CONTENTS &
ABSTRACTS

In
English. Summaries in Estonian

Proceedings of the Estonian Academy of Sciences.

Physics * Mathematics

Volume 54 No. 4
December 2005

** **

On
additive generalization of Voronoï’s theory to algebraic number fields; 195–211

Alar Leibak

**Abstract.** A generalization of Voronoï’s theory to perfect quadratic
form over algebraic number fields is studied. This generalization follows
Koecher’s idea (see *Math. Ann*., 1960, **141**, 384–432) of using the
minimums of Tr*f** *(*X*) for the
positive definite quadratic form *f** *(*X*). As a result some useful properties of perfect quadratic
forms are presented and the upper and lower bounds of Hermite’s constant are
proved.

**Key words: **perfect quadratic forms, algebraic number fields.

The
complete enumeration of binary perfect forms over the algebraic number field Q_{}; 212–234

Alar Leibak

**Abstract.** A complete list of binary perfect forms with coefficients in Q_{} is presented and the theoretical background of
Voronoï's algorithm of perfect quadratic forms with coefficients in a
totally real normal algebraic number field is discussed.

**Key words: **perfect quadratic forms, Voronoï's algorithm, totally real
normal algebraic number fields.

Properties
of 2-dimensional time-like ruled surfaces in the Minkowski space _{}; 235–242

Murat Tosun, Ismail Aydemir,
and Nuri Kuruoglu

**Abstract.** Some results, which are well known for the ruled surfaces in
the Euclidean space _{} are generalized here to the case of _{} In particular, it is shown that a time-like ruled surface in _{} is developable if and only if it has zero Gaussian curvature;
moreover, it is then minimal if and only if it is totally geodesic.

**Key words:** ruled surfaces, Minkowski spaces.

A
new approximation to the variance of the ANOVA estimate of the intraclass
correlation coefficient; 243–254

Tanel Kaart

**Abstract. **The simple random one-way linear model and ANOVA estimators of the
intraclass correlation coefficient are examined. New approximation for the
sampling variance of the intraclass correlation coefficient is derived and its
minimum, corresponding to a balanced data set, is established. The theoretical
results are checked with simulation experiments. In addition, the effect of
data set imbalance and structure on the accuracy of intraclass correlation
coefficient estimators is studied by modelling.

**Key words:** variance analysis, intraclass correlation coefficient,
heritability, sampling variance, data imbalance, optimal design.

Contents of volume 54; 255–256