In spite of its 150 years history, the problem of classifying all finite-dimensional division algebras over a field k is still unsolved whenever k is not algebraically closed. The present thesis concerns some different aspects of this problem, and the related problems of classifying all composition and absolute valued algebras. A tripartition of the class of all fields is given, based on the dimensions in which division algebras over a field exist. Moreover, all finite-dimensional flexible real division algebras are classified. This class includes in particular all finite-dimensional commutative real division algebras, of which two different classifications, along different lines, are presented. It is shown that every vector product algebra has dimension zero, one, three or seven, and that its isomorphism type is determined by its adherent quadratic form. This yields a new and elementary proof for the corresponding, classical result for unital composition algebras. A rotation in a Euclidean space is an orthogonal map that locally acts as a plane rotation with a fixed angle. All pairs of rotations in finite-dimensional Euclidean spaces are classified up to orthogonal similarity. A description of all composition algebras having an LR-bijective idempotent is given. On the basis of this description, all absolute valued algebras having a one-sided unity or a non-zero central idempotent are classified.

這一篇論文裡，我們討論如何計算開放式有限容量等候系統的穩定機率。其中到達時間和服務時間的機率分配都是Coxian分配。我們利用向量表示法（Product-Form Method）求解穩定機率，並建立C_{k}/C_{m}/1/4與C_{k}/C_{m}/1/6的穩定機率之表格。在使用向量表示法的過程中，計算所需的時間與系統容量無關。因此，在我們計算穩定機率的經驗中，當N>100時，我們可以明顯感覺出向量表示法比一般傳統方法有更快的計算速度。 / In this thesis, we study the C_{k}/C_{m}/1/N open queueing system with finite capacity, N. We use the product-form method to solve the steady-state probabilities and give tables of numerical results in examples of C_{k}/C_{m}/1/4 and C_{k}/C_{m}/1/6. The merit of this method is that the computation time is independent of N. In our computational experiments, we have observed that when the capacity size of queueing system, N>100, the computing efficiency of the product-form method is much better than that of a traditional method.

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Neste trabalho abordaremos alguns tÃpicos matriciais e determinantes e sua aplicaÃÃo no Ensino MÃdio. Em especial a Matriz de Gram em uma transformaÃÃo linear que pode ser apli-cada, por exemplo, para calcular a Ãrea de um triÃngulo em funÃÃo dos seus lados e tambÃm o Gramiano (determinante da Matriz de Gram) que permite calcular o volume de um paralele-pÃpedo. Ambos podem ser aplicados no ensino mÃdio. Nesse trabalho tembÃm fazemos uma generalizaÃÃo do produto vetorial e algumas de suas propriedades envolvendo determinantes. Por fim mostramos a Identidade de Lagrange. / In this paper we discuss some topics and determinants matrix and its application in high school. In particular, the Gram matrix in a linear transformation which can be applied, for example, to calculate the area of a triangle in terms of their sides and also Gramiano (Gram matrix determinant) for calculating the volume of a parallelepiped. Both can be applied in high school. In this work we tembÃm a generalization of the vector product and some of its properties involving determinants. Finally we show the identity of Lagrange.