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Showing papers in "Tohoku Mathematical Journal in 1971"



Journal ArticleDOI
TL;DR: In this article, the authors introduced the notion of free nilpotent Lie algebras, which they interpreted as a quotient algebra of the free semi-simple Lie algebra by a suitable ideal, and investigated the relationship between the derivations of the former and the latter.
Abstract: Some results have been given by several authors as to the derivations of Lie algebras. Namely, E. Schenkman and N. Jacobson [1] proved that every non-zero nilpotent Lie algebra has an outer derivation. The author [4] sharpened this theorem and showed that every nilpotent Lie algebra over a field of characteristic 0 possesses an outer derivation in the radical of its derivation algebra. On the other hand G. Leger [2] has given a necessary and sufficient condition for a Lie algebra to have an outer derivation, and in [3] he has shown that if a Lie algebra has no outer derivations and its center is not zero then it is not solvable and its radical is nilpotent and is not quasi-cyclic. Moreover, S. Togo [6] proved that such a Lie algebra coincides with its derived algebra and he proceeded the studies concerning to the Lie algebra which has outer derivations. The purpose of this paper is to add some results to them. In § 2 we shall introduce the notion of the free nilpotent Lie algebra, and interpret a nilpotent Lie algebra as a quotient algebra of the free nilpotent Lie algebra by a suitable ideal. We shall also investigate the relationship between the derivations of the former and of the latter. In § 3, we shall study some applications of the results by Leger [2]. As it is well known, every semi-simple Lie algebra has no outer derivations. It is also known that there exists a solvable Lie algebra with null center which has no outer derivations. However, it seems to the author that it is unknown whether there exists a Lie algebra with non-zero center which has no outer derivations. In § 4 we shall give an example of such a Lie algebra of dimension 41 and with one dimensional center.

59 citations




Journal ArticleDOI
TL;DR: In this article, the authors propose a method to solve the problem of "uniformity" and "uncertainty" in the context of health care, and propose a solution.
Abstract: §

34 citations













Journal ArticleDOI
TL;DR: In this paper, the authors show how to generate inequalities of the Wirtinger type for elliptic differential equations and how to use the integral relations obtained to establish comparison and oscillation theorems for ellipses.
Abstract: was first observed by Beesack [2]. In this paper we show how his method can be modified and extended. In addition to generating inequalities of the Wirtinger type, the integral relations obtained are also useful in establishing comparison and oscillation theorems for elliptic equations. This approach provides yet another connection between some of the methods used in ordinary differential equations and those employed by a number of authors [3,5-8, 11-14] in the study of Sturmian theorems for elliptic differential equations and systems. A variable point of w-dimensional Euclidean space R will be denoted by x=(xu , xn). Let G be a bounded domain of R n with piecewise smooth boundary 3G, and let Gλ be an open set of R n such that GcGλ. Throughout this paper we shall adopt the Einstein summation convention in which Latin indices z, /, ky etc. will take values 1, 2, , m while Greek indices a and β will take values 1,2, , n. All functions considered will be real-valued with domain G or Gj. We consider the system of linear second order differential inequalities