The methods and algorithms of XFEL data analysis for protein molecules are discussed. Experimental data on the structure and spatial distribution of the electron density in biomacromolecules and their complexes, algorithms, data analysis and integration of X-ray scattering, electron microscopy and molecular modeling techniques to study the structure and dynamics of biological macromolecules and their complexes are discussed as well.

We study the existence conditions for a double-deck structure of a boundary layer in typical problems of incompressible fluid flow along surfaces with small irregularities (periodic or localized) for large Reynolds number. We obtain characteristic scales (a power of a small parameter included in a solution) which lead to the double-deck structure, and we obtain a formal asymptotic solution of a problem of a flow inside an axially-symmetric pipe and a two-dimensional channel with small periodic irregularities on the wall. We prove that a quasistationary solution of a Rayleigh-type equation (which describes the flow oscillation on the “upper deck” of the boundary layer with the double-deck structure, i.e. in the classical Prandtl boundary layer) exists and is stable. We obtain a formal asymptotic solution with the double-deck structure for the problem of fluid flow along a plate with small localized irregularities such as hump, step or small angle. We construct a numerical solution algorithm for all equations which we obtained and we show the results of their applications.

A specific measure of short-range orientational order in dipolar systems is introduced. This can be used for analyzing computational results aiming at constructing phase diagrams and revealing microscopic structure of phases of magnetic systems. 

A specific measure of short-range orientational order in dipolar systems is introduced. This can be used for analyzing computational results aiming at constructing phase diagrams and revealing microscopic structure of phases of magnetic systems.

In order to model the processes taking place in systems with Josephson contacts, a differential equation on a torus with three parameters is used. One of the parameters of the system can be considered small and the methods of the fast-slow systems theory can be applied. The properties of the phase-lock areas – the subsets in the parameter space, in which the changing of a current doesn’t affect the voltage — are important in practical applications. The phaselock areas coincide with the Arnold tongues of a Poincare map along the period. A description of the limit properties of Arnold tongues is given. It is shown that the parameter space is split into certain areas, where the tongues have different geometrical structures due to fastslow effects. An efficient algorithm for the calculation of tongue borders is elaborated. The statement concerning the asymptotic approximation of borders by Bessel functions is proven. 

In order to model the processes taking place in systems with Josephson contacts, a differential equation on a torus with three parameters is used. One of the parameters of the system can be considered small and the methods of the fast-slow systems theory can be applied. The properties of the phase-lock areas – the subsets in the parameter space, in which the changing of a current doesn’t affect the voltage — are important in practical applications. The phaselock areas coincide with the Arnold tongues of a Poincare map along the period. A description of the limit properties of Arnold tongues is given. It is shown that the parameter space is split into certain areas, where the tongues have different geometrical structures due to fastslow effects. An efficient algorithm for the calculation of tongue borders is elaborated. The statement concerning the asymptotic approximation of borders by Bessel functions is proven.

We discuss physical parameters of quantum Penning nanotraps. In the case of 3:(-1) resonance between transverse frequencies of the trap we describe the reproducing measure on symplectic leaves corresponding to irreducible representations of non-Lie symmetry algebra with qubic commutation relations. Nonhomogeneity of the magnetic field and anharmonicity of the electric potential of the trap, after double averaging, generate an effective Hamiltonian which becomes a second order ordinary differential operator in the irreducible representation. We obtain an integral formula for asymptotical eigenstates of the perturbed 3:(-1) resonance Penning trap via the eigenfunctions of this operator, as well via the coherent states and the reproducing measure.

This is a short review of a few recent papers on the peptide bond in protein molecules. The break down of the planar structure of the peptide bond is discussed.

We consider the problem of constructing semiclassical asymptotic expansions of discrete spectrum and the corresponding stationary states of one-dimensional Schrödinger operator in the case of resonance tunneling. We consider two basic models: tunneling in an asymmetric double-well potential on a line and momentum tunneling of a particle in a potential field on a circle. For an asymmetric double-well potential we obtain the criterion of resonance tunneling, i. e. the necessary and sufficient conditions of stationary states bilocalization. We obtain explicit asymptotic formulas for the tunneling energy splitting in the case of high energy levels and for energies close to the minima of the potential. In the general case of dynamic tunneling we proposed a general method to find the asymptotic estimates for the tunneling energy splitting. In the case of the particle on a circle our method yields an asymptotic formula for the tunneling splitting, which is applicable in the case of analytical potential as well as in the case of finite smoothness. As an example, we consider the problem of momentum tunneling of the quantum pendulum.

We introduce a new method for modeling of heat transfer in the thermo-field emission nanocathode. The base of our model is the modified Stefan problem with the special conditions on the free boundary and on the tip of the cathode (Nottingham effect). We use a modification of the phase field system for the numerical simulation. Using numerical simulation we analyze the Nottingham effect influence on the propagation of the interface between the phases in the cathode.

The aim of the article is to construct a solution for the problem of the optimal recovery (in the mean-square sense) of a measurable square-integrable (with respect to the Lebesgue measure) function defined on a finite-dimensional compact set. We prove optimal recovery procedure and establish conditions of its unbiasedness and consistency. Furthermore, an 𝜀-optimal stochastic recovery procedure is proposed and proved.