The groups of reduced and reduced unitary norms of tamely ramified Henselian division algebras are described. Important properties of reduced and reduced unitary Whitehead groups of division algebras over function fields of p-adic curves after extension scalars to their completions with respect to special discrete valuations and of skew fields of the so-called noncommutative p-adic rational functions are obtained.

We conseder a linear quasiperiodic control system closed by the phase variables-linear feedback. It is assumed that the average coefficient matrix has a block representation. The control problem of an asynchronous multifrequency spectrum is solved.

We find a closed-form classical solution of the homogeneous biwave equation with Cauchy conditions, a boundary condition on the lateral boundary, and a nonlocal integral condition involving the values of the solution at interior points of the domain. A classical solution is understood as a function that is defined everywhere in the closure of the domain and has all classical derivatives occurring in the equation and conditions of the problem.

We introduce and characterize the class of graphs in which every connected dominating set is a (connected) neighbourhood set and a class of graphs whose all connected induced subgraphs have equal minimum neighbourhood set and minimum connected neighbourhood set cardinalities. Assuming P ≠ NP, we also prove that the minimum connected neighbourhood set problem cannot be approximated within a logarithmic factor in polynomial time in their common subclass, the class of simplicial split graphs.

The problem of sequential testing of simple hypotheses for time series with a trend is considered in case of missing observations. The sequential test is constructed and its performance characteristics are analysed. Numerical results of experiments are given.

An explicit solution of first-kind one-type integral equation with logarithms in the kernel reduces to a successive solution of the characteristic singular equation and the Volterra equation. The solution is given in closed form, depending on the index of the singular equation and the solvability of the Volterra equation.

In this paper, the initial boundary value problem for the simplest inhomogeneous second-order non- trictly hyperbolic equation with the mixed Dirichlet and Neumann boundary conditions in a quadrant is fully investigated and solved. By means of the method of characteristics we have obtained its classical solution in analytic explicit form and have proved the necessity and the sufficiency of the established requirements and the smoothness of the original data (the right hand-side of the quation, initial and boundary data) to ensure its unambiguous solvability everywhere in a variety of classical solutions. The requirements on the smoothness of the data of this problem are by “one” are higher than if we have solved the similar first- or secondorder mixed problem for the hyperbolic equation of semi-infinite string vibrations.

In the work new criteria of self-returning of any point of n-ary groups concerning the elements of sequence of tops of 2k-gons of the n-ary groups are established. It is proved that centroids of 2k- gons of G possessing that property that any point of n-ary groups of self-returning concerning the elements of sequence of tops of this 2k-gons coincides with the G k-square centroid.

We have shown that the stationary group of the isotropic vector of the four- dimensional pseudo- uclidean space, which is the subgroup of the rotation group SO(3.1) of this space, is the group of motion of the first- and second-kind horospheres in the three-dimensional extended Lobachevsky space and act transitively on horospheres.

For massless Dirac particles, the general mathematical study of the tunneling proccess of a particle through the effective potential barrier generated by the Schwarzschild black hole is made. Results are significantly different for two situations: first when a particle falls on the barrier from within and second when a particle falls on the barrier from the outside. The study is based on the use of 8 Frobenius solutions of the related second-order differential equations with the second-rank non- regular singularities. The mathematical structure of the derived asymptotic relations is exact, however the analytical expressions for the involved convergent powers series are not known. So, a further study should be based on the numerical summation of the series.

The first results of simulation of an electromagnetic calorimeter for registration of soft photons on the Nuclotron beam with the energy of 3.5 GeV on the basis of Geant4 and UrQMD packages are discussed in the paper. The theoretical assumption is made on the relation between Bose – Einstein condensate and abnormal soft photon production on the basis of the SVD-2 data (Joint Institute for Nuclear Research, Dubna, Russian Federation). This is an actual task for heavy nucleus collisions.

A method for increasing the output energy of an extracavity Raman laser with a double-pass pump is proposed. The method is based on the repeated recovery of a depleted pump in the zone of SRS generation and is demonstrated by the example of eye-safe Raman lasers with a KGW crystal pumped at geometries of E Ng, and Е Nm by pulsed multimode radiation of a Nd:KGW-laser with a 4F3/2 → 4I13/2 working transition. When returning 70 % of a depleted pump in the Raman laser, an efficiency of generation of eye-safe radiation at wavelengths of 1507 and 1538 nm increases by approximately 11 %.

The electron energy band structure and optical properties of various phases of tin sulfide were theoretically estimated by computer simulation. All the investigated materials were found to be indirect-gap semiconductors with a band gap ranging from 0.17 to 2.4 eV. The band gap in the range of 1.0–1.5 eV and the absorption coefficient near the fundamental absorption edge of more than 105 cm–1 in the cubic and orthorhombic phases of tin sulfide with a stoichiometric composition of SnS make them promising for solar energy conversion.

The research results of UV 266 nm gold ablation are presented. It is shown that the deposit structure on the surface around ablation pits sharply depends on a pit depth. As the pit depth is increased, gold micro- and nanoparticles acquire a more developed surface structure and the surface around the pits gets deep black color – “black” gold appears. Some features and possible mechanisms of forming “black” gold structures at ablation over the 266 nm powerful nanosecond laser radiation range are also considered.

We consider a problem of optimizing the output of a batch of parts and intensities of its processing with tool blocks on a multiposition equipment under non-stationary demand and predetermined time intervals. The batch content does not vary from one interval to another. The objective function is the sum of production cost, storage cost of excess parts, and penalties for unmet demand. The production cost depends on processing intensities. A decomposition method for solving the problem is proposed.

Two optimization problems in logical design are considered: decomposition of Boolean functions and state assignment of a finite automaton. A common approach to those problems is suggested. This approach is connected with the search of a maximal cut in a graph with weighted edges. Heuristic methods based on this approach to solve the problems are suggested.