CONTENTS &
ABSTRACTS

In English. Summaries in Estonian

Proceedings of the Estonian Academy of Sciences.

Physics * Mathematics

** **

Volume 50 No. 1
March 2001

**Editorial;
3–4**

**Summation
methods of trigonometric Fourier series defined by the Zak transform; 5–15**

Andi KIVINUKK and Julia VERHOVITÒ

**Abstract.** We investigate the order of approximation by some new-type
summation methods of trigonometric Fourier series. These methods are defined by
the Zak transform.

**Key words:** trigonometric Fourier series, Zak transform,
sinc function, modulus of continuity, Lanczos’ filter.

**Parallel
and semiparallel space-like surfaces in pseudo-Euclidean spaces; 16–33**

Elena SAFIULINA

**Abstract.** Parallel submanifolds in
pseudo-Euclidean spaces are characterized locally by the system _{} Submanifolds
satisfying the integrability condition _{} of this system are
called semiparallel; geometrically they are 2nd-order envelopes of the parallel submanifolds. The existence and
geometry of such two-dimensional Riemannian submanifolds (surfaces) are
investigated and their complete classification is given. Moreover, it is shown
that in _{} with _{} do exist not totally geodesic minimal
semiparallel space-like surfaces.

**Key words:** parallel surfaces, semiparallel surfaces, space-like surfaces,
minimal surfaces.

**Application
of orthogonal transformations in the revised simplex method; 34–41**

Evald ÜBI

**Abstract.** Unlike the classical method, orthogonal transformations are used for
computing more precise simplex tables. These transformations are stored as
products. With these transformations the basic matrix is turned to the
triangular form, which is used instead of the reciprocal matrix. The values of
initial and dual variables are found from the triangular system. The performed
algorithm MULT is appropriate for solving ill-conditioned problems and it is
more labour-consuming than the classical revised simplex method. Examples of
solving the ill-conditioned problems are given.

**Key words:** linear programming, least squares method, ill-conditioned problems.

**General
solution of a system of differential equations modelling a class of
exactly-solvable potentials; 42–48**

Axel SCHULZE-HALBERG

**Abstract.** We obtain the general solution of a system of differential
equations introduced by Ge et al. (*Phys.
Rev. A*, 2000, **62**,
052110–052117). This solution yields a class of exactly-solvable potentials and
can be used to calculate the ground state for the class of these potentials.

**Key words:** Schrödinger equation,
exactly-solvable potential.

**Numerical
stability in a model of the layered structure of thermohaline fields; 49–62**

Peep MIIDLA, Kalev RANNAT, and Jaak HEINLOO

**Abstract. **The paper presents a
set-up and the stability criteria for numerical realization of the model of the
vertical structure of thermohaline fields caused by double diffusion effects.
The model considers the turbulence accompanying the environment layering
process as rotationally isotropic in the sense suggested by the theory of rotationally
anisotropic turbulence. It is formulated through the set of three nonlinear
differential equations for the environment’s temperature, salinity, and
turbulence kinetic energy. The numerical realization of the model is based on
an implicit difference method and the uniform rectangular time-space grid. The
numerical stability criterion of the algorithm for the model’s approximated
solution is estimated by the maximum values of the solutions of system
components.

**Key words: **turbulence, double
diffusion, numerical stability.

** **

CHRONICLE

**Estonian
Academy of Sciences: meeting the challenges for 2000–2004.**

**Framework of
activities; 63–64**