The classical Farkas theorem of the alternative is considered, which is widely used in various areas of mathematics and has numerous proofs and formulations. An entirely new elementary proof of this theorem is proposed. It is based on the consideration of a functional that, under Farkas’ condition, is bounded below on the whole space and attains a minimum. The assertion of Farkas’ theorem that a vector belongs to a cone is equivalent to the fact that the gradient of this functional is zero at the minimizer.

We deduce sufficient maximality conditions for graph surfaces on two-step Carnot groups with sub-Lorentzian structure.

In this paper we consider a completely Liouville integrable Hamiltonian system with two degrees of freedom, which describes the dynamics of two vortex filaments in a Bose-Einstein condensate enclosed in a harmonic trap. For vortex pairs of positive intensity detected bifurcation of three Liouville tori into one. Such bifurcation was found in the integrable case of Goryachev-Chaplygin-Sretensky in the dynamics of a rigid body. For the integrable perturbation of the physical parameter of the intensity ratio, identified bifurcation proved to be unstable, which led to bifurcations of the type of two tori into one and vice versa.

In this communication, a condition based on the deactivation of the trigger mechanism of creep is proposed for the liquid-glass transition of an amorphous substance. This mechanism is confined to the atom delocalization process, which in silicate glasses represents the ultimate displacement of the bridging oxygen atom in the Si-O-Si bridge due to the local low-activation elastic strain of the silicon-oxygen network.

The formulation of problems on the equilibrium of a nonlinearly elastic solid with continuously distributed dislocations is proposed for the case of superposition of large strains. The numerical results showing the effect of distributed dislocations on the stress-strain state of the beam are presented.

Cationic copolymerization of cholesterol vinyl ether with N-allenylpyrrolidone yielded co-oligomers with molecular mass of 1200-2100. The polymerization of N-allenylpyrrolidone involves both 1,2- and 2,3-positions of the allenyl substituent to give four types of units as a result of prototropic isomerization of the initially formed structures. In the developed method, the composition of co-oligomers can be controlled and, hence, their hydrophilic/hydrophobic balance, solubility, and membranotropic properties can also be controlled to change the potential biological activity of the products.

A new method was proposed to produce high-flexural-strength corundum ceramics from α-Al₂O₃ samples synthesized by heat treatment of boehmite AlOOH at 1300 °C for 5 h. It was shown that hydrothermal treatment of MDGA-grade hydrargillite at 200 °C in a 0.4 wt % magnesium acetate solution for 3 days gives finely crystalline boehmite, which is a feedstock for synthesizing corundum ceramics with a flexural strength of 400 MPa.

New data were obtained on the effect of small additions of zirconium oxide nanoparticles on the strength properties of powder matrices based on pure aluminum, including those with the addition of 1.5% copper cold pressed and sintered in forevacuum. It is shown that even with an insignificant concentration of nano-additives, there is a noticeable increase in the mechanical properties of materials: tensile strength, bending, compression, and yield strength.

Based on a study of pyrite from the Uzelga Cu-Zn volcanogenic massive sulfide (VMS) deposit (South Urals) the age of ore mineralization was first determined with the direct age-dating method, based on the fraction of radiogenic helium, incorporated into the pyrite crystal lattice from submicron inclusions of U and Th minerals. Taking into account the measurement errors, the obtained age of 377 ± 8 Ma (MSWD = 1.2) is quite consistent with the independent age dates available for the ore mineralization (Late Eifelian-Early Givetian, 385-390 Ma).

Lubavinskoe Au-bearing ore deposit is situated to submeridional deep layered tectonic zone. In these zones the ore veins are associated with small Mesozoic stocks of granodiorite and rocks of the dyke complexes of the sublatitudinal spread. According to geochemical features these intrusions are correspond to adacits. The proportion isotopic ratio of the oxygen and strontium at granodiorites indicates to their origin due to mantle-core interaction. Calculated oxygen isotope composition δ¹⁸ in the fluid in equilibrium with quartz in the productive phase 220-280 °С changes from 3,59 to 9.59‰, which corresponds to water of magmatic fluid nature. This is confirmed by the isotopic composition of sulfar for sulphides δ³⁴S, ‰ from +0.7 to +6.7‰, corresponding to hydrothermal orogenic ore deposits.

The mantle xenoliths of lherzolite composition from Mesozoic alkaline-ultrabasic diatremes of the Jetty Oasis were studied. The studied xenoliths represent the mantle section of the East Antarctic Craton down to depths of 60-80 km. The osmium isotope composition of these nodules testifies to the beginning of the formation of the lithospheric mantle in the considered region at about 2400 Ma. The absence of any signs of Early Archean lithosphere points either to partial destruction of the lithosphere at the convergent boundary of the plates in the Late Archean or to thermal erosion of the Archean lithosphere under the deep-seated plume impact at the Mesozoic time during rift propagation.

This paper reports on the physical-chemical parameters and vertical distribution of different types of fluid inclusions in quartz from the Au-bearing interval of the Kola Superdeep Borehole. We assume that gold ore mineralization was formed when a deep flux of CO₂ interacted with brines at depths of 10 205-9269 m due to tectonic factors. In this paper, we discussed the role of these processes in the formation of the orogenic gold deposits including the gold mineralization of the Southern Pechenga structural zone.

Based on the results of satellite monitoring, the anomalies of spatiotemporal distributions of wildfire areas, as well as the volumes of caused emissions of carbon-containing trace gases (CO, CO₂) and aerosols (PM_2.5), for the territory of European Russia and Eastern and Central Europe from 2005 to 2018 are detected. It is found that, for 2005-2013, the annual areas of wildfires shrank for the territory of European Russia by almost a factor of 10, and the volumes of emissions of harmful gas traces into the atmosphere decreased by a factor of 7. However, starting from 2014, they were recorded to increase. The seasonal (spring and summer) maxima occurred in the distributions of the burned areas and the emission volumes in the time period under study. An anomalously large number of fire sources was recorded on the territory of Ukraine in 2014-2018 (70% of all fires in Eastern and Central Europe) where the specific density of fire sources, their relative areas, and the relative volumes of emissions increased by a factor of 3-5.7 compared to 2010-2013 and grew by a factor of 5-7.5 in August 2014-2018 compared to other territories of Eastern, Central Europe, and European Russia.

Developed multiplex PCR test-system for identification of transgenic cis-regulatory elements (promoter, insulator) from genetic constructions based on pBC1 Milk expression vector incorporated into the Mus musculus genome. Test-system was validated by genotyping producent strains generated by linearized vector DNA microinjection into fertilized zygote with occasional transgene genome integration, expressing the following transgenes in mammary gland, in milk: Human lactoferrin protein in different length, heat-shock protein HSP-70, Firefly luciferase protein, lysozyme protein. Reaction conditions selection described in details for better reproducibility.

A microbiotesting technique is described, which involves recording changes in pH, redox potential, electrical conductivity, optical density, light scattering and luminescence intensities, as well as other parameters of samples with viable test microorganisms, incubated in a liquid nutrient medium in the presence and in the absence of tested factors. The results of the analysis using this system of pro- and antibiotic activity of various oil products, as well as weak electromagnetic fields of the megahertz range are presented. It is shown that the proposed method, in comparison with standard methods, allows to obtain more detailed and objective information, and also requires less time and labor.