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