A function defined on normed vector spaces X is called convex with respect to the set LĈ := LĈ (X,R ) of
Lipschitz continuous classically concave functions (further, for brevity, LĈ -convex), if it is the upper envelope of some subset of functions from LĈ. A function f is LĈ -convex if and only if it is lower semicontinuous and bounded from below by a Lipschitz function. We introduce the notion of LĈ -subdifferentiability of a function at a point, i. e., subdifferentiability with respect to Lipschitz concave functions, which generalizes the notion of subdifferentiability of classically convex functions, and prove that for each LĈ -convex function the set of points at which it is LĈ -subdifferentiable is dense in its effective domain. The last result extends the well-known Brondsted – Rockafellar theorem on the existence of the subdifferential for classically convex lower semicontinuous functions to the more wide class of lower semicontinuous functions. Using elements of the subset LĈ _θ ⊂ LĈ, which consists of Lipschitz continuous functions vanishing at the origin of X we introduce the notions of LĈ _θ -subgradient and LĈ _θ -subdifferential for a function at a point.
The properties of LĈ -subdifferentials and their relations with the classical Fenchel – Rockafellar subdifferential are studied. Considering the set LČ := LČ (X,R ) of Lipschitz continuous classically convex functions as elementary ones we define the notions of LČ -concavity and LČ -superdifferentiability that are symmetric to the LĈ -convexity and LĈ -subdifferentiability of functions. We also derive criteria for global minimum and maximum points of nonsmooth functions formulated in terms of LĈ _θ -subdifferentials and LČ _θ -superdifferentials.

Herein, for one-step random Markov processes the comparison of the operator and combinatorial methods based on the use of functional integrals is performed. With the combinatorial approach, the transition from the stochastic differential equation to the functional integral is used. This allows us to obtain the expression for the mean population size in terms of the functional integral. With the operator approach, the transition to the functional integral is performed via the creation and annihilation operators. It is shown that the mean values calculated using the functional integrals arising in the combinatorial and operator approaches coincide.

In this paper, we consider the first mixed problem for the one-dimensional Klein – Gordon – Fock type equation in a half-strip. Meanwhile, the existence and uniqueness of a solution of arbitrary smoothness is researched. While solving this problem using the method of characteristics, equivalent second type Volterra integral equations appear. The existence of a unique solution in the class of n times continuously differentiable functions is proven for these equations when initial functions are smooth enough. Moreover, it is shown that for the smoothness of the solution of the initial problem it is necessary and sufficient that the matching conditions for the given functions be fulfilled if they are sufficiently smooth. The method of characteristics, used for problem analysis, is reduced to separating the total area of the solution on subdomains in each of them so that the solution of the subproblem is constructed with the help of the initial and boundary conditions. Then, the obtained solutions are glued in common points, and the received glued conditions are the matching conditions. This approach permits to construct both exact and approximate solutions. The exact solutions can be found when it is possible to solve the equivalent Volterra integral equations. Otherwise, one can find an approximate solution of the problem either in analytical or numerical form. Along with this, when constructing an approximate solution, the matching conditions turn out to be essential, which must be taken into account when using numerical methods for solving the problem.

This paper is devoted to the development of a mathematical tool for obtaining the Bayesian estimations of the parameters of multidimensional regression objects in their finite-dimensional multidimensional-matrix description. Such a need arises, particularly, in the problem of dual control of regression objects when multidimensional-matrix mathematical formalism is used for the description of the controlled object. In this paper, the concept of a one-dimensional random cell is introduced as a set of multidimensional random matrices (in accordance with the “cell array” data type in the Matlab programming system), and the definition of the joint multidimensional-matrix Gaussian distribution is given (the definition of the Gaussian one-dimensional random cell). This required the introduction of the concepts of one-dimensional cell of the mathematical expectation and two-dimensional cell of the variance-covariance of the one-dimensional random cell. The integral connected with the joint Gaussian probability density function of the multidimensional matrices is calculated. The two formulae of the total probability and the Bayes formula for joint multidimensional-matrix Gaussian distributions are given. Using these results, the Bayesian estimations of the unknown coefficients of the multidimensional-matrix polynomial regression function are obtained. The algorithm of the calculation of the Bayesian estimations is realized in the form of the computer program. The results represented in the paper have theoretical and algorithmic generality.

In this paper, we represent new examples of constructing model problems of the mechanics of a deformable solid using a fractional differentiation apparatus. The solutions to boundary problems of mechanics are found, in which the defining differential equations have a fractional order. In particular, such problems as a model of a “fractal” oscillator, a model problem on the dynamic of wave propagation in rock, model problems on the deformation of wave propagation in deformable viscoelastic media (a semi-infinite viscoelastic rod) for various viscoelasticity models are considered. When building the solutions, the Mainardi algorithm and the Laplace transformation are used. Model solutions for the considered problems are built. Asymptotic solutions of wave propagation equations in viscoelastic media under different viscoelasticity models are obtained.

In this work, the quantum-mechanical problem of the motion of two material points of different masses on a three-dimensional sphere with a non-fixed position of the center of mass of the system is formulated on the basis of the previously solved classical problem. It is shown that the established Schrödinger equation includes two different reduced masses, depending on the distance between the points. For the case of the interaction potential of points, depending only on the distance between them, this equation allows the separation of variables into a radial, depending on the relative distance and both the reduced masses and the spherical part. The equation for the spherical part depends only on one of the above reduced mass and allows one to formulate and solve the problem of a rigid rotator - the distance between the points is fixed. The solution and spectrum of the problem of a rigid rotator are found. It is shown that the spectrum of the system has an upper limit that does not depend on the distance between points, in contrast to the spectrum in a flat space.

It is Petras who first developed the P-symmetric theory for a spin 1/2 particle with an anomalous magnetic moment within the general Gel’fand – Yaglom approach. Recently, similarly it was introduced a P-asymmetric wave equation for a spin 1/2 particle which describes a particle with an electric dipole moment. In this paper, we study solutions of the equation for the P-asymmetric particle in presence of external magnetic fields. It turns out that the energy spectra are the same for P-asymmetric and P-symmetric particles. To clarify this coincidence, we demonstrate that there exists a simple transformation relating these two models, by which one wave equation can be reduced to the form of the other. Meanwhile, expressions for wave functions and P-reflection operators are different in these two theories. We extend this approach to the model in which both P-symmetric and P-asymmetric sectors are presented. The main result is the same, namely, there exists a simple, more general as compared with the mentioned above transformation relating the P-symmetric model and the model with two sectors, and expressions for wave functions and P-reflection operators are different in these two bases. We demonstrate that in the presence of an external uniform magnetic field, the energy spectra in the model with two sectors coincide with those in the P-symmetric theory. Thus, we develop a general theory for the P-asymmetric model and the model with two sectors within the Petras approach.

In this work, the process of transformation of an annular beam in a Bessel-like field due to diffraction during propagation in a free space over long distances and due to the focusing effect is investigated. A number of models of annular fields are considered, including an analytical model in the form of a polynomial function in a bounded region of space, as well as an experimentally implemented model based on a scheme with two axicons. A comparison is made of the transverse and longitudinal intensity distributions for these models, and a high degree of stability of the structure of the longitudinal distribution of the axial intensity to a change in the model of the annular field is found. This distribution is characterized by the presence of an intense maximum with an asymmetric profile, the appearance of which is not connected with lens focusing. In the initial region of the pointed maximum, the process of formation of a Bessel beam from an annular beam arises, and a sharp increase in intensity takes place. It is also established that the focusing of an annular field at large distances essentially differs from focusing at short distances. In the case of large distances, the increase of the axial intensity does not take place in the vicinity of the focal plane, but much closer to the transmitter, and here the great increase of intensity caused by direct focusing is not identified. The transverse profile of a Bessel-like beam is calculated at large distances. It is shown that this profile is characterized by a small number of lateral rings, and the axial maximum and the first ring contain more than 90% of the light power. The problem of generation of a model annular field by a Fourier-type resonator with a special transparency mirror is considered.

The basis of the modern methods of protecting products, goods, documents, and equities are holographic security technologies, the development of which indicates that they are improving, new equipment and materials appear, new methods of hologram recording are being developed, their manufacturability is increasing, identification and authentication are being simplified. Combined images are widely used in modern holography. When creating combined images, various optical effects arise, such as moire, parallax, color change, etc., which, in combination with each other, as well as with other images (microtext, hidden images, sequential numbering, marking, coding, chemical indicators), permit to use them both to protect documents and to obtain an original artistic effect. This article discusses combined protective elements based on a relief-phase hologram with a deposited polymer carrier layer containing a latent image visible in polarized light. This protective element is named crystallogram. In the process of developing a crystallogram, the synthesis of monomers and the preparation of an anisotropic polarizable composition was mastered, a layer of polymerizable liquid crystals (PLC) was obtained with a contrast visualization of a latent image. A technology was developed for combining the relief-phase hologram with the deposited polymer carrier layer with the subsequent blocking of the polarizable liquid crystals (PLC) layer by protective varnish layers.

The design and modeling of a metasurface is carried out, which makes it possible to transform an incident linearly polarized electromagnetic wave into a transmitted wave with elliptical polarization close to circular. At the same time, the reflection coefficient of the wave is close to zero at the resonant frequency, since the metasurface is similar to the free space in its wave resistance. The resonant elements of the meta-surface (meta-atoms) are two-turn planar spirals with balanced dielectric and magnetic properties. Such spirals exhibit radically different properties with respect to waves with right and left circular polarization. The metasurface as a polarization converter has strong chiral properties, since it contains planar spirals of only one direction of twisting, and can be manufactured within the framework of printed circuit board technologies.

This paper presents guidelines for modeling the capacity of electronic household waste collection points. These points are used as infrastructure elements with a multi-stage logistic support scheme for the electronic waste disposal process. This paper includes theoretical and methodological information on the procedure for placing points of waste collection in cities using the processes of determining the parameters of waste accumulation, calculating the design capacity of warehouses at these points, and developing routes for the transportation of waste to the places of their disposal. We represent the dependence of the logistic support costs, including the costs of maintaining waste collection points, and waste disposal to utilization facilities, on the duration of the waste accumulation period. A mathematical model for optimizing the logistic support costs is developed, which takes into account the most important parameters of the waste disposal system, namely, the topology of the collection points, the intensity of waste accumulation, the configuration of the routes, and the vehicle carrying capacity. Using the example of the Vietnamese capital, the city of Hanoi, the required number of waste collection points is calculated, the volume of waste accumulation at each point is determined, the optimal period of waste accumulation, in which the total costs for logistic support for the disposal process will be minimal, is determined. Recommendations on the organization of waste transportation, depending on the actual level of filling the capacity of collection and accumulation points, are given.