Let z = f(x, y) be a surface in three-dimensional Euclidean space. Consider a neighborhood V of this surface, whose points satisfy the inequality | f(x, y) - z| < Q  ^-Y, where 0 < у < 1 and Q  is a sufficiently large positive integer. In the works of Huxley, Beresnevich, Velani, the distribution of rational points in V has been started. In this article, we study the distribution of points with real conjugate algebraic coordinates ᾱ = α₁α₂α₃ in V. For some c₁ = c₁(n), a lower bound is obtained in the form of c₂ Q ^n+1-Y for the number of algebraic numbers of degree n ≥ 3 and of height at most c₃ Q.

In this article, we study the question of the solvability of an analogue of the Cauchy problem for ordinary differential equations with fractional Riemann-Liouville derivatives on the unbounded right-hand side in certain function spaces. The solvability conditions of the problem under consideration in given function spaces, as well as the existence conditions of a unique solution are presented. The study uses the method of reducing the problem to the second-kind Volterra equation, the Schauder principle of a fixed point in a Banach space, and the Banach-Cachoppoli principle of a fixed point in a complete metric space.

For the first time, a pulse laser treatment method was developed and demonstrated for the formation of antireflective coatings based on composite materials containing polymer with carbon nanotubes. The effect of the composite surface in the visual and near-IR regions modification by the pulse laser treatment on light reflectivity has been studied. The possibility of creating innovative non-reflective surfaces of composite samples in the visual and near-IR ranges is demonstrated.

Nowadays neuroscience strongly demands application of the mathematical methods for description of many neurophysiological and neurochemical processes among which the synaptic transmission outstands. One of the main problems in synaptic transmission modelling is the lack of the accurate values of dynamic parameters of biomolecules and complexes taking part in this process.

The goal of this study is to elaborate the method for evaluation of synaptic transmission parameters that cannot be measured directly (so-called hidden parameters) and apply its results for investigation of the main stages of synaptic transmission in neuronets of hippocampus.

The method is based on the parametric identification of the synaptic transmission deterministic model, which includes equations for description of inhibitors action on the main biochemical participants. We used three inhibitors: cilnidipine, 1.2-        bis(2-aminophenoxy)ethane-N,N,N',N'-tetraacetic acid tetrakis(acetoxymethyl ester) (BAPTA-AM), 6-cyano-7-nitroquinoxaline-2,3-dione (CNQX). The parametric identification was performed by minimization of deviation of modeled field excitatory postsynaptic potential from those measured in rat hippocampus slices with microelectrode technique when inhibitors were applied.

The results of the parametric identification of proposed model show that the model can adequately describe the generation of field excitatory postsynaptic potentials and their inhibition. The elaborated method afforded to evaluate the numerical meanings of eleven synaptic transmission hidden parameters. Using these parameters we have modelled the key synaptic transmission stages and got the time courses of the main biochemical participants: calcium ions in presynaptic bouton, SNARE complexes, synaptic vesicles in different states, glutamate in the synaptic cleft and open channels of AMPA receptor on the postsynaptic membrane. Thus, we propose method of hidden parameters evaluation that can be applied for different synaptic contacts in the brain of mammalians.

The spherically symmetric potential is considered, whose dependence on the distance r is described by the smooth composition of Coulomb at r < r₀ and oscillator at r > r₀ potentials. The boundary distance r₀ is determined by the parameters of these potentials. The exact continuous solution of the radial Schrodinger equation is expressed in terms of the confluent hypergeometric functions. The discrete energy levels are obtained. The graphic illustrations for the energy spectrum and the radial wave functions are presented.

Anodic materials with spinel structure Li₂MTi₃O₈ (M - divalent metal) are promising for storing electricity from renewable energy sources, for portable electronics and electric vehicles. In this work, mesoporous lithium titanates Li₂MTi₃O₈ (M - Co, Cu, Zn) powders with spinel structure were prepared by the method of self-propagating high-temperature synthesis (SHS) from the glycine-citrate-nitrate mixtures and, for comparison, by the sol-gel method. Their crystal structure, phase composition, thermal stability, microstructure and dispersion were studied. It was established that the SHS method of preparation of lithium titanates has several advantages over the sol-gel method because there is no need to use any solvents; it is possible to reduce particle aggregation, to increase specific surface and to reduce bulk density of the obtained powders.

Coatings of oligomethylsiloxane resin (OMES) and hydrolyzed heptadecafluorotetrahydrodecyltrimethoxysi-lane (HFS) oligomers were obtained by the spin-coating method on polyurethane (PU) - modified steel substrates. Their wettability and tribological properties were studied. It was found that the modification of the PU layer with silicon organic compounds increases its hydrophobicity and wear resistance. The water contact angles are 63.1° for a PU coating, 81.2° for PU/OMES and 115.8° for PU/HFS.

In aqueous solution, in the acidic environment (pH <3), water-soluble complexes are formed based on polyacrylic acid (PAA) and polyethyleneglycol with a molecular weight of more than 6000 or on polyvinyl alcohol (PVA). The conditions for obtaining hydrogel based on polyacrylic acid and polyvinyl alcohol with a high degree of swelling (about 500 g of water/g of polymer) were determined: pH <3, temperature 120-140 °С. The degree of interaction of polymers is directly proportional to the content of polyvinyl alcohol in the initial mixture in the range of the molar PVA and PAA ratios from 1 : 1 to 10 : 1, the temperature and duration of heat treatment and is inversely proportional to the molecular weight of polyacrylic acid. The equilibrium degree of hydrogel swelling increases at the pH of the swelling medium above 6, and the swelling rate depends on the solvent diffusion into the hydrogel and is determined by the size of its particles and the swelling conditions (washing, mixing, etc.).

In order to assess the differential expression of genes, seedlings of narrow-leaved and yellow lupine treated with different Fusarium isolates were studied by the method of SRAP-analysis. As a result, PCR fragments correlated with tolerance to infection were found. The corresponding genetic determinants are likely involved in the resistance (tolerance) control of lupine plants to Fusarium.

Feasibility of producing brazzein plant sweetener by the cell-free biosynthesis procedure as an alternative to the classical submerged fermentation method was assessed in the present investigation. Chimeric RNA polymerase of T7 bacteriophage, S30-cell extract of Escherichia coli and multicopy plasmid vector pET42mut with the inserted brazzein gene were included into the protein synthesis. The completed research resulted in the first successful demonstration of the sweet protein biosynthesis in the cell-free system. The volumetric brazzein yield under optimized process conditions was 2 mg/ml of the reaction mixture, exceeding 57 times the maximum values that had been achieved in the previous studies applying whole-cell expression systems.

Using an experimental model of aluminum neurotoxicosis, it was established that under conditions of chronic administration of aluminum chloride to rats, oxidative stress develops and inhibits the redox potential of the glutathione system in the mitochondrial and postmitochondrial fractions of the cerebral hemispheres. It was shown that the ingestion of N-acetylcysteine, as well as its combined use with coenzyme A biosynthesis precursors (D-panthenol or D-pantetin) against the background of aluminum neurotoxicosis, leads to a marked decrease in the production of reactive oxygen species by mitochondria, a decrease in the production of thiobarbituric acid reactive substances, and normalization of GSH content and its biosynthesis in brain tissue. The results indicate a high efficiency of the biosynthesis precursor of glutathione N-acetylcysteine in the prevention of oxidative stress in the chronic model of aluminum neurotoxicosis, which may be the rationale for its use as a modulator of mitochondrial redox status in the development of neurodegenerative pathology.

An effective method for treatment of chronic tonsillitis has been developed. It is based on combined (sequential) exposure of the tonsils to laser radiation of violet (405 nm wavelength) and red (650 wavelength) spectral regions at a power density of 25 mW/cm² for 3 minutes for each amygdala for 7 days. It was shown that laser therapy according to the above method has positive anti-inflammatory, decongestant and antimicrobial effects. Moreover, the antimicrobial effect of laser radiation is due to the excitation of endogenous porphyrin and flavin photosensitizers, localized mainly in pathogenic microflora. The high efficiency of generation of singlet oxygen by these compounds is capable of realizing the bactericidal action of light without introducing exogenous photosensitizers.

The structure of ultrafine-grained materials obtained by deformation methods under the influence of ultrasonic vibrations is investigated; new methods of intense plastic deformation have been developed with the imposition of ultrasonic vibrations, allowing the formation of an ultrafine-grained structure in extended materials; optimal modes of plastic deformation and heat treatment were determined, allowing one to achieve the highest thermal stability and ductility in combination with high strength of materials. The designs of ultrasonic oscillatory systems for obtaining extended bulk nanostructured materials were developed, and a technological scheme of producing an ultrafine-grained wire using ultrasonic processing with equal-channel angular broaching was tested. It is shown that this process of deformation processing is industrially applicable, and can be combined into a single technological cycle with wire drawing. The results can be used in engineering, instrument making and cable industry.

The effect of hybrid filling of poly (ethylene terephthalate) (PET) with short glass fibers (GF) and multi-walled carbon nanotubes (MWCNTs) on the static and dynamic mechanical properties of composites, as well as the structural features and rheological behavior of melts were investigated. Compounding materials was carried out by the PET melt blending with the use of the reaction extrusion method. The studied composites contained from 15 to 60 wt. % GF, the concentration of MWCNTs was 0.5 and 1.6 wt. %. It was shown that the PET hybrid filling led to a synergistic increase in the strength, the elastic modulus and the dynamic shear modulus of composites under tension and bending in a wide temperature range. MWCNT additives intensified interfacial adhesion, had a nucleating effect on the crystallization of the binder, and increased its thermal stability.

For the first time, silicon carbide nanoparticles were obtained by spray pyrolysis of chlorotrimethylsilane droplets in an aerosol reactor with argon flow. It was shown that with an increase in the reactor wall temperature to 1100 °C, the sample purity degree grows, and the average size of the nanoparticles is 6-40 nm.

The key points of modernization of the national economy are identified: widespread development of market infrastructure, minimization of direct government intervention; cooperation and integration of interested business entities and the creation of cooperative integration corporations (firms) and cluster systems; end-to-end export orientation of domestic agribusiness; a high level of labor stimulation and its direct connection with production results.