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The creation of a domain wall pair in the magnetic core of amorphous glass-coated bistable microwires with circular cross-section is studied. Energy balance is formulated in the spirit of particle decay, equalizing magnetic field work and energy of two domain walls, emitted in opposite directions. The critical nucleation field is evaluated and compared with observations.

Diagnostics and decomposition of atmospheric disturbances in a planar flow are considered and applied to numerical modelling with the direct possibility to use in atmosphere monitoring especially in such strong events which follow magnetic storms and other large scale atmospheric phenomena. The study examines a situation in which the stationary equilibrium temperature of a gas may depend on a vertical coordinate, which essentially complicates the diagnostics. The relations connecting perturbations for acoustic and entropy (stationary) modes are analytically established and led to the solvable diagnostic equations. These equations specify acoustic and entropy modes in an arbitrary stratified gas under the condition of stability. The diagnostic relations are independent of time and specify the acoustic and the entropy modes. They provide the ability to decompose the total vector of perturbations into acoustic and non-acoustic (entropy) parts uniquely at any instant within the total accessible heights range. As a prospective model, we consider the diagnostics at the height interval 120–180 km, where the equilibrium temperature of a gas depends linearly on the vertical coordinate. For such a heights range it is possible to proceed with analytical expressions for pressure and entropy perturbations of gas variables. Individual profiles of acoustic and entropy parts for some data are illustrated by the plots for the pure numerical data against those obtained by the model. The total energy of a flow is determined for both approaches and its vertical profiles are compared.

The main result of this work is the estimation of the entropy mode accompanying a wave disturbance, observed at the atmosphere heights range of 90–120 km. The study is the direct continuation and development of recent results on diagnosis of the acoustic wave with the separation on direction of propagation. The estimation of the entropy mode contribution relies upon the measurements of the three dynamic variables (the temperature, density, and vertical velocity perturbations) of the neutral atmosphere measured by the method of the resonant scattering of radio waves on the artificial periodic irregularities of the ionospheric plasma. The measurement of the atmosphere dynamic parameters was carried out on the SURA heating facility. The mathematical foundation of the mode separation algorithm is based on the dynamic projection operators technique. The operators are constructed via the eigenvectors of the coordinate evolution operator of the transformed system of balance equations of the hydro-thermodynamics.

Two models of two- and three-layers of a magnetic nanoparticle are presented applying the Heisenberg theory with the parameter (exchange integral and closest neighbors) values for surface layers, which differ from the bulk layer values. The difference in the numbers of closest neighbors is taken into account. The corresponding distribution and partition functions are constructed. The magnetization of the particles is calculated by the conventional transition to thermodynamics. The results are illustrated by plots, representing magnetization curves and hysteresis loops for the layer contributions. The magnetization curves for both models are compared.

The problem of wave identification is formulated as applied to the results of measurements of the temperature and the density of the neutral atmosphere in the range height 90–120 km by the artificial periodic irregularities (APIs) technique. The technique is based on the resonant scattering of radio waves by artificial periodic irregularities of the ionospheric plasma emerging in the field of a standing wave arising from the interference of the incident and reflected waves from the ionosphere. APIs were created using SURA heating facility (named as SURA experiment). The acoustic wave theory is reformulated on the base of data which can be observed in the given experimental setup. The basic system of equations is reduced so that it accounts only upward and downward directed waves, ignoring entropy mode. The algorithm of wave identification based on usage of dynamic projection operators for such a reduced case is proposed and explicit form of projection operators is derived. Its application to finite number dataset via Discrete Fourier Transform (DFT) is described and results of its application to the DFT-transformed set of experimental observation of the temperature and density perturbations are presented. The result yields hybrid amplitudes, that allow us to calculate energy of the directed waves that enter the observed superposition. The problem of entropy mode detection is discussed, the corresponding projecting operators for the full evolution system are built and a way to apply the method to quantification of it is proposed.

A problem of analytical-numerical modeling of the plane wave propagation from an ocean surface to the atmosphere is considered. We are interested in the thermosphere impact of tsunami waves. We suppose that the transport of energy and the momentum from lower atmosphere to thermosphere heights is due to acoustic waves. A set of expressions for the atmosphere variables (pressure, velocity, entropy) as a function of the atmosphere parameters, time and height is derived and illustrated by plots. The surface water wave parameters, typical for tsunami also enter final expressions for the atmosphere and ionosphere variables.

A new algorithm is presented for evaluating the velocity field of the heat convection flow from a vertical solid flat plate using the temperature field. Using the coordinate transformation of Cartesian coordinates, to coordinates defined via streamlines, and visible in such coordinate approximations, it is possible to express the basic flow fields in terms of the temperature gradients only. After discretization we formulate approximated sufficient finite-difference formulas to evaluate the velocity field using the experimental data.

We propose a new inverse problem formulation based on the hydrodynamics consideration of a gas/water fluid that results in planetary waves diagnostics. We analyze such a possibility beginning from a simplest version of geophysical hydrodynamics, written in the 𝛽-plane model. The problem of diagnostics is solved approximately after expansion with respect to the transverse basis functions applying projecting to Rossby and Poincare waves in each transverse subspace that contains its superposition. The corresponding discrete version of the operators is built to be applied to the observation data.

We consider a generalization of the projection operator method for the case of the Cauchy problem in 1D space for systems of evolution differential equations of first order with variable coefficients. It is supposed that the dependence of coefficients on the only variable 𝑥 is weak, that is described by the introduction of a small parameter. Such problem corresponds, for example, to the case of wave propagation in a weakly inhomogeneous medium. As an example, we specify the problem to adiabatic acoustics in waveguides with a variable cross-section. Projection operators are constructed for the Cauchy problem to fix unidirectional modes. The method of successive approximations (perturbation theory) is developed and based on the pseudodifferential operators theory. The application of projection operators adapted for the case under consideration allows deriving approximate evolution equations corresponding to the separated directed waves.

A complete – system of Maxwell equations is splitting into independent subsystems by means of a special dynamic projecting technique. The technique relies upon a direct link between field components that determine correspondent subspaces. The explicit form of links and corresponding subspace evolution equations are obtained in conditions of certain symmetry, it is illustrated by examples of spherical and quasi-one-dimensional waves.

The problem of the propagation of acoustic wave disturbance initiated by a boundary condition is used to simulate a disturbance of atmospheric gas caused by a rise of water masses. The boundary condition is a function of a dynamic variable that is defined on the border of the problem domain. In this work, it is chosen in such a way that its parameters and form correspond to disturbances in the gas layer produced by a tsunami wave at the air−water interface. The atmosphere is approximately described as a 1D multilayer gas media with an exponential structure of density in each layer. The boundary conditions are set at the interface between water−air and gas layers. These determine the direction of propagation and the ratio of dynamic variables characterizing an acoustic wave. The relationship between such variables (pressure, density, and velocity) is derived by means of projection operators on the subspaces of the z-evolution operator for each layer. The universal formulas for the perturbation of atmospheric variables in an arbitrary layer are obtained in frequency and time domains. As a result, explicit expressions are derived that determine the spectral composition and vertical velocity, by the stationary phase method, of the acoustic disturbance of the atmosphere at an arbitrary height, including the heights of the ionosphere. In return, this can be used to calculate the ionospheric effect. The effect is described by the explicit formula for electron density evolution, which is the solution of the diffusion equation. This forms a quick algorithm for early diagnostics of tsunami waves.

The free convection heat transfer from an isothermal vertical plate in open space is investigated theoretically. In contrast to conventional approaches we use neither boundary layer nor self-similarity concepts. We base on expansion of the fields of velocity and temperature in a Taylor Series in x coordinate with coefficients being functions of the vertical coordinate (y). In the minimal version of the theory we restrict ourselves by cubic approximation for both functions. The Navier-Stokes and Fourier-Kirchhoff equations that describe the phenomenon give links between coefficient functions of y that after exclusion leads to the ordinary differential equation of forth order (of the Mittag-Leffleur type). Such construction implies four boundary conditions for a solution of this equation while the links between the coefficients need two extra conditions. All the conditions are chosen on the basis of the experience usual for free convection. The choice allows us to express all the theory parameters as functions of the Rayleigh number and the temperature difference. To support the conformity of the theory we derive the NusseltRayleigh numbers relation that has the traditional form. The solution in the form of velocity and temperature profiles is evaluated and illustrated for air by examples of plots of data.

This paper presents calculation of the electron-impurity scattering coefficient of Bloch waves for one dimensional Dirac comb potential. The impurity is also modeled as delta function pseudopotential that allows explicit solution of the Schrodinger equation and scattering problem for Bloch waves.

A general problem of monochrome plane electromagnetic wave reflection and refraction at the interface between the conducting medium and the dielectric is formulated and solved by symbolic computation for given incident wave polarization. The conductivity account via the Ohm law directly in the Maxwell equation leads to a complex wavenumber and hence complex amplitudes of the reflected and refracted waves. Atomic absorption is taken into account via the imaginary part of permittivity. The general formula for the time-averaged Pointing vector in the conducting media as a function of the medium parameters and the incident angle is derived and used for the refraction angle definition. The result is compared with textbooks and recent publications. The dependence of intensity as a function of the angle to the interface is determined also via the Pointing vector as a function of the incident wave and medium parameters.

We consider a boundary regime problem for 1D wave propagation in a metamaterial medium with simultaneously negative dielectric permittivity and magnetic permeability. We apply a projecting operator method to the Maxwell system in the time domain that allows the space of the linear propagation problem to be split into subspaces of directed waves for the relations of a given material with general dispersion. After projection, the equations for directed waves have a maximally simplified form, which is most convenient for numerical and analytical integration. Matrix elements of the projectors act as integral operators. For a given nonlinearity and dispersion we derive a general system of interacting right/left waves with combined (hybrid) amplitudes. The result is specified for the Drude metamaterial model for both permittivity and permeability coefficients and the Kerr nonlinearity. We also discuss and investigate singular solitary wave solutions of the system as a limit stationary elliptic system related to some boundary regimes.

A brief history and a mathematical description of the dynamic projection operators technique is presented. An example of the general Cauchy problem for evolution equations in 1+ 1 dimensions is studied in detail. A boundary regime propagation is formulated in terms of operators and illustrated by the simplest one-dimensional diffusion equation. The problem of temperature waves is discussed.

A problem of wave identification is formulated. We propose a diagnostic analysis of medium disturbances based on distinguishing of components of a wave vector that is specific for each kind of the wave mode. Mathematically it is realized by projection operator technique. An example is considered in conditions of a one-dimensional Cauchy problem for a conventional wave equation in the matrix form and its version with weakly x-dependent coefficients as a demonstration of the method application for the simplest adiabatic theory of one-dimensional acoustics. The case of acoustics in a gas with a dissipation account is also discussed from the point of view of the wave and entropy mode diagnostics.

A novel solution of the free convection boundary problem is represented in analytical form for velocity and temperature for an isothermal vertical plate, as an example. These fields are built as a Taylor Series in the x coordinate with coefficients as functions of the vertical coordinate (y). We restrict ourselves by cubic approximation for both functions. The basic Navier-Stokes and Fourier-Kirchhoff equations and boundary conditions give links between coefficients and connected with free convection heat transfer phenomenon which define the analytical form of the solution as a function of the Grashof number only. In the solution the non zero velocity of a fluid flow through a leading edge of the plate is taken into account. The solution in the form of velocity and temperature profiles is numerically evaluated and illustrated for air.

We consider a formulation of the Cauchy problem for the Kolmogorov equation which corresponds to a localized source of particles to be scattered by a medium with a given scattering amplitude density. The multiple scattering amplitudes are introduced and the corresponding series solution of the equation is constructed. We investigate the integral representation for the first series terms, its estimations and values of the photon number of finite and point receivers. Application to the LIDAR problem and X-ray beam scattering for orthogonal and inclined to a layer is considered.

In this paper we apply the finite-difference method to solve a parabolic equation of a general problem of short wave diffraction in a conducting medium. It is based on the implicit Runge-Kutta method of the second order combined with the iterative procedure. We also present new numerical simulations for X-rays focusing using the mentioned approach. We consider CRLs with a parabolic profile with a radius of curvature up to 0.2 mm. The main goal of this work is to elaborate an X-ray calculator for a PC which would present new possibilities compared to conventional ones. The correspondent code is written in FORTRAN to obtain the focal distance and diffraction spot profiles. Simulations for two cases were performed, the first one with 33 Al lenses for X-ray energy 15 keV, the results showed that we needed to consider more than 50000 points in each direction which forced us to consider a one-dimensional simulation only. For the second case we performed a simulation for several lenses, up to 15 Al lenses to perform the 2-d simulation. We have good agreement with the experimental data for the focal distance, and for the intensity at the focal plane while, for the spot size, we have smaller FWHM for the Gaussian beam at the detector than in the experimental data. We believe that the FWHM we have is smaller as our lenses are ideal without any defects.

An approximate analytical solution of a two dimensional problem for stationary NavierStokes, continuity and Fourier-Kirchhoff equations describing a free convective heat transfer from an isothermal cone is presented. The problem formulation is based on assumptions typical for natural convection: non-compressibility and the Boussinesq approximation. The solution is based on Frobenius expansions at the vicinities of two points: the initial point and the singular point of the boundary layer equation. Numerical matching of the expansions and Nusselt number evaluations are traced.