In English. Summaries in Estonian

Proceedings of the Estonian Academy of Sciences

Physics * Mathematics


Volume 49 No. 1 March 2000


On a function of reducibility of a class of four-dimensional semiparallel submanifolds; 3–11


Abstract. Semiparallel submanifolds Mm in Euclidean space En  are the second-order envelopes of symmetric orbits, which are determined by the system  of differential equations with integrability conditions  . The last system characterizes the semiparallel submanifold Mm in En. In the paper a M4 function describing the reducibility properties of the second-order envelopes  of a normally non-flat, reducible symmetric orbit  with a Veronese component, which is a Veronese surface in E5, is introduced and some geometrical properties of the M4 are pointed out.

Keywords: semiparallel submanifolds, symmetric orbits, Veronese surfaces.


Wavelet coefficients of functions of generalized Lipschitz classes; 12–20


Abstract  The connections between the smoothness of a function in the neighbourhood of a given point and its wavelet coefficients are studied. The results presented here are localized versions of the main result earlier obtained by J. Lippus [Sampling Theory and Applications (Marvasti, F. A., ed.). Riga, 1995, 167–172].

Keywords: wavelet coefficients, local smoothness.


On the construction of smoothing splines by quadratic programming; 21–27


Abstract. The problem of minimization of a smoothing functional under inequality constraints, which has a solution in the form of a natural spline, is reduced to the problem of quadratic programming with a positive semidefinite matrix. Using the results of quadratic programming, we obtain the modified simplex method for the solution of this problem by adding-removing interpolating knots of a spline.

Key words: smoothing spline, quadratic programming, simplex method.


Trichotomous-noise-induced phase transitions for the stochastic Hongler model; 28–39


Abstract. The effect of environmental instability, in the form of a three-level Markovian noise, on the Hongler system is calculated. An explicit formula for the stationary probability distribution is obtained. The well-known  dichotomous noise can be regarded as a special case of the trichotomous noise. As a rule, the system variable has three specific values where the probability density distribution can be singular. The dependence of the behaviour of the stationary probability density on the noise parameters is investigated in detail and illustrated by a phase diagram.

Key words: open systems, stochastic Hongler model, environmental variance, random telegraph process, phase transitions.


Temperature distribution in a semi-infinite atmosphere subjected to cosine varying collimated radiation; 40–57


Abstract. Accurate numerical solutions are presented for the radiation field in a semi-infinite, two-dimensional, plane-parallel, absorbing-emitting but nonscattering grey atmosphere subjected to cosine varying collimated incident boundary radiation. The kernel of the integral equation for the emissive power is approximated by a sum of exponents. After this approximation the integral equation can be solved exactly. The solution contains the well-known Ambarzumian–Chandrasekhar H-function. Some methods to determine this function are considered in detail.

This approach allowed of finding the accurate values for the emissive power and the radiative flux at arbitrary optical depths in the atmosphere. The calculations show that the radiative flux may have a maximum at certain values of the spatial frequency in the atmosphere and that the region where the emissive power reaches a constant value may lie very deep in the atmosphere.

Keywords: two-dimensional radiative transfer, H-function, emissive power, radiative flux.


Instructions to authors; 58–62

Copyright Transfer Agreement; 63–64