**CONTENTS
& ABSTRACTS**

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

Kaarin RIIVES

**Abstract. **Semiparallel
submanifolds *M ^{m }*in
Euclidean space

**Keywords:** semiparallel submanifolds, symmetric
orbits, Veronese surfaces.

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

Jüri LIPPUS

**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

Natalia BUDKINA

**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

Romi MANKIN, Ain AINSAAR, and Astrid HALJAS

**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

Tõnu VIIK

**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