Complex number

Abstract: This paper concerns conditions for the approximation of functions in certain general spaces using radial-basis-function networks. It has been shown in recent papers that certain classes of radial-basis-function networks are broad enough for universal approximation. In this paper these results are considerably extended and sharpened.

Abstract: Using mathematical tools developed by Hermann Weyl, the Wigner classification of group‐representations and co‐representations is clarified and extended. The three types of representation, and the three types of co‐representation, are shown to be directly related to the three types of division algebra with real coefficients, namely, the real numbers, complex numbers, and quaternions. The author's theory of matrix ensembles, in which again three possible types were found, is shown to be in exact correspondence with the Wigner classification of co‐representations. In particular, it is proved that the most general kind of matrix ensemble, defined with a symmetry group which may be completely arbitrary, reduces to a direct product of independent irreducible ensembles each of which belongs to one of the three known types.

Abstract: This book investigates the geometry of quaternion and octonion algebras. Following a comprehensive historical introduction, the book illuminates the special properties of 3- and 4-dimensional Euclidean spaces using quaternions, leading to enumerations of the corresponding finite groups of symmetries. The second half of the book discusses the less familiar octonion algebra, concentrating on its remarkable "triality symmetry" after an appropriate study of Moufang loops. The authors also describe the arithmetics of the quaternions and octonions. The book concludes with a new theory of octonion factorization. Topics covered include the geometry of complex numbers, quaternions and 3-dimensional groups, quaternions and 4-dimensional groups, Hurwitz integral quaternions, composition algebras, Moufang loops, octonions and 8-dimensional geometry, integral octonions, and the octonion projective plane.

Abstract: A. Tarski [4] has given a decision method for elementary algebra. In essence this comes to giving an algorithm for deciding whether a given finite set of polynomial inequalities has a solution. Below we offer another proof of this result of Tarski. The main point of our proof is accomplished upon showing how to decide whether a given polynomial f(x, y) in two variables, defined over the field R of rational numbers, has a zero in a real-closed field K containing R.1 This is done in ?2, but for purposes of induction it is necessary to consider also the case that the coefficients of f(x, y) involve parameters; the remarks in ?3 will be found sufficient for this point. In ?1, the problem is reduced to a decision for equalities, but an induction (on the number of unknowns) could not possibly be carried out on equalities alone; we consider a simultaneous system consisting of one equality f(x, y) = 0 and one inequality F(x) $ 0. Once the decision for this case is achieved, at least as in ?3, the induction is immediate. Entering into our considerations are the field R of rational numbers and an arbitrary real-closed field K: the argument proceeds uniformly for all K. Because of this, one gets for real-closed fields a principle analogous to the so-called "Principle of Lefschetz." This principle asserts that results of a certain kindthe kind occurring, for example, in A. Weil's Foundations of Algebraic Geometry (see [6; pp. 242-245])-which are true for the field of complex numbers automatically hold also for an arbitrary algebraically closed field of characteristic 0. The corresponding principle for real-closed fields, which we may call the "Principle of Tarski," says that any sentence of elementary algebra which holds in one real-closed field also holds in every real-closed field. In particular it is true that any polynomial f(xi, ... , xn) e K[x1 , ... , x ], K a real-closed field, has on any n-dimensional closed interval a maximum and a minimum. In ?6(b) we illustrate the principle by showing that if an algebraic variety defined over a real-closed field carries any points with coordinates in K, then it also carries one such point which is nearest to the origin. Our proof may have some bearing on the actual construction of a decision machine. Some remarks on this point are made in ?6(e). Thanks are due to Professor Tarski for valuable comments on the paper.