ABSTRACT. A computer program is introduced, which allows to determine statistically optimal approximation using the "Asymptotic Parabola" fit, or, in other words, the spline consisting of polynomials of order 1,2,1, or two lines ("asymptotes") connected with a parabola. The function itself and its derivative is continuous. There are 5 parameters: two points, where a line switches to a parabola and vice versa, the slopes of the line and the curvature of the parabola. Extreme cases are either the parabola without lines (i.e

the difference between the 0.975 and 0.025 percentiles (divided by 2*1.96) as the accuracy estimate of a given parameter in the bootstrap method. The variability of the semi-regular pulsating star Z UMa is analyzed. The presence of multicomponent variability of an object, including, four periodic oscillations (188.88(3), 197.89(4) days and halves of both) and significant variability of the amplitudes and phases of individual oscillations are shown. Approximation using the parabolic spline is only slightly better than the asymptotic parabola, for our sampling of the complete interval. It is expectedly better for larger subintervals. The use of different complementary methods allows to get a statistically optimal phenomenological approximation.

ABSTRACT. The methods for determination of the characteristics of the extrema are discussed with an application to irregularly spaced data, which are characteristic for photometrical observations of variable stars. We introduce new special functions, which were named as the "Wall-Supported Polynomial" (WSP) of different orders. It is a parabola (WSP), constant line (WSL) or an "asymptotic" parabola (WSAP) with "walls" corresponding to more inclined descending and ascending branches of the light curve. As the interval is split generally into 3 parts, the approximations may be classified as a "nonpolynomial splines".These approximations extend a parabolic/linear fit by adding the "walls" with a shape, which asymptotically corresponds to the brightness variations near phases of the inner contact. The fits are compared to that proposed by Andronov (2010Andronov ( , 2012 and Mikulasek (2015) and modified for the case of data near the bottom of eclipses instead of wider intervals of the light curve. The WSL method is preferred for total eclipses showing a brightness standstill. The WSP and WSAP may be generally recommended in a case of transit eclipses, especially by exoplanets. Other two methods, as well as the symmetrical polynomials of statistically optimal order, may be recommended in a general case of non-total eclipses.The method was illustrated by application to observations of a newly discovered eclipsing binary GSC 3692-00624 = 2MASS J01560160+5744488, for which the WSL method provides 12 times better accuracy.

We developed the software package MAVKA for the determination of characteristics of extrema (moment of extremum, magnitude) and their errors. The program realizes the application of 11 basic functions for approximation of extrema. We tested all these methods in two parts. In the first part we used generated data sets (various smooth curves with noise). We investigated deviations between generated and computed values of moments of extremum and magnitude, as well as execution time for different extrema parameters. In the second part we used real observations of different variable stars using photometric and visual observations from different databases.

ABSTRACT. We made our CCD-observations of GSC 3950-00707 by using the telescope Celestron-14 of Vihorlat Observatory and Astronomical Observatory on Kolonica Saddle. The moments of minima were calculated by using the symmetrical polynomial fit. We also analyzed the observations from automated surveys ASAS-SN and found 3 mean minima by using trigonometrical polynomial fit. The analysis of our observations and data from the surveys allows to conclude that it is the W UMa-type variable and its published period value is not accurate. We analyzed the O-C curve and corrected the elements.