Showing papers in "Periodica Mathematica Hungarica in 1977"

Panconnected graphs II

Abstract: A graphG of orderp is said to bepanconnected if for each pairu, v of vertices ofG, there exists a,u, v-path of lengthl inG, for eachl such that dG(u, v)≦l≦p − 1, whered G (u, v) denotes the length of a shortestu, v-path inG. Three conditions are shown to be sufficient for a graphG of orderp to be panconnected: (1) the degree of each vertex ofG is at least (p+2)/2; (2) the sum of the degrees of each pair of nonadjacent vertices ofG is at least (3p−2)/2; (3) the graphG has at least $$\left( {\begin{array}{*{20}c} {p - 1} \\ 2 \\ \end{array} } \right) + 3$$ edges. It is also shown that each of these conditions is best possible. Additional results on panconnectedness are obtained including a characterization of those completen-partite graphs which are panconnected.

On an invariant theory of the finsler spaces

Abstract: In a Finsler space/ ,n of n dimensions a geometrical association of a local orthonormal frame has a fundamental signification. All geometrical objects of F n referred to such a frame will be expressed by invariants which describe geometrical properties of F n. In this direction, L. B~WALD developed an invariant theory of twodimensional Finsler spaces in his papers [1, 2]. Also there are recently several papers concerned with three-dimensional Finsler spaces [6, 7, 8, 10, 11]. Especially in the paper [8] one of the authors gives a systematical description of the general theory of three-dimensional non-Riemannian Finsler spaces based on A. Mo6~'s invariant frame and discusses some special spaces. In the present paper we shall restrict the notion of a non-Riemannian Finsler space by introducing the notion of a strongly non-Riemannian ~'insler space. The category of these spaces seems to be sufficiently large for applications. For the strongly non-Riemannian Finsler space we give a general method for an invariant determination of an orthonormal frame 2~. The twodimensional case of Berwald and the three-dimensional case of Mo6r and ~a t su moto are obtained by taking n ~ 2 and n ~ 3 in our ease, respectively, The movement equations of the frame ~ enable us to define geometrically the hand v-connections of F n, and the integrability conditions of these equations yield the fundamental equations of the space. Thus the geometry of such spaces can be completely determined in this way. The idea of the present paper was suggested by MmON prompted by the opportuni ty to write the review of M~TSUMOTO'S paper [8] for Zentr~lblatt fiir Mathematik and ihre Grenzgebiete, and the publication is the fruit of the acquaintance of two authors in Debrecen.

Positive functionals on BP*-algebras

Abstract: A new class of locally convex algebras, called BP*-algebras, is introduced. It is shown that this class properly includes MQ*-algebras which were introduced and studied by the first author andR. Rigelhof [10]. Among other results, it is proved that each positive functional on a BP*-algebraA is admissible but not necessarily continuous as shown by an example. However, ifA, in addition, is either (i) a Q-algebra, or (ii) has an identity and is barrelled, or (iii)A is endowed with the inductive limit topology, then each positive functional onA is continuous.