The problem of characterization of bilinear inequality in discrete form is solved.

Two transforms of functions on a half-line are considered. It is proved that their composition gives a concave majorant for every non-negative function. In particular, this composition is an identical transform on the class of non-negative concave functions. Applications of this result in the operator theory of Hilbert space and in the theory of quantum systems are mentioned. Several open problems are formulated.

The problem of the detonation formation as a result of unconfined flame propagation is solved numerically. The mechanism of detonation formation is distinguished. It is related to the local formation of shock waves du- ring the linear stage of development of flame front perturbations formed on the surface of the expanding flame front. General criteria of the establishment of the conditions for the detonation transition via the proposed mechanism are formulated.

Regularities and mechanisms of coalescence of Au nanodroplets and sintering of solid Au nanoparticles have been investigated by using molecular dynamics and some theoretical models. It has been established that the characteristic time of coalescence t is proportional to radius r0 of initial nanodroplets. Both the above conclusion and some quantitative estimations of the proportionality coefficient between t and r0 agree with Frenkel’s theory (1946) though this theory was rut forward to describe coalescence of macroscopic droplets.

The method allows determining the mechanical characteristic of nanoobjects was presented. A heterogeneous material consisting of a nanophase and a binder phase was considered, the mass and volume concentrations of components were given. Heterogeneous material is reduced to homogeneous by averaging methods while the mechanical characteristics will be associated with averaged ones. Assuming that the mechanical characteristics of the binder and averaged homogeneous materials are known from mechanical tests, the system of equations allow us to determine the mechanical characteristics of nanoobjects included in this heterogeneous material. It is believed that the mechanical characteristics of bonding and averaged homogeneous materials make it possible to obtain equations of equations that allow one to determine the mechanical characteristics of nano-objects present in this heterogeneous material. Classical mechanical tests were carried out, describing the uniaxial stress and strain states of materials, which made it possible to obtain an analytical form the dependences of the mechanical characteristics of nanophases depending on their size. Specific examples are given for silica dioxide nanoparticles (Aerosil and Tarkosil powders).

Oligomers of cholesterol vinyl ether (CVE) with methyl vinyl sulfide synthesized in the presence of cationic catalyst ВF3 · OEt2 in up to 81% yield (Mn up to 5700) demonstrate optical activity exceeding that of CVE homopolymer, which is probably due to both conformational effects of the oligomer chain and chirality induction during the polymerization.

The creation of methods for complete and cost-effective collection of water droplets from an aerosol which arises as a by-product of the low-potential heat uptake from industrial devices, is one of the key tasks of rational use of water resources contributing to the improvement of the environment near large industrial enterprises. This paper shows how the application of materials with extreme wettability and a specific surface topography in spray separators can significantly increase the efficiency of water collection.

U-Pb age, oxygen and hafnium isotopic ratios in zircon from rocks of ocean core complexes at Mid-Atlantic Ridge have been studied using SHRIMP and MC-LA-ICP-MS techniques. U-Pb dating revealed four group of zircons: 1) 0,6-1,7 Ma, 2) 6,7-11,2 Ma, 3) 12,9-17,6 Ma, 4) ≥200 to 2044 Ma. The δ18O values range from 4,74 to 7,2‰ and are distinct for zircon grains of different ages. Hafnium isotopic ratio for zircon aged from 0,6 to 17,6 Ma corresponds or is close to that of MORB from Central Atlantic. The oxygen and hafnium isotopic compositions of zircon elder than 280 Ma correspond to those of the sialic continental crust. A hypothesis of involvement of the ancient “pre-Atlantic” sialic (≥280 млн лет) and “old Atlantic” (7-17 Ma) crusts in a generation of the contemporary (“young”) oceanic crust during formation of the slow-spreading Mid-Atlantic Ridge has been proposed.

The first results on the dissolution of flat-faced diamond crystals of octahedral habit in Fe-Ni-S melt at 3,5 GPa and 1400 °С are presented. It has been established that as a result of dissolution, flat-faceted diamond crystals are transformed into curve-faced individuals with morphological features similar to kimberlite diamonds. It is concluded that similar forms of natural diamonds could have been formed in reducing domains of the Earth’s mantle before entering the kimberlite magma.

A scenario is proposed for replenishing the Earth’s exosphere with atomic hydrogen of cosmic origin. An assessment was made and the coincidence of the total atomic hydrogen content in the exosphere with the number of protons (after thermolization converted into hydrogen ions) precipitated in the SAA zone during the year according to the data of the Meteor M and NOAA‑19 satellites was confirmed. The observed coincidence indicates that the rates of replenishment of hydrogen due to precipitation from radiation belts and dissipation into outer space coincide in order of magnitude. It is concluded that the exosphere hydrogen is mainly of cosmic origin and its main source is the thermalized protons of galactic cosmic rays, solar cosmic rays and partially solar wind.

The lithalsa, located on the first terrace of the Sentsa River in Okinsky district of the Republic of Buryatia, were studied. The most typical features of isotopic diagrams, obtained from the ice core from the 20-meter borehole, is their stage structure: three isotope minima at depths of approximately: 3-5, 9-12, and 18-21 m are separated by two distinct isotope maxima. Most likely, this is due to the cycle of the triple flooding of the growing permafrost mound and the subsequent active evaporation of the lake water - ​the main source of water in the ice core of lithalsa. Based on the detailed radiocarbon dating of organic material from the upper layers of the highest and surrounding lithalsas, the time of their formation was established. The active accumulation of organic material in dried heaving soils occurred from 0,5 to 0,2 ka BP. This is the time of the beginning of the formation of permafrost, active ice formation and the associated heaving process and lithalsa growth. The age of at least 3 of 4 studied mounds is not older than 200 years, it coincides with the cooling in the early 19th century.

A new algorithm is used for detailed numerical study of the evolution of isotropic swell in a homogeneous ocean. It is shown that the Zakharov-Filonenko spectrum occurs in an explosive manner in a short time. The Kolmogorov constant of the solution is estimated numerically.

The present study showed that the recombinant analogue of SLURP‑1 effectively inhibits the growth of a 3D model of tumors - multicellular spheroids from human epidermoid carcinoma A431 cells and human lung adenocarcinoma A549 cells. The combined application of SLURP‑1 with gefitinib (inhibitor of epidermal growth factor receptor (EGFR)) leads to the synergistic antiproliferative effect on spheroids from A431 cells. The results obtained suggest the possibility for design of first-in-class anticancer drugs based on recombinant SLURP‑1.

Nigrostriatal dopaminergic neurons (DNs) involved in the regulation of motor function, are characterized by high plasticity. Indeed, at the death of up to 50% of DNs in Parkinson’s disease, the survived neurons provide normal regulation. This study was aimed to determine whether proteins of the vesicular cycle, syntaxin Ia (Syn Ia), synaptotagmin I (Syt I), Rab5a and complexins I and II (Cmpx I and II), are involved in the mechanisms of neuroplasticity in the substantia nigra, which mainly contains cell bodies and processes of the DNs. In neurotoxic models of Parkinson’s disease in mice, it was shown that at the degeneration of up to 50% of DNs, the content of Syt I, Syn Ia, Cmpх I and II involved in vesicle exocytosis does not change in the substantia nigra as whole, while this is compensatory increased in individual survived DNs. Thus, the data obtained in this study suggest that the impairment of motor behavior that occurs at the death of half of nigrostriatal DNs is not caused by the impairment of the production of vesicular cycle proteins in surviving DNs.