We apply the operator approach to a stochastic system belonging to a class of death-birth processes, which we introduce utilizing the master equation approach. By employing DoiPeliti formalism we recast the master equation in the form of a Schr & ouml;dinger-like equation. Therein appearing pseudo-Hamiltonian is conveniently expressed in a suitable Fock space, constructed using bosonic-like creation and annihilation operators. The kernel of the associated time evolution operator is rewritten using a functional integral, for which we propose an approximate method that allows its analytical treatment. The method is based on the expansion in eigenfunctions of the Hamiltonian generating given functional integral. In this manner, we obtain approximate values for the probabilities of the system being in the first and second states for the case of the pure birth process.

This paper explores the numerical intergator of ODE based on combination of Appelroth’s quadratization of dynamical systems with polynomial right-hand sides and Kahan’s discretization method. Utilizing Appelroth’s technique, we reduce any system of ordinary differential equations with a polynomial right-hand side to a quadratic form, enabling the application of Kahan’s method. In this way, we get a difference scheme defining the one-to-one correspondence between the initial and final positions of the system (Cremona map). It provides important information about the Kahan method for differential equations with a quadratic right-hand side, because we obtain dynamical systems with a quadratic right-hand side that have movable branch points. We analyze algebraic properties of solutions obtained through this approach, showing that (1) the Kahan scheme describes the branch points as poles, significantly deviating from the behavior of the exact solution of the problem near these points, and (2) it disrupts algebraic invariant variety, in particular integral relations describing the relationship between old and Appelroth’s variables. This study advances numerical methods, emphasizing the possibility of designing difference schemes whose algebraic properties differ significantly from those of the initial dynamical system.

—Lagutinskii’s theory of integration of dynamic systems is reformulated for arbitrary linear systems of hypersurfaces. The following problems are considered. Some dynamic system and some linear system of algebraic hypersurfaces are given. It is necessary to determine whether any integral curve lies on one of the hypersurfaces of the linear system. In the affirmative case, it is necessary (1) to formulate an equation for this hypersurface and (2) prove the existence of the integral of motion and write an explicit expression for it. An example is constructed showing that the hypersurfaces of the initial linear system and lines of the level of the integral may not coincide.

The discussion revolves around the most recent outcomes in the realm of approximating functional integrals through calculations. Review of works devoted to the application of functional integrals in quantum mechanics and quantum field theory, nuclear physics and in other areas is presented. Methods obtained by the authors for approximate calculation of functional integrals generated by nonrelativistic Hamiltonians are given. One of the methods is based on the expansion in eigenfunctions of the Hamiltonian. In an alternate approach, the functional integrals are tackled using the semiclassical approximation. Methods for approximate evaluation of functional integrals generated by relativistic Hamiltonians are presented. These are the methods using functional polynomial approximation (analogue of formulas of a given degree of accuracy) and methods based on the expansion in eigenfunctions of the Hamiltonian, generating a functional integral.

The paper presents the results of an experimental and numerical study of some properties of multimode liquid crystal waveguide structures. The nematic 4-cyano-4'-pentylbiphenyl (5CB) was used as a liquid crystal. A description of the experiments performed and some of the techniques used are given. Scattering diagrams are presented that characterize the features of propagation in a multimode liquid-crystal waveguide of one and many modes. It is shown that when an external repetitively pulsed electric field is switched on, the attenuation and size of inhomogeneities decrease. To explain this effect, the classical theory of liquid crystal director fluctuations is used. For the first time some properties of a liquid-crystal waveguide are described with explicit allowance for the two-dimensional Frederiks model. The two-dimensional nature of the liquid crystal director reorientation effects in our case (including under the action of an electric impulse-periodic field) required the involvement of the three-dimensional theory of light scattering in an LC waveguide and, as a consequence, the study of two-dimensional scattering diagrams in experiments, which make it possible to consider the two-dimensional nature of the behavior of the LC director. The relevance and importance of such studies are related both to the practical use and prospects of using liquid crystal materials in various high-speed and low-energy integrated-optical devices, for example, in communication elements, modulators, and sensors.

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.

At the temperature 293 K, the influence of two types of nanoimpurities (carbon multiwall nanotubes and C60 fullerene) both separately and together on the dielectric properties of Shell oil transformer oil has been studied. It has been shown that these impurities do not significantly effect on the value of the dielectric permittivity of Shell oil, but more significantly increase its conductivity. It has been found that in the presence of nanotubes inside Shell oil, the dependence of its electrical conductivity on the fullerene concentration is nonmonotonic. The samples with the fullerene concentration 100 ppm have the highest conductivity. At the fullerene concentration 300 ppm, the conductivity of Shell oil with the impurities of carbon nanotube and C60 fullerene becomes almost equal to the electrical conductivity of Shell oil only with the impurities of carbon nanotubes. It has been suggested that C60 fullerene can be used to reduce the electrical conductivity of Shell oil with magnetic nanoparticles required to increase the cooling efficiency of transformers under the action of their own magnetic field.

In this paper, we numerically investigate a two-dimensional differential equation describing the motion of a director of a nematic liquid crystal for the case of an alternating external electric field. The presence of the previously discovered accumulation effect has been confirmed by numerical modeling. A comparison is made with the case of a constant electric field, and also a qualitative comparison with an experiment is given. Incomplete agreement with experimental data indicates the need for further research. However, it should be noted that the constructed mathematical model of the phenomenon allows at this stage to obtain estimates that are sufficiently acceptable for experiment and correctly predict the dynamics of processes in liquid crystals. An analysis of the features of the propagation of quasi-waveguide modes in a liquid crystal waveguide showed that, in the case of dynamic processes, such effects as power exchange between coupled modes, leakage of modes, re-emission of modes into modes of a different order, etc., can be observed. The programs for numerical solution and computer modeling of two-dimensional parabolic partial differential equation were developed both in FORTRAN and C/C++. The results obtained are important for further investigation of dynamic processes inside non-stationary liquid crystal layers, both from a theoretical point of view for understanding kinetic processes in liquid crystals and from a practical point of view when organizing and conducting different experimental research.

This paper examines the problem of detecting skin malignancies, in particular, melanoma, from the analysis of dermoscopic images using deep learning methods. For this purpose, a deep convolutional neural network architecture was developed, which was used to process dermoscopic images of various skin lesions contained in the HAM10000 data set. The studied images were previously cleared of noise and other contaminants for processing by neural networks. In addition, since the disease classes are unbalanced, a number of transformations have been made to balance them. At the first stage, the images were divided into two classes: melanoma and benign tumor. At the second stage, all images of skin injuries were grouped into seven classes. Computer experiments on the use of the constructed deep neural network on the data obtained in this way have shown that the proposed approach provides 91%.

We consider the construction of the fundamental function and Abelian differentials of the third kind on a plane algebraic curve over the field of complex numbers that has no singular points. The algorithm for constructing differentials of the third kind is described in Weierstrass's Lectures. The article discusses its implementation in the Sage computer algebra system. The specificity of this algorithm, as well as the very concept of the differential of the third kind, implies the use of not only rational numbers, but also algebraic ones, even when the equation of the curve has integer coefficients. Sage has a built-in algebraic number field tool that allows implementing Weierstrass's algorithm almost verbatim. The simplest example of an elliptic curve shows that it requires too many resources, going far beyond the capabilities of an office computer. Then the symmetrization of the method is proposed and implemented, which solves the problem and allows significant economy of resources. The algorithm for constructing a differential of the third kind is used to find the value of the fundamental function according to the duality principle. Examples explored in the Sage system are provided.

The paper investigates the problem of recognizing human emotions by voice using deep learning methods. Deep convolutional neural networks and recurrent neural networks with bidirectional LSTM memory cell were used as models of deep neural networks. On their basis, an ensemble of neural networks is proposed. We carried out computer experiments on using the constructed neural networks and popular machine learning algorithms for recognizing emotions in human speech contained in the RAVDESS audio record database. The computational results showed a higher efficiency of neural network models compared to machine learning algorithms. Accuracy estimates for individual emotions obtained using neural networks were 80%. The directions of further research in the field of recognition of human emotions are proposed.

In this paper, the matrix-valued functional integrals generated by the Dirac equation with relativistic Hamiltonian are considered. The Dirac Hamiltonian contains scalar and vector potentials. The sum of the scalar and vector potentials is equal to zero, i.e., the case of pseudospin symmetry is investigated. In this case, a Schrödinger-type equation for the eigenvalues and eigenfunctions of the relativistic Hamiltonian generating the functional integral is constructed. The eigenvalues and eigenfunctions of the Schrödinger-type operator are found using the Sturm sequence method and the reverse iteration method. A method for the evaluation of matrix-valued functional integrals is proposed. This method is based on the relation between the functional integral and the kernel of the evolution operator with the relativistic Hamiltonian and the expansion of the kernel of the evolution operator in terms of the found eigenfunctions of the relativistic Hamiltonian.

Experimentally observed leaky modes of a dielectric waveguide are characterised by a weak tunnelling of the light through the waveguide and its long-time propagation along the waveguide. Traditional mathematical models of leaky waveguide modes meet some contradictions resolved using additional considerations. We propose a model of leaky modes in a waveguide free from the above contradictions, akin to the quantum mechanical model of the "pseudo-stable" Gamow-Siegert states. By separating variables, from the complete problem for plane inhomogeneous waves we obtain a non-self-adjoint Sturm-Liouville problem to determine the complex coefficient of the phase delay of the studied mode. The solution of the complete wave problem determines the propagation cone for the leaky mode of the waveguide, inside which there are no contradictions. Thus, solution is in qualitative agreement with experimental data.

An approximate evaluation of matrix-valued functional integrals generated by the relativistic Hamiltonian is considered. The method of evaluation of functional integrals is based on the expansion in the eigenfunctions of Hamiltonian generating the functional integral. To find the eigenfunctions and the eigenvalues the initial Hamiltonian is considered as a sum of the unperturbed operator and a small correction to it, and the perturbation theory is used. The eigenvalues and the eigenfunctions of the unperturbed operator are found using the Sturm sequence method and the reverse iteration method. This approach allows one to significantly reduce the computation time and the used computer memory compared to the other known methods.

The main contribution of the program is in the analytic approach to the setting of dielectric constant for all discretization points of a grid in the computational domain according to their locations with respect to the molecular surface allowing also taking into account possible cavities. Assigned values of dielectric constant are used in a solver of the linear Poisson-Boltzmann equation for a protein-water system using a finite difference method. (C) 2019 Elsevier B.V. All rights reserved.

The problem of the existence of explicit and at the same time conservative finite difference schemes that approximate a system of ordinary differential equations is investigated. An autonomous system of nonlinear ordinary differential equations on an algebraic manifold V is considered. A difference scheme for solving this system is called conservative, if the calculations of this scheme do not go beyond V, i.e., preserve it exactly. An explicit scheme is understood as such a difference scheme in which a system of linear equations is required to proceed to the next layer. We formulate the problem of constructing an explicit conservative scheme approximating a given autonomous system on a given manifold. For the case of 1-manifold, a solution to this problem is given and geometric obstacles to the existence of such difference schemes are indicated. Namely, it is proved that the scheme exists only if the genus of the integral curve is 1 or 0.

In the first part of the article we give a brief review of various approaches to symbolic integration of ordinary differential equations (Liouvillian approach, power series method) from the point of view of numerical methods. We aim to show that all higher transcendental functions were considered in the past centuries as solutions of such differential equations, for which the application of the computational techniques of that time was particularly efficient. Nowadays the finite differences method is a standard method for integration of differential equations. Our main idea is that now all transcendental functions can be considered as solutions of such differential equations, for which the application of this method is particularly efficient. In the second part of the article we consider an autonomous system of differential equations with algebraic integrals of motion and try to find a totally conservative difference scheme. There are only two cases when the system can be discretized by explicit totally conservative scheme: integrals specify an elliptic curve or unicursal curve. For autonomous systems describing the Jacobi elliptic functions we construct the finite differences scheme, which conserves all algebraic integrals and defines one-to-one correspondence between the layers. We can see that this scheme truly describes the periodicity of the motion.

The mixing of the levels of a compound nucleus in the field of a high-intensity light wave is considered. The cross sections for inelastic neutron scattering and radiative capture are computed with allowance for this effect. The effect of the electron shell on the effective charge for the dipole interaction between the neutron + nucleus system and the external electromagnetic field is taken into consideration.

A new method for the evaluation of the characteristics of the solution of a system of stochastic differential equations is presented. This method is based on the representation of a probability density functionpthrough a functional integral. The functional integral representation is obtained by means of the Onsager-Machlup functional technique for a special case when the diffusion matrix for the SDE system defines a Riemannian space with zero curvature.

Maxwell equations describe the propagation with diffraction of waveguide modes through a thin-film waveguide lens. If the radius of the thin-film lens is large, then the thickness of the lens varies slowly in the yz plane. For this case we propose the model, which is based on the assumption of a small change in the electromagnetic field in a direction y . Under this assumption the vector diffraction problem is reduced to a number of scalar diffraction problems. The solutions demonstrate the vector nature of the electromagnetic field, which allows us to call the proposed model a quasi-vector model.