In English. Summaries in Estonian

Proceedings of the Estonian Academy of Sciences.

Physics * Mathematics


Volume 49 No. 4 December 2000


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


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


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


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 analiza, 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.



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


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.