CONTENTS &
ABSTRACTS

In English. Summaries
in Estonian

Proceedings of the
Estonian Academy of Sciences.

Physics *
Mathematics

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Volume 49 No. 4
December 2000

**2-Semiparallel surfaces in space forms. 2.
The general case; 203–214**

Ülo LUMISTE

**Abstract. **A surface in *n*-dimensional space form is said to be
2-semiparallel if _{} All such surfaces are
classified (Theorem 6) and geometrically described. Also their existence is
established, together with their arbitrariness in some subclasses (Proportions
4 and 5).

**Key words:** 2-semiparallel
surfaces, locally Euclidean surfaces.

**Numerical computation of weakly
singular integrals; 215–224**

**Enn TAMME**

**Abstract.** Numerical evaluation of weakly
singular integrals with composite quadrature rules on graded grids is
considered. The dependence of the error of the quadrature rule on nonuniformity
of the grid is studied. The conditions under which a quadrature rule with
nonuniform grid converges in case of a singular integrand with the same rate as
the same rule with uniform grid does in case of a smooth integrand are
discussed. Theoretical results are verified by numerical examples in case of
the composite Gaussian quadrature rule and composite Simpson’s rule.

**Key words:** weakly singular
integral, composite quadrature rule.

**Two Tauberian remainder theorems for the Cesàro method of
summability; 225–232**

Ivar TAMMERAID

**Abstract.** The Tauberian remainder theorems
for the Cesàro method of summability are studied using two different methods
of the proof. Theorem 1 is proved by applying the method of summability with a
given rapidity. Theorem 2 is proved using several consequences of one basic
Wiener’s Tauberian remainder theorem proved by A. Beurling. Both theorems
are connected with the hypothesis of G. Kangro (see Tammeraid, I. *Metody algebry i anal*iza, III. 1988,
113–114).

**Key words:** Tauberian
remainder theorems.

**Exactly solvable potential with a pronounced
maximum; 233–250**

Matti SELG

**Abstract.** An analytic method for solving the
one-dimensional Schrödinger equation is applied to confining potentials with
well-defined maxima. Exactly solvable substitute for the original potential is
constructed, which consists of several smoothly joined Morse-type (ordinary,
reversed, or pseudo-Morse) components. The analytic procedures of calculating
the complete spectrum of the bound and quasi-bound states as well as the
corresponding stationary wavefunctions are described, and an exact formula for
the wavefunction’s phase shift is derived. The efficacy of the approach is
demonstrated on the rotationally highly excited excimers _{} in the electronic state _{}

**Key words:** exactly solvable
potentials, quasi-bound states, phase shift.

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**Ultraviolet irradiance in meteorologically
contrasting summers of 1998 and 1999 in Estonia; 251–265**

Kalju EERME, Uno VEISMANN, and Rutt KOPPEL

**Abstract. **Regular monitoring of
solar erythemal ultraviolet (UV) irradiance at Tõravere (Estonia) was performed
in two years with notably different weather conditions in summer (herein the
months May–August). Erythemal UV characteristics were compared and their
dependence on meteorological conditions was studied. The erythemal doses over
the first half of the summer were similar for both years, but over the second
half the dose for 1999 was 28% higher. The strongest affecting factor in both
summers was low cloudiness. Its partial correlation with the daily UV doses in
the extremely cloudy summer of 1998 (– 0.70) was not
much higher than in the fine weather summer of 1999 (– 0.65)
when the noon solar elevation contributed two times more than in cloudy 1998
(about 0.6 and 0.3). The contribution of total ozone was close to zero in 1998
and quite weak (– 0.44) in 1999. The daily distributions of
erythemal UV index values in case of moderate partial cloudiness often give
higher values than in clear sky conditions and the daily doses are higher.

**Key words: **erythemal UV
irradiance, atmospheric total ozone, UV monitoring, low cloudiness, snow cover,
UV dose, UV index, albedo.

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