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A new type of cosmological models for the Universe that has no Beginning and evolves from the infinitely distant past is considered.

These models are alternative to the cosmological models based on the Big Bang theory according to which the Universe has a finite age and was formed from an initial singularity.

In our opinion, there are certain problems in the Big Bang theory that our cosmological models do not have.

In our cosmological models, the Universe evolves by compression from the infinitely distant past tending a finite minimum of distances between objects of the order of the Compton wavelength $\lambda_C$ of hadrons and the maximum density of matter corresponding to the hadron era of the Universe. Then it expands progressing through all the stages of evolution established by astronomical observations up to the era of inflation.

The material basis that sets the fundamental nature of the evolution of the Universe in the our cosmological models is a nonlinear Dirac spinor field $\psi(x^k)$ with nonlinearity in the Lagrangian of the field of type $\beta(\bar{\psi}\psi)^n$ ($\beta = const$, $n$ is a rational number), where $\psi(x^k)$ is the 4-component Dirac spinor, and $\psi$ is the conjugate spinor.

In addition to the spinor field $\psi$ in cosmological models, we have other components of matter in the form of an ideal liquid with the equation of state $p = w\varepsilon$ $(w = const)$ at different values of the coefficient $w (−1 < w < 1)$. Additional components affect the evolution of the Universe and all stages of evolution occur in accordance with established observation data. Here $p$ is the pressure, $\varepsilon = \rho c^2$ is the energy density, $\rho$ is the mass density, and $c$ is the speed of light in a vacuum.

We have shown that cosmological models with a nonlinear spinor field with a nonlinearity coefficient $n = 2$ are the closest to reality.

In this case, the nonlinear spinor field is described by the Dirac equation with cubic nonlinearity.

But this is the Ivanenko–Heisenberg nonlinear spinor equation which W.Heisenberg used to construct a unified spinor theory of matter.

It is an amazing coincidence that the same nonlinear spinor equation can be the basis for constructing a theory of two different fundamental objects of nature — the evolving Universe and physical matter.

The developments of the cosmological models are supplemented by their computer researches the results of which are presented graphically in the work.

We consider a finite-dimensional optimization problem, the formulation of which in addition to the required variables contains parameters. The solution to this problem is a dependence of optimal values of variables on parameters. In general, these dependencies are not functions because they can have ambiguous meanings and in the functional case be nondifferentiable. In addition, their domain of definition may be narrower than the domains of definition of functions in the condition of the original problem. All these properties make it difficult to solve both the original parametric problem and other tasks, the statement of which includes these dependencies. To overcome these difficulties, usually methods such as non-differentiable optimization are used.

This article proposes an alternative approach that makes it possible to obtain solutions to parametric problems in a form devoid of the specified properties. It is shown that such representations can be explored using standard algorithms, based on the Taylor formula. This form is a function smoothly approximating the solution of the original problem for any parameter values, specified in its statement. In this case, the value of the approximation error is controlled by a special parameter. Construction of proposed approximations is performed using special functions that establish feedback (within optimality conditions for the original problem) between variables and Lagrange multipliers. This method is described for linear problems with subsequent generalization to the nonlinear case.

From a computational point of view the construction of the approximation consists in finding the saddle point of the modified Lagrange function of the original problem. Moreover, this modification is performed in a special way using feedback functions. It is shown that the necessary conditions for the existence of such a saddle point are similar to the conditions of the Karush – Kuhn – Tucker theorem, but do not contain constraints such as inequalities and conditions of complementary slackness. Necessary conditions for the existence of a saddle point determine this approximation implicitly. Therefore, to calculate its differential characteristics, the implicit function theorem is used. The same theorem is used to reduce the approximation error to an acceptable level.

Features of the practical implementation feedback function method, including estimates of the rate of convergence to the exact solution are demonstrated for several specific classes of parametric optimization problems. Specifically, tasks searching for the global extremum of functions of many variables and the problem of multiple extremum (maximin-minimax) are considered. Optimization problems that arise when using multicriteria mathematical models are also considered. For each of these classes, there are demo examples.

The numerical solving of the system of high-temperature radiative gas dynamics (HTRGD) equations is a computationally laborious task, since the interaction of radiation with matter is nonlinear and non-local. The radiation absorption coefficients depend on temperature, and the temperature field is determined by both gas-dynamic processes and radiation transport. The method of splitting into physical processes is usually used to solve the HTRGD system, one of the blocks consists of a joint solving of the radiative transport equation and the energy balance equation of matter under known pressure and temperature fields. Usually difference schemes with orders of convergence no higher than the second are used to solve this block. Due to computer memory limitations it is necessary to use not too detailed grids to solve complex technical problems. This increases the requirements for the order of approximation of difference schemes. In this work, bicompact schemes of a high order of approximation for the algorithm for the joint solution of the radiative transport equation and the energy balance equation are implemented for the first time. The proposed method can be applied to solve a wide range of practical problems, as it has high accuracy and it is suitable for solving problems with coefficient discontinuities. The non-linearity of the problem and the use of an implicit scheme lead to an iterative process that may slowly converge. In this paper, we use a multiplicative HOLO algorithm named the quasi-diffusion method by V.Ya.Goldin. The key idea of HOLO algorithms is the joint solving of high order (HO) and low order (LO) equations. The high-order equation (HO) is the radiative transport equation solved in the energy multigroup approximation, the system of quasi-diffusion equations in the multigroup approximation (LO₁) is obtained by averaging HO equations over the angular variable. The next step is averaging over energy, resulting in an effective one-group system of quasi-diffusion equations (LO₂), which is solved jointly with the energy equation. The solutions obtained at each stage of the HOLO algorithm are closely related that ultimately leads to an acceleration of the convergence of the iterative process. Difference schemes constructed by the method of lines within one cell are proposed for each of the stages of the HOLO algorithm. The schemes have the fourth order of approximation in space and the third order of approximation in time. Schemes for the transport equation were developed by B.V. Rogov and his colleagues, the schemes for the LO₁ and LO₂ equations were developed by the authors. An analytical test is constructed to demonstrate the declared orders of convergence. Various options for setting boundary conditions are considered and their influence on the order of convergence in time and space is studied.

Тhe aim of this work is to assess the mechanical behavior of coronary vessels and angioplasty balloons during the angioplasty procedure, based on intravascular ultrasound (IVUS) and angiography data obtained for each patient individually. To treat atherosclerosis, a serious chronic inflammatory disease of the arteries characterized by the formation of atherosclerotic plaques, which causes vessel narrowing and impairs blood supply to tissues and organs, modern medical practice employs a minimally invasive endovascular procedure known as balloon angioplasty. Key aspects of modeling this procedure include understanding the behavior of patients’ arteries and the balloons used during the intervention. Based on intravascular ultrasound and angiography data, a biomechanical model of an artery affected by atherosclerosis is developed. A finite element model of the arterial segment is constructed, accounting for its nonlinear hyperelastic behavior. To simulate the behavior of the angioplasty balloon, a mathematical model of the balloon is developed and validated against experimental data. An assessment of the stress-strain state of a specific patient’s coronary artery and angioplasty balloon is performed. For a full-scale simulation of the angioplasty process, a mathematical model is developed that incorporates all three objects considered above: personalization of the artery model through the use of real patient data; a finite element model of the artery built based on of the personalized model, accounting for its nonlinear behavior; a finite element model of the angioplasty balloon. The developed mathematical models and the results obtained from them will further allow us to derive dependencies of key angioplasty parameters. These dependencies can be used to improve angioplasty techniques based on intravascular imaging data. Furthermore, the application of mathematical modeling methods will help reduce the number of clinical trials in this field.

Many nonlinear control systems by nonsingular transformation variable {condition-control} happen to canonical Brunovsky form. The different questions dare in canonical form to theories of control, then inverse change variable is realized return to source variable. In work on base this ideology are studied transformations to symmetries space {time-condition-control}.

Numerical solutions for the two-dimensional reaction-diffusion equation with nonlocal nonlinearity are obtained. The solutions reveal formation of dissipative structures. Structures arising from initial distributions with one and several centers of localization are considered. Formation of extending circular structures is shown. Peculiarities of formation and interaction of extending circular structures depending on  nonlocal interaction are considered.

The paper describes a new algorithm of construction of the nonlinear arithmetic triangle on the basis of numerical simulation and the binary system. It demonstrates that the numbers that fill the nonlinear arithmetic triangle may be binomial coefficients of a new type. An analogy has been drawn with the binomial coefficients calculated with the use of the Pascal triangle. The paper provides a geometrical interpretation of binomials of different types in considering the branching systems of rays.

The work deals the implementation of a three-dimensional mathematical model of the nonlinear time-varying temperature control and a numerical method of solving it.

In this paper, we propose the unified procedure for the development and calibration of mathematical model for multilane urban road traffic flow. We use macroscopic approach, describing traffic flow with the system of second-order nonlinear hyperbolic equations (for traffic density and velocity). We close the resulting model with the equation of vehicle flow as a function of density, obtained empirically for each segment of road network using data from traffic detectors and vehicles’ GPS tracks. We verify the developed new model and calibration methods by using it to model segment of Moscows Ring Road.

Optical systems with two-dimensional feedback demonstrate wide possibilities for studying the nucleation and development processes of dissipative structures. Feedback allows to influence the dynamics of the optical system by controlling the transformation of spatial variables performed by prisms, lenses, dynamic holograms and other devices. A nonlinear interferometer with a mirror image of a field in two-dimensional feedback is one of the simplest optical systems in which is realized the nonlocal nature of light fields.

A mathematical model of optical systems with two-dimensional feedback is a nonlinear parabolic equation with rotation transformation of a spatial variable and periodicity conditions on a circle. Such problems are investigated: bifurcation of the traveling wave type stationary structures, how the form of the solution changes as the diffusion coefficient decreases, dynamics of the solution’s stability when the bifurcation parameter leaves the critical value. For the first time as a parameter bifurcation was taken of diffusion coefficient.

The method of central manifolds and the Galerkin’s method are used in this paper. The method of central manifolds and the Galerkin’s method are used in this paper. The method of central manifolds allows to prove a theorem on the existence and form of the traveling wave type solution neighborhood of the bifurcation value. The first traveling wave born as a result of the Andronov –Hopf bifurcation in the transition of the bifurcation parameter through the сritical value. According to the central manifold theorem, the first traveling wave is born orbitally stable.

Since the above theorem gives the opportunity to explore solutions are born only in the vicinity of the critical values of the bifurcation parameter, the decision to study the dynamics of traveling waves of change during the withdrawal of the bifurcation parameter in the supercritical region, the formalism of the Galerkin method was used. In accordance with the method of the central manifold is made Galerkin’s approximation of the problem solution. As the bifurcation parameter decreases and its transition through the critical value, the zero solution of the problem loses stability in an oscillatory manner. As a result, a periodic solution of the traveling wave type branches off from the zero solution. This wave is born orbitally stable. With further reduction of the parameter and its passage through the next critical value from the zero solution, the second solution of the traveling wave type is produced as a result of the Andronov –Hopf bifurcation. This wave is born unstable with an instability index of two.

Numerical calculations have shown that the application of the Galerkin’s method leads to correct results. The results obtained are in good agreement with the results obtained by other authors and can be used to establish experiments on the study of phenomena in optical systems with feedback.