The largest Lyapunov exponents of linear differential systems dx/dt = μA(t)x,  x∈Rⁿ, t ≥ 0 with the real parameter-multiplier μ are considered. It is proven that a function ƒ: R→R̅   

is the largest Lyapunov exponent of some linear differential system with а real parameter-multiplier if and only if it fits the next four conditions: 1) it belongs to the (*,G_δ)  Baire class; 2) it vanishes at zero; 3) it is nonnegative on some real semi-axis; 4) if it is not identically equal to +∞ on any real semi-axis, then there exists such a real number b that the inequality ƒ(μ) ≥ bμ holds for all μ∈R.

Two examples of linear differential equations with continuous coefficients on the time semi-axis were constructed, such that spectra of upper characteristic frequencies of zeros and signs of the first equation are the set of rational numbers from the segment [0, 1] and spectra of upper characteristic frequencies of zeros and signs of solutions of the second equation consist of the set of irrational numbers from the segment [0, 1] and zero. 

The purpose of the study is to establish the analytical properties of solutions of nonlinear differential equations describing the planar motion of four bodies. 50 sets of constant values of interparticle interactions in the problem of four bodies in the plane are found, at which the components of the general solution are the meromorphic functions, as well as 15 sets, at which the corresponding systems have no Painlevé property. The results obtained can be applied in the analytic theory of differential equations, as well as for solving the problems of cosmic dynamics. 

We consider a finite-sheeted covering surface of the sphere of genus zero. We built the algorithm of construction of the conformal homeomorphism of this surface on the sphere by a given branch point and permutations describing the sheets gluing order. 

Using the Cauchy matrix, the formulas for calculation of the linear differential system of the exact upper bound of the upward mobility of the lower Bohl exponents and of the exact lower bound of the downward mobility of the upper Bohl exponents of its solutions under small perturbations of the coefficients of the system are obtained. It is proved that under small perturbations of the coefficients, the first of the mentioned bounds is upward stable, but is downward unstable, and the second one is downward stable, but is upward unstable. 

The result on the order of convergence of the approximate formula is obtained for evaluation of the mathematical expectation of one class of special-type functionals of the Wiener process. The formula is based on the use of sampling the time interval and the quadrature formulas exact for third-degree functional polynomials. 

The Stewart – Levine model is considered, which describes the dynamics of unstable strains of two micro-organisms, provided that a specific consumption rate of a substrate by both the plasmid-bearing organism and the plasmid-free organism is given by the Mono function. For the case when the half-saturation constants are equal, the reduction of the third-order differential system describing the considered model to a nonlinear differential equation of the first-order is realized. For such a system we built the software modules that allow simulating its solutions properties which depend on the input parameters. The coefficient relations, at which the third-order differential system has an analytical solution, are found, and the visualization of solutions for the certain sets of parameters is given. 

In the article we proved the Khintchine theorem in the case of divergence in the three-dimensional Euclidean space while considering only irreducible polynomials of degree exactly n. In the course of proof we built a regular system of triples of conjugate real algebraic numbers of degree exactly n in the three-dimensional Euclidean space. All results are obtained using the methods of metric number theory. 

Tensor and matrix formulations of the relativistic wave equation providing a description both of an electromagnetic field and a massless Kalb – Ramond field with the zero helicity are given. It is shown that this equation is a particular case of the Dirac – Kähler system. It opens new possibilities for applications of the Dirac – Kähler field in the string theory. 

The features of the two-photon absorption signal in lead tungstate crystals (PbWO4) in the “pump – probe” experiment are considered. The differences in the spectral dependences of the recorded two-photon absorption effect in the presence of ionizing radiation of a crystal and without it are discovered. The method of utilizing the effect to generate a time stamp of interaction of ionizing radiation with a scintillator is proposed. 

The features of plasma formation in double-pulse laser ablation in liquid have been studied to optimize the process of nanoparticles synthesis. On the basis of spectroscopic plasma diagnostics the spatial structure and the time range of laserinduced plasma emission have been revealed and the composition of its component has been determined. 

Atmospheric pressure air plasma jets within dc, pulsed and self-oscillatory current regimes are realized. It is shown that the main mechanism of inactivation of bacteria Staphylococcus aureus is the effect of chemically active molecules of NO, NO₂ and HNO₂. The method of IR absorption spectroscopy is used to investigate chemical active component concentrations. The optimal regime of discharge inducing plasma jets, which is more suitable for production of bactericidal components, is found. 

The theoretical investigation of the single-frequency oscillation in all-optical gain optoelectronic oscillator based on fiberoptic delay lines is performed. It is shown that there is no need in microwave phase shifters within the optoelectronic oscillator loop in order to provide low phase noise and spurious level oscillations. Threshold of the dynamical instabilities in the all-optical gain optoelectronic oscillator is calculated. It is shown that the reproducibility of the oscillation frequency is provided by means of the continuous tuning of the loop gain during the switching-on. 

The problem under consideration is to find a synchronizing sequence of a minimal size for a logical network having flipflop primitives of type D as memory elements. A novel method is proposed, which is based on the formulation of the task as the Boolean satisfiability problem solved with any standard SAT-solver. The method is based on forming the conventional conjunctive normal form representation for combinational block, implementing excitation functions of the flip-flops.