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 Trf (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