Physical applications of second-order linear differential equations that admit polynomial solutions

Conditions are given for the second-order linear differential equation P3 y" + P2 y'- P1 y = 0 to have polynomial solutions, where Pn is a polynomial of degree n. Several application of these results to Schroedinger's equation are discussed. Conditions under which the confluent, biconfluent, and the general Heun equation yield polynomial solutions are explicitly given. Some new classes of exactly solvable differential equation are also discussed. The results of this work are expressed in such way as to allow direct use, without preliminary analysis.Comment: 13 pages, no figure

Observations of Lensed Relativistic Jets as a Tool of Constraining Lens Galaxy Parameters

The possibility of using lensed relativistic jets on very small angular scales to construct proper models of spiral lens galaxies and to independently determine the Hubble constant is considered. The system B0218+357 is used as an example to illustrate that there exists a great choice of model parameters adequately reproducing its observed large-scale properties but leading to a significant spread in the Hubble constant. The jet image position angle is suggested as an additional parameter that allows the range of models under consideration to be limited. It is shown that the models for which the jet image position angles differ by at least $40^o$ can be distinguished between themselves during observations on very small angular scales. The possibility of observing the geometric properties of lensed relativistic jets and measuring the superluminal velocities of knot images on time scales of several months with very long baseline space interferometers is discussed.Comment: 11 pages, 3 figures, Will be published in the Astronomy Letters, V.37, PP.483-490, 201

Modeling the Images of Relativistic Jets Lensed by Galaxies with Different Mass Surface Density Distributions

The images of relativistic jets from extragalactic sources produced by gravitational lensing by galaxies with different mass surface density distributions are modeled. In particular, the following models of the gravitational lens mass distribution are considered: a singular isothermal ellipsoid, an isothermal ellipsoid with a core, two- and three-component models with a galactic disk, halo, and bulge. The modeled images are compared both between themselves and with available observations. Different sets of parameters are shown to exist for the gravitationally lensed system B0218+357 in multicomponent models. These sets allow the observed geometry of the system and the intensity ratio of the compact core images to be obtained, but they lead to a significant variety in the Hubble constant determined from the modeling results.Comment: 26 pages, 9 figures, will be published in the Astronomy Letters, 2011, v.37, N4, pp. 233-24

Results of study of long-fiber flax’s collection material by the parameters of economically valuable traits

Cultivated common flax (Linum usitatissimum L.) is a traditional Russian technological crop of complex usage. Estimation of seasonal effects in economically valuable traits (productivity of straw and seeds; content of fiber and its quality) of long-fiber flax under conditions of Volga-Vyatka region (Kirov region) in 2014 - 2017 is presented in the article. Objects of study - 140 varieties of long-fiber flax different in ecological and geographic origin. Stable and plastic traits are selected according to correlation coefficient. Strong “genotype - environment” interactions were expressed by seed productivity, number of bolls and seed per plant; moderate interactions - by productivity of straw and fiber; weak interactions - by plant length, fiber content, 1000-grain mass. Strong correlation is pointed out between straw productivity and total and technical plant height (r = 0.71±0.09.. ,0.78±0.08) in average for years of study. Sources are selected for most important directions of long-fiber flax breeding: for straw productivity - Sinel, Vizit (Russia). Merilin (The Netherlands Soglasie (Belarus), Heiya 11, Heiyal2 (China); for seed productivity - Priboj, Sinichka, Er-27 (Russia), Ottawa (Canada), Soglasie (Belarus), Belinka, Merilin (The Netherlands); for fiber content- Mirazh, Peresvet, Tverskoj, Pskovsky-93, Dobrynya (Russia); Merilin (The Netherlands); Charivny, Zaryanka (Ukraine); for fiber durability - Soglasie (Belarus), Sinel, Pskovsky-93 (Russia), Kamenyar, Baltuchaj (Ukraine), B-179 (Lithuania); for fiber flexibility - Flax of Heilonjiang № 7, Huaguang 2, Heiya 11, Heiya 13 (China); Honkei-25 (Japan), Kamenyar (Ukraine), Priboj (Russia). By a complex of economically valuable traits the following genotypes are selected: Sinel, TOST 3, Peresvet (Russia), Flax of Heilonjiang № 13 and Heiya 12 (China), Merilin (The Netherlands)

Evaluation of technological and bakery properties of winter rye in dependence on predecessor in adaptive-landscape crop farming

Results of perennial studies on influence of predecessors on grain quality parameters of winter rye Kirovskaya 89 and Falenskaya 4 are presented in the article. Winter rye was sowed after such kinds of fallow as pure, manure, green manure (lupine, melilot, red clover, annual grasses), occupied (field pea + oat; radish + vetch + oat; clover). It was established that predecessor could influence on grain quality of winter rye due to its impact on agro-physical, biological properties of soil, and phyto-sanitary state of sowings. Test weight of grain varied by years and by predecessors in Kirovskaya 89 from 564 up to 707 g/l; in Falenskaya 4 - from 650 up to 710 g/l. At the average for years of investigations, both varieties formed high-test weight grain according to State Standard. Grain of variety Kirovskaya 89 belongs to II class of quality on a base of parameter “falling numbef’ at any predecessor - its grain does not demand subgrading (bread of good quality). Variety Falenskaya 4 forms grain of I class in majority of years and was characterized as improver for subgrading for grain with low qualities. 1000 grain mass in variety Kirovskaya 89 was significantly higher at green manure fallow, in variety Falenskaya 4 - at pure fallow. Protein content was low than 11.5% at any predecessor and type of fallow that does not lowered bakery properties of flour. Total bakery value of variety Kirovskaya 89 at pure fallow was 3.09 points, increased at green manure, and occupied fallows up to 3.51.3.57 points

Estimation of the mass of the very massive galaxy cluster SRGe CL2305.2-2248 from strong lensing

SRGe CL2305.2–2248 (SPT-CL J2305–2248, ACT-CL J2305.1–2248) is one of the most massive galaxy clusters at high redshifts (z≃0.76) and is of great interest for cosmology. Deep images have been obtained at the Russian–Turkish 1.5-m telescope (RTT-150) for an optical identification of this cluster. In combination with the open Hubble Space Telescope archival data, they have made it possible to select candidates for gravitationally lensed images of distant blue galaxies in the form of arcs and arclets. The observed giant arc near the brightest cluster galaxies allows the Einstein radius to be estimated, 9.8±1.3 arcsec. The photometric redshift of the lensed source has been found (zs=2.44±0.07). Its use in combination with the Einstein radius estimate has made it possible to independently estimate the mass of SRGe CL2305.2–2248 by extrapolating the strong-lensing results to large radii and using model density profiles in relaxed clusters. This extrapolation leads to mass estimates smaller than those obtained from X-ray and microwave observations by a factor of ∼1.5−3. A probable cause of this discrepancy may be the galaxy merger process, which is also confirmed by the morphology of SRGe CL2305.2–2248 in the optical range

Spectroscopic redshift measurements for galaxy clusters from the Lockman Hole survey with the eROSITA telescope onboard the SRG observatory

We present the first results of our program of optical observations for galaxy clusters from the Lockman Hole X-ray survey with the eROSITA telescope onboard the SRG space observatory. We present the results of spectroscopic redshift measurements for 11 galaxy clusters that have been identified optically among the extended X-ray sources of the SRG/eROSITA survey using data from optical and infrared sky surveys. The spectroscopic observations were carried out in late 2019–early 2020 with the 1.6-m AZT-33IK telescope of the Sayan Observatory and the 6-m SAO RAS telescope (Bolshoi Teleskop Azimutalnyi, BTA)