cairemos numa mesma equação algébrica, que será chamada de equação das configurações centrais. Mostraremos, em seguida, que as equações de configurações centrais são equivalentes a um outro conjunto de equações algébricas, que servem também para calcular as chamadas configurações centrais, porém, com estas equações as simetrias do problema ficam mais claras, às vezes. Faremos algumas aplicações diretas destas equações algébricas. Uma subclasse interessante da classe das configurações centrais são as chamadas de equações diferenciais empilhadas, nas quais um subconjunto próprio dos corpos também forma uma configuração central. Nos dois últimos capítulos veremos alguns exemplos de configurações centrais deste tipo, em especial aquelas onde podemos retirar uma massa e ainda ter uma configuração central. / In this work we present some aspects of the Newtonian n--body problem. We study the case of two bodies, which have a straightforward solution, although we can not get all the variables as functions of the time. For n greater or equal to 3 we show that there is no method to integrate this problem by quadratures. We can have just some information about the general case, as the Lagrange-Jacobi\'s Identity the Sundman-Weierstrass\'s theorem and others. We will see some cases of particular solutions, which will be called homographic solutions. In these solutions the geometric shape of initial configuration of the bodies is preserved during the movement. We will see necessary conditions on the initial positions that turn possible to obtain these solutions. We show a relation between these particular solutions and critical points of an application, that associate the total energy and total angular momentum of the system. In these several cases, we will fall in same algebraic equation, which we called of the central configurations equations. We show that the central configurations equations are equivalent to another set of algebraic equations, which are also used to compute the central configurations, but with these equations the symmetries of the problem become clearer. We will make some direct applications these algebraic equations. An interesting subclass of the class of central configurations are called stacked differential equations, in which a proper subset of the bodies form a central configuration too. In the last two chapters we will see some examples of central configurations of this kind, especially those where we can remove a mass and still have a central configuration.

依據1957年韓文學會的《韓國語大辭典》 的分類，在韓國語詞彙中漢字詞占總詞彙的53%，與之相比，非漢字詞彙占47%。由此可見，由於韓國屬於漢字文化圈，所以韓國人在學習華語的時候，與非漢字文化圈的人相比，存在著許多優勢。不過實際上韓國學生在學習華語時經常遇到很多困難，而且有時候並不能精確地使用詞彙。我們發現韓語中部分的漢字詞與相對應的華語詞彙存在著同形同義和同形異義的現象，雖然同形同義詞只是在語法上有些微的差異，但這些差異會成為韓國學習者學習華語的困擾。不僅在學習華語時會產生誤解和誤用，甚而會影響華語交際。由於韓國學習者的漢字基礎常常會誤導他們，所以他們在學習與運用華語時，已有的韓語漢字基礎會對學習產生負遷移。 在第二語言學習中甘瑞瑗(2002) 指出，詞彙習得和詞彙教學是很重要的一環。對韓國學生來說，掌握華韓語之間漢字詞的關連性是能否有效運用華語的關鍵之一。因此，筆者認為，比較和分析「台灣華語8000詞」和與之相應的韓語漢字詞，具有學習上的幫助。 本文旨在以「台灣華語8000詞」 為中心，對照《韓國漢字語辭典》找出兩者之間的同形漢字詞，並把這些同形漢字詞分為同形同義詞、同形部分異義詞與同形完全異義詞三類，具體地分析台灣華語詞和韓語漢字詞的異同。接著以個案研究的方式，探討韓語漢字詞在韓國學生學習華語詞彙時是否帶來正遷移的現象；並以問卷調查的方式來檢驗韓國學生已認識的韓語漢字詞，是否也對華語詞彙學習造成負遷移的影響。 最後，根據個案調查及問卷研究結果，分別對華韓同形同義詞、同形部分異義詞與同形完全異義詞等三類，提出華語詞彙教學建議。 / According to the research of the Chinese Character Society which get published 1957 in the Korean Dictionary, 53% of the Korean vocabulary is based on the Chinese language. This high percentage demonstrates the great impact of the Chinese culture on the Korean language over a long time. Today, Korean learners of the Chinese language may take advantage of this historical and linguistic fact when compared to learners from Western countries. However, in practice, Korean learners still have great difficulties in acquiring the correct usage of a variety of Chinese words in spite of lexical similarities with their mother tongue. Interferences from the Korean language usage on the learners’ target language are an obvious fact. In a first approach, compared with the homographic vocabulary of the Chinese language, Chinese loan words in the Korean language can be classified into three main categories according to their semantic congruency: 1. homosemantic words: homographic words in both languages share principally the same lexical meaning (同形同義詞); 2. semantic congruent words: homographic words in both languages share a congruent basic meaning but lexical meaning differs in certain properties (同形部分異義詞); 3. semantic incongruent words: homographic words in both languages principally do not share a common lexical meaning (同形完全異義詞). The reason may be due to historical meaning changes in both languages. Semantic differences in basically semantic congruent words and semantic incongruency of homographic words both handicap correct vocabulary acquisition of the Chinese language by Korean learners and complicate their correct comprehension and correct usage of the Chinese language. The relevance of correct vocabulary acquisition was already pointed out by the research of Gan Ruiyuan (甘瑞瑗,2002). The present study wants to do a fresh approach in the study of the basic homographic vocabulary of Chinese and Korean languages in its significance for Chinese language teaching to Korean students. To do this, it compares the semantic features of the Chinese basic vocabulary listed in the Taiwanese dictionary 8000 Words in Chinese with their Korean homographics listed in the Dictionary of the Sino-Korean language and classifies the results according to the three categories of semantic congruency mentioned above. Semantic incongruent features are discussed regarding their difficulty both in acquisition and in the correct usage for Korean learners of the Chinese language. In addition, a short learner’s enquiry wants to give further insights into phenomena of language interference which appear in the usage of Chinese homographic vocabulary by Korean students. Finally, the study wants to give some practical suggestions for teaching Chinese homographic vocabulary to Korean students.

In this thesis, central configurations, regularization of Simultaneous binary collision, linear stability of Kepler orbits, and index theory for symplectic path are studied. The history of their study is summarized in section 1. Section 2 deals with the following problem: given a collinear configuration of 4 bodies, under what conditions is it possible to choose positive masses which make it central. It is always possible to choose three positive masses such that the given three positions with the masses form a central configuration. However, for an arbitrary configuration of 4 bodies, it is not always possible to find positive masses forming a central configuration. An expression of four masses is established depending on the position x and the center of mass u, which gives a central configuration in the collinear four body problem. Specifically it is proved that there is a compact region in which no central configuration is possible for positive masses. Conversely, for any configuration in the complement of the compact region, it is always possible to choose positive masses to make the configuration central. The singularities of simultaneous binary collisions in collinear four-body problem is regularized by explicitly constructing new coordinates and time transformation in section 3. The motion in the new coordinates and time scale across simultaneous binary collision is at least C^2. Furthermore, the behavior of the motion closing, across and after the simultaneous binary collision, is also studied. Many different types of periodic solutions involving single binary collisions and simultaneous binary collisions are constructed. In section 4, the linear stability is studied for the Kepler orbits of the rhombus four-body problem. We show that, for given four proper masses, there exists a family of periodic solutions for which each body with the proper mass is at the vertex of a rhombus and travels along an elliptic Kepler orbit. Instead of studying the 8 degrees of freedom Hamilton system for planar four-body problem, we reduce this number by means of some symmetry to derive a two degrees of freedom system which then can be used to determine the linear instability of the periodic solutions. After making a clever change of coordinates, a two dimensional ordinary differential equation system is obtained, which governs the linear instability of the periodic solutions. The system is surprisingly simple and depends only on the length of the sides of the rhombus and the eccentricity e of the Kepler orbit. In section 5, index theory for symplectic paths introduced by Y.Long is applied to study the stability of a periodic solution x for a Hamiltonian system. We establish a necessary and sufficient condition for stability of the periodic solution x in two and four dimension.