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.