Showing papers in "Tohoku Mathematical Journal in 1980"

Surfaces of revolution with prescribed mean curvature

Abstract: In this paper we study a surface of revolution in the Euclidean three space Λ. The generating curve of the surface satisfies a nonlinear differential equation which describes the mean curvature. The purpose of this note is to solve the differential equation by an elementary method. Solutions are represented explicitly by generalized FresnePs integrals which involve the mean curvature. Therefore, for a given continuous function H(s), we can construct a 3-parameter family of surfaces of revolution admitting H(s) as the mean curvature. We shall remark that these computations are different from the one in Delaunay [1]. About 140 years ago, he solved the differential equation under the constancy of the mean curvature and gave a method of geometric constructions for such surfaces. For the proof, he first obtains a solution of an evolute of the generating curve. By making use of this solution, he found a representation formula of the generating curve. Therefore these solutions hold only on some intervals on which the evolute can be defined. It seems not to be simple to obtain a global solution from his method. Our calculations will easily find global solutions and the corollary of the main theorem of this note describes all complete surfaces of revolution with constant mean curvature. During this research I stayed in Kδln and received many nice advices from Professors Peter Dombrowski and Helmut Reckziegel. By their suggestions, my original computations could be simplified and generalized. And also I shall mention that Reckziegel got interested in drawing graphs of these generating curves by making use of the computer and obtained many beautiful pictures. He permitted me with favor to include some of the graphs by his programming in this paper. The author wishes to express his deep gratitude to both of them for helpful

Stability theory for countably infinite systems of differential equations

Abstract: New stability results for a class of countably infinite systems of differential equations are established. We consider those systems which may be viewed as an interconnection of countably infinitely many free or isolated subsystems. Throughout, the analysis is accomplished in terms of simpler subsystems and in terms of the system interconnecting structure. This approach makes it often possible to circumvent difficulties usually encountered in the application of the Lyapunov approach to complex systems with intricate structure. Both scalar Lyapunov functions and vector Lyapunov functions are used in the analysis. The applicability of the present results is demonstrated by means of several motivating examples, including a neural model.

On growth and decay of solutions of perturbed retarded linear equations

Abstract: (P) * = L(xt) + f(tf xt) Roughly speaking, we will try to answer the following question: if the perturbation f(t, xt) is small, are the solutions of (P) "close" to the solutions of (L), for large values of ti It is clear that we can compare the solutions of the above systems in many ways, depending on what we mean by "close". Many works have been done in the case where it is required that the difference between the solutions of (L) and the solutions of (P) tends to zero, as t goes to infinity. This can be found, for example in Cooke [2], Evans [5], Kato [8]. In this paper we are interested in the case where the relative error, between the solutions of (L) and the solutions of (P), goes to zero, as t goes to infinity. For ordinary differential equations, results in this direction can be found in Szmidt [11], Onuchic [9], Coppel [3], Brauer and Wong [1], Rodrigues [10], etc. Szmidt uses the topological method of Wazewski and Onuchic [9] uses admissibility theory introduced by Massera and Schaffer. More precisely, we will be concerned with the following problems:

A normal integral basis theorem for dihedral groups

Abstract: MAIN THEOREM 1.1. Let P be a projective ZDn-module. Then P is free if and only if P is free as a ZCn-module. Let A be an order in a finite dimensional semi-simple Q-algebra QA. C{A) denotes the locally free class group of A. Let B £ QA be a maximal order containing A. Then the kernel D{A) of the natural homomorphism of C(A) onto C{B) does not depend on the choice of B. Viewing projective ZZ)Λ-modules as ZCw-modules we obtain the restriction map res: C(ZDn)^C(ZCn). It is well known that res (D(ZDn)) £ D(ZCn). For an arbitrary finite group G, every projective ZG-module is locally free and vice versa ([17]). Hence Main Theorem can be reformulated as

On the existence of simple Liapunov functions for linear retarded difference-differential equations

Abstract: : Liapunov functions of simple form have been used for the study of stability properties of difference-differential equations. In this paper, we provide necessary and sufficient conditions for the existence of such functions. (Author)