CONTENTS &
ABSTRACTS

In
English. Summaries in Estonian

Proceedings of the Estonian Academy of Sciences.

Physics * Mathematics

** **

Volume 53 No. 3
September 2004

On the linear spline collocation for
pseudodifferential equations on the torus; 139–147

Juha Anttila, Jyri Hämäläinen, and Jukka Saranen

**Abstract.** We examine the spline collocation method for a class of
pseudodifferential equations on a two-dimensional torus. In the analysis, we
assume nonuniform mesh, continuous piecewise linear splines, and nodal point
collocation. By employing the “Arnold–Wendland trick”; we are able to carry out
the stability and convergence analysis. The results show quasioptimal order
estimates for the convergence of the collocation solution.

**Key words:** boundary elements,
collocation.

On a method of the construction of smoothing
histosplines;
148–155

Natalia
Budkina

**Abstract.** The problem of the approximation of a given histogram by a
function from Sobolev space under inequality constraints for area matching
conditions is considered. The smoothing problem is reduced to the problem of
linear programming with some nonlinear restrictions.

Key words: smoothing
problem, histospline simplex method.

Pseudodifferential calculus on the 2-sphere; 156–164

Ville Turunen

**Abstract.** We show how pseudodifferential equations on the unit
sphere of the 3-dimensional Euclidean space can be studied using the spherical
harmonic Fourier series on the symmetry group of the sphere.

**Key words: **pseudodifferential
operations, symbol calculus, asymtotic expansions, spherical harmonics rotation
group.

A
convergence theorem for approximate methods of tangent hyperbolas; 165–176

Indrek Kaldo and Otu Vaarmann

**Abstract.** For solving an operator equation *F*(*x*) = 0, where
*F* is a nonlinear operator from a Banach space *X* into another
Banach space *Y*, approximate variants of the method of tangent hyperbolas
are developed, provided *F* is twice Frechet-differentiable and its first
derivative has the uniformly bounded inverse. A local convergence theorem is
provided for the methods under consideration and their computational aspects
are briefly discussed.

**Key words:** nonlinear equation, Banach space, Newton’s method, cubically
convergent method, approximate variants of methods, midpoint method.

Optimization
problems with points of discontinuity and discrete arguments; 177–185

Ants Tauts

**Abstract. **Minimization of such
functions is considered, where some arguments are related to the final function
by intermediate functions with discontinuity points, but other arguments have
only 0 and 1 for the allowed values, although the theoretical generalization
allows also intermediate values. Both of the circumstances create difficulties
in the use of the gradient method. We solve the first problem by approximation,
primarily by a square polynomial obtained using the integral form of the least
squares method, and later by the partial sums of orthogonal series of the wave
function treated with the logarithmic averages method. The second problem can
be solved with the help of the planes, which have been taken in the *n*-dimensional
space in such a way that any allowed point on the side of the space relative to
this plane is better than all the points on the other side.

**Key words:** optimization, discontinuity points, discrete values, least
squares method, wave functions, logarithmic average.

On the description of stochastic systems; 186–200

Jaak Heinloo

**Abstract.** A set-up of the systemic
description of a (stochastic) system, particular states of which form (on the
lowest determination level of the system state) a population of random events,
is presented. Elementary and hierarchic stochastic systems are considered and
the structure of their description is formulated. As an application of the
conception, the set-up of the description of liquid media motion is considered.

**Key words:** probability, system,
liquid media.