Abstract:
The perturbed systems of sines, which appear when solving some partial differential equations by the Fourier method, are considered in this paper. Basis properties of these systems in weighted Sobolev spaces of functions are studied. 1. Introduction When solving many problems in mathematical physics by Fourier method (see e.g., [1–4]), there appear perturbed systems of sines and cosines of the following form: where , are real parameters. Using Fourier method requires the study of basis properties of the above systems in Lebesgue and Sobolev spaces of functions. Relevant investigations date back to the well-known works by Paley and Wiener [5] and Levinson [6]. For , basis properties of these systems in spaces , , are completely studied in [7–12]. The case of weighted was considered by E. I. Moiseev in [13, 14]. Basis properties of some perturbed systems of exponents in Sobolev spaces are studied in [15–19]. Further references include [20–23]. Our paper is devoted to the study of basis properties of these systems in weighted Sobolev spaces. Unlike previous works, we offer a different method of investigation. 2. Auxiliary Facts Let and be weighted Lebesgue and Sobolev spaces with the following norms: where , , . Denote by the following direct sum: where is a complex plane. The norm in this space is defined as follows: , where . The following easily provable lemmas play an important role in obtaining our main result. Lemma 1. Let . Then the operator performs an isomorphism between the spaces and ; that is, the spaces and are isomorphic. Proof. First we show the boundedness of this operator. We have the following: Applying Hlder's inequality, we obtain the following: Consequently, where . Let us show that . Let ; that is, where , . Differentiating this equation, we obtain a.e. on . It follows that . From (9) it directly follows that a.e. on and this implies that . Show that . Let be an arbitrary function. Assume . It is clear that and . Then by Banach theorem we find that the operator has a bounded inverse. This proves Lemma 1. Now let us prove the following lemma. Lemma 2. Let and . Then for all , where . Proof. Let ,？？ . We have the following: As and , then . Similarly, we find that and . It is easy to see that and, moreover, This proves the lemma. From results of the paper [24] it follows the validity of the following lemma. Lemma 3. Let , , , and in . Then the series converges absolutely. 3. Main Result Theorem 4. Let , . Then system (2) forms a basis for if and only if system (1) forms a basis for , where , . Proof. First let us assume that the system

Abstract:
In the present paper a criterion for basicity of exponential system with linear phase is obtained in Sobolev weight space . In solving mathematical physics problems by the Fourier method, there often arise the systems of exponents of the form where and are continuous or piecewise-continuous functions. Substantiation of the method requires studying the basis properties of these systems in Lebesgue and Sobolev spaces of functions. In the case when and are linear functions, the basis properties of these systems in , , were completely studied in the papers [1–9]. The weighted case of the space was considered in the papers [10, 11]. The basis properties of these systems in Sobolev spaces were studied in [12–14]. It should be noted that the close problems were also considered in [15]. In the present paper we study basis properties of the systems (1) and (2) in Sobolev weight spaces when , , where is a real parameter. Therewith the issue of basicity of system (2) in Sobolev spaces is reduced to the issue of basicity of system (1) in respective Lebesgue spaces. Let and be weight spaces with the norms respectively, where , . Denote by the direct sum , where is the complex plane. The norm in this space is defined by the expression , where . The following easily provable lemmas play an important role in obtaining the main results. It holds the following. Lemma 1. Let , ; . Then the operator realizes an isomorphism between the spaces and ; that is, the spaces and are isomorphic. Proof. At first prove the boundedness of the operator . We have Having applied the Holder inequality, hence we get where Consequently where Let us show that . Put ; that is, where , . By differentiating this equality, we get , a.e. on . Hence it follows that . From (11) it directly follows that a.e. on , and so . Show that ( is the range of values of the operator ). Let be an arbitrary function. Let . It is clear that and . Then from the Banach theorem we get that the operator is boundedly invertible. The lemma is proved. The following lemma is also valid. Lemma 2. Let and , . Then for all , where Proof. Let , . We have Since and , then , . It is easy to see that and moreover . The lemma is proved. In obtaining the basic results we need the following main lemma. Lemma 3. Let , and , , be a real parameter, . Let have the expansion in the space . Then it is valid Proof. As it follows from Lemma 2, . At first consider the case when , . In this case the system is minimal in (see [4]). Then from the results of the paper [16], the Hausdorff-Young inequality is valid for this system; that is,

Abstract:
The chemical and microbiological hydrolyses of epoxide compounds of acetylene series have been comparatively carried out. It has been shown that as distinguished from chemical one in microbiological hydrolysis along with corresponding optically active glycols, ketoalcohols of acetylene series are also formed. It has been also defined that the synthesized glycols of acetylene series have bactericide properties related to sulfate-reducing bacteria at concentration of 100 - 200 mg/l. It has been established that an introduction of electron-acceptor chlorine atoms in molecules influence on decreasing of bactericide activity of acetylene glycols.

Abstract:
The methods of preparation of endiine and endiallene diols by interaction of cis-1,4-dibrombutene and cis-1,4-dichlor-rbutene with monosubstituted acetylene alcohols in presence of the catalytic systems consisting of one-iodide copper, triethylamine and K_{2}СО_{3} in a medium of dimethylformamide have been developed. It has been shown that unlike 1,4-dibrombutene, the nucleophilic substitution reaction with 1,4-dichlorbutene proceeds by acetylene-allene isomerization with formation of endiallene diols. It has been established that the endiine diols can be used in thin organic synthesis (in the reactions of oxidation, splitting, dehydration, epoxidation, hydrolysis, 1,2-cycloaddition and hypochlorination) with the aim of preparation of practically useful substances. It has been revealed during hydrolysis of epoxide compounds by the chemical and microbiological methods that in the course of microbiological hydrolysis (Aspеrgillus niger), the optically active trans-structured diols are formed.

Abstract:
The properties of a ball-shaped semiconductor particles and metal particles with a semiconductor thin film on the surface thereof are established. So the dimensionless thermoelectric figure of merit of a material consisting of a large number of these particles is equal to 10 - 100.

Abstract:
A task of mapping a hexagonal grid to different types of helical surfaces including nanocones, nanotubes and nanoscrolls by unfolding a given surface to a carbon layer plane has been solved. Basing on these models, polyhedric models with all atomic bonds being constant and equal to 1.42? as in a flat carbon layer have been built, and an algorithm of coloring all faces of such models has been developed. Received models can be utilized for visual demonstration of the helical growth of nanotubes, nanocones, nanofiber and other nanoobjects, and also for physical properties calculation.

Abstract:
The paper presents the results of experimental studies of the effect of irradiation with electromagnetic waves terahertz frequencies of molecular spectrum of emission and absorption of nitric oxide and atmospheric oxygen on animals in a state of acute and chronic immobilization stress, a condition regarded as a model of violations in different parts of the vascular system including microcirculation specific for different parts of the vascular system, including the microcirculation characteristic of various diseases of the cardiovascular system.

Abstract:
Many papers have used fluorescent probe diffusion to infer membrane viscosity but the measurement is actually an assay of the free volume of the membrane. The free volume is also related to the membrane tension. Thus, changes in probe mobility refer equally well to changes in membrane tension. In complicated structures like cell membranes, it appears more intuitive to consider variations in free volume as referring to the effect of domains structures and interactions with the cytoskeleton than changes in viscosity since tension is a state variable and viscosity is not.

Abstract:
Sex determination system in birds is characterized by a homo-(Neognatae) and heteromorphic (Paleognatae) sex chromosomes. Heterogametic sex is female (ZZ/ZW system). DMRT1 gene is a gene regarded as a main male sex determining factor in this group of animals. The question remains about the participation of other factors (HEMOGEN, AMH etc.) in appearance of testis, and the role of steroid hormones in formation of ovaries. Complete sex inversion is not typical for species with genotypic sex determination (GSD), although the effect of estrogen metabolites is noted for birds. For birds epigenetic mechanisms of regulation (methylation of DNA and non-coding RNA) have been described for sex controlling genes such as CYP19A1 and DMRT1.

Abstract:
In this paper, we discuss estimating anisotropy of air density irregularities (ratio of characteristic horizontal and vertical scales) from satellite observations of bi-chromatic scintillations of a double star whose components are not resolved by the detector. The analysis is based on fitting experimental auto- and cross-spectra of scintillations by those computed using the 3-D spectral model of atmospheric irregularities consisting of anisotropic and isotropic components. Application of the developed method to the scintillation measurements of the double star α-Cru by GOMOS (Global Ozone Monitoring by Occultation of Stars) fast photometers results in estimates of anisotropy coefficient of ~15–20 at altitudes 30–38 km, as well as other parameters of atmospheric irregularities. The obtained estimates of the anisotropy coefficient correspond to small-scale irregularities, close to the buoyancy scale.