## Mathematical Structures and Modeling ¹1(53)

 Mathematical Structures and Modeling. - Omsk : OmSU, 2020. ¹1(53), 152 p. ISSN (print): 2222-8772 ISSN (online): 2222-8799 For researchers, post-graduate students and senior students. Journal issue in one file

The Fundamental Mathematics and Physics

The article contains precise formulas for finding of the essential spectra that are revolted with asymptotically constants on infinity in coefficients with use of rather compact and rather small indignations on infinity in Lebesgue spaces of $$L^{p}$$. These formulas are new analogs of the classical theorem of Weyl.
Keywords: perturbation theory of operators, essential spectra, ordinary differential operators

M.N. Boldyreva, A.A. Magazev, I.V. Shirokov
Classification of Yang - Mills Fields Admitting Integrals of Motion for the Wong Equations

In the paper, we investigate the gauge fields that are characterized by the existence of non-trivial integrals of motion for the Wong equations. For the gauge group $$SU(2)$$, the class of fields admitting only the isospin first integrals is described in detail. All gauge non-equivalent Yang - Mills fields admitting a linear integral of motion for the Wong equations are classified in the three-dimensional Euclidean space.
Keywords: Yang--Mills field, the Wong equations, integral of motion, gauge transformation

M.N. Podoksenov, V.V. Chernykh
Autoisometries and Autosimilarities of Lie Algebra $$\mathcal{A}(1)\oplus \mathcal{R}^2$$

We consider four-dimensional Lie algebra $$\mathcal{A}(1)\oplus \mathcal{R}^2$$ endowed with Lorentzian scalar product. We find all the one-parameter groups of isometries and similarities, which are simultaneously automorphisms of Lie algebra, and also we find the conditions of existence of such one-parameter group. Conditions of existence are associated with the location of ideals with respect to isotropic cone.
Keywords: Lie algebra, Lorentzian metrics, automorphism, similarity, isometry

The electromagnetic field in electrical exploration problems is often represented as integrals with a fast-oscillating nucleus. When calculating these integrals on a computer, it is necessary to deform the contour of integration into the plane of the complex variable. The article studies the allowable deformation region of the integration contour in the case of a non-uniform medium, in which strong and weak solutions of electromagnetic field are analytical. The source of the field is a vertical dipole. A similar problem was solved for a horizontally layered medium with a harmonious electrical or magnetic dipole as a source.
Keywords: Electrical exploration, electromagnetic field of vertical electric or magnetic dipole, fast-oscillating integrals, deformation contour, complex plane, absence of singular points, deformation domain

On Geometry of Finsler Causality: For Convex Cones, There Is No Affine-Invariant Linear Order (Similar to Comparing Volumes)

Some physicists suggest that to more adequately describe the causal structure of space-time, it is necessary to go beyond the usual pseudo-Riemannian causality, to a more general Finsler causality. In this general case, the set of all the events which can be influenced by a given event is, locally, a generic convex cone, and not necessarily a pseudo-Reimannian-style quadratic cone. Since all current observations support pseudo-Riemannian causality, Finsler causality cones should be close to quadratic ones. It is therefore desirable to approximate a general convex cone by a quadratic one. This can be done if we select a hyperplane, and approximate intersections of cones and this hyperplane. In the hyperplane, we need to approximate a convex body by an ellipsoid. This can be done in an affine-invariant way, e.g., by selecting, among all ellipsoids containing the body, the one with the smallest volume; since volume is affine-covariant, this selection is affine-invariant. However, this selection may depend on the choice of the hyperplane. It is therefore desirable to directly approximate the convex cone describing Finsler causality with the quadratic cone, ideally in an affine-invariant way. We prove, however, that on the set of convex cones, there is no affine-covariant characteristic like volume. So, any approximation is necessarily not affine-invariant.
Keywords: space-time geometry, Finsler spaces, causality

Applied Mathematics and Modeling

A.V. Eremeev, A.V. Spirov
Estimates from Evolutionary Algorithms Theory Applied to Directed Evolution

The field of evolutionary computation emerged in the area of computer science due to transfer of ideas from biology and developed independently for several decades, enriched with techniques from probability theory, complexity theory and optimization methods. Our aim is to consider how some recent results form the theory of evolutionary computation may be transferred back into biology. It has been noted that the non-elitist evolutionary algorithms optimizing Royal Road fitness functions may be considered as models of evolutionary search for the synthetic enhancer sequences “from scratch”. This problem asks for a tight cluster of supposedly unknown motifs from the initial random (or partially random) set of DNA sequences using SELEX approaches. We apply the upper bounds on the expected hitting time of a target area of genotypic space in order to upper-bound the expected time to finding a sufficiently fit series of motifs in a SELEX procedure. On the other hand, using the theory of evolutionary computation, we propose an upper bound on the expected proportion of the DNA sequences with sufficiently high fitness at a given round of a SELEX procedure. Both approaches are evaluated in computational experiment, using a Royal Road fitness function as a model of the SELEX procedure for regulatory FIS factor binding site.
Keywords: runtime analysis, SELEX procedure, Royal Road function, enhancer

L.A. Volodchenkova, A.K. Guts
Mycelial Growth Analysis Based on the Chanter-Thornley Model

The Chanter-Thornley mycelial growth model is analyzed with point of view of the mathematical bifurcation theory. It was shown that one can observe not only the continuous growth of the mycelium, but also its saltatory evolution.
Keywords: mycelium, Chanter-Thornley model, saltatory growth, bifurcation theory

Bautin S. P., Bugaenko A. A., Krutova I. Yu.
Particular Solutions of a Linearized System of Gas Dynamics Equations in the Absence of Gravity Taking into Account the Actions of the Coriolis Force

For the system of gas dynamics equations, taking into account the action of the Coriolis force, linearization is carried out in case when the effect of gravity is not taken into account. Particular solutions in the form of running waves are found.
Keywords: system of equations of gas dynamics, Coriolis force, linearization, exact solutions

A study is made of the parametric singularity of the Cauchy integral that arises when the sides of a closed loop come together. Formulas are obtained that relate the values of the Cauchy integral on different sides of an extremely narrow contour.
Keywords: Cauchy integral, Sokhotsky-Plemel formulas, parametric singularity of the Cauchy integral

The existence of a parametric singularity of a three-dimensional analogue of the Cauchy integral over a closed surface is proved. This feature arises when the sides of a closed surface approach each other. Formulas are obtained that relate the values of the analogue of the Cauchy integral on different sides of an extremely narrow surface.
Keywords: Cauchy integral, parametric singularity of the Cauchy integral

Griselda Acosta, Eric D. Smith, Vladik Kreinovich
How to Explain That Changes in Elderlies Depression Level Are Uniformly Distributed

Changes in the elderlies depression level result from a large number of small independent factors. Such situations are ubiquitous in applications. In most such cases, due to the Central Limit Theorem, the corresponding distribution is close to Gaussian. For the changes in the elderlies depression level, however, the empirical distribution is far from Gaussian: it is uniform. In this paper, we provide a possible explanation for the emergence of the uniform distribution.
Keywords: elderlies depression, uniform distribution, changes in depression level

Computer Science

L.A. Volodchenkova, D.V. Kozirev
Creation of the Server Side of the Programm for Remote Data Storage

In this article the programm interface (API) of the server side of the cloud data storage is created.
Keywords: cloud storage, server side, software interface

A.I. Gorev, E.G. Gorev
About the Applicability of Existing Data Processing Algorithms to Big Data

The processes of obtaining BIG DATA during operation of computer technologies, the purpose of which is not to create data arrays, are defined. However, the use of the obtained data can have a great positive effect.
Keywords: big data, structured data, automated information systems

Education

Griselda Acosta, Eric D. Smith, Vladik Kreinovich
Confirmation Bias in Systems Engineering: A Pedagogical Example

One of the biases potentially affecting systems engineers is the confirmation bias, when instead of selecting the best hypothesis based on the data, people stick to the previously-selected hypothesis until it is disproved. In this paper, on a simple example, we show how important it is to take care of this bias: namely, that because of this bias, we need twice as many experiments to switch to a better hypothesis.
Keywords: confirmation bias, systems engineering, hypothesis testing