CONTENTS & ABSTRACTS

In English. Summaries in Estonian

Proceedings of the Estonian Academy of Sciences.

Physics * Mathematics

Volume 54 No. 3 September 2005

Relationship between join and betweenness geometries; 131–153

Ülo Lumiste

Abstract. A treatment of join geometry was elaborated by Prenowitz and Jantosciak in a voluminous monograph of 1979 (Join Geometries. A Theory of Convex Sets and Linear Geometry), which in most part deals with the theory of convex sets but touches also upon linear geometry and the betweenness relation. The latter relation was taken as the only basic notion (besides the notion of point) by the Estonian mathematicians J. Sarv, J. Nuut, and A. Tudeberg (Humal) in their treatment of the foundations of geometry in the 1930s. A solid betweenness geometry as a theory of betweenness models was worked out by the author of the present paper in 1964 but it appeared in publications not widely available. On the basis of the 1979 monograph, the author analyses the relationship between these two geometries. First, betweenness geometry is recapitulated, and then the more general interimity and betwixtness geometries are introduced. It is proved that a betweenness geometry is at the same time an ordered join geometry, and conversely, an exchanged join geometry is a betwixtness geometry, but the more special ordered join geometry coincides with betweenness geometry. In higher than two dimensions the latter is Desarguesian and leads to a convex region in a linear space over an ordered skew field.

Key words: join geometry, betweenness model, convex region, Desarguesian space.

Decomposition of discrete-time nonlinear control systems; 154–161

Ülle Kotta

Abstract. The goal of the paper was to extend the results on the decomposition of the state equations of continuous-time nonlinear systems into the discrete-time domain. The results on accessible–nonaccessible decomposition mimic those of the continuous-time case. Decomposition is carried out in the vector space of differential one-forms. The results on observable–unobservable decomposition are not carried over to the discrete-time domain, in general, since the observable space cannot always be locally spanned by exact one-forms whose integrals would define the observable state coordinates. We conjecture that for reversible systems the observable space is integrable.

Key words: nonlinear systems, discrete-time systems, decomposition, accessibility, observability, vector spaces, one-forms, algebraic methods.

Optimality of theoretical error estimates for spline collocation methods for linear weakly singular Volterra integro-differential equations; 162–180

Inga Parts

Abstract. Two spline collocation methods for solving linear weakly singular Volterra integro-differential equations are considered. A result on the superconvergence at the collocation points is proved and optimality of several theoretical error estimates is demonstrated by extensive numerical experiments. Based on numerical results, a conjecture about the theoretical error estimates at the collocation points is stated for the cases not covered by known theorems.

Key words: weakly singular, Volterra, integro-differential equations, spline collocation, superconvergence.

About the complexity evaluation of large structured objects; 181–192

Tõnu Lausmaa

Abstract. An algebraic informational measure is presented for evaluating the intrinsic complexity of large structured objects on the basis of the distribution of the basic property of their elements. The method is based on the notion of partition, formed by classifying the object elements according to their property values. Partition is evaluated by the notion of extropy, which characterizes its information content. The discrete property function is extrapolated into a continuous property curve and an extropic measure called the extropy index is calculated for the object on the basis of this curve. To test the proposed method, it was applied to evaluate the structural complexity of state economies on the basis of the population income distribution. The comparison of the extropy index with the GDP index and the human development index gave the correlation coefficients of 0.77 and 0.86, respectively.

Key words: extropy, partition, structural complexity, informational measure, economy evaluation.