# Математика. Рубрика в журнале - Вестник Южно-Уральского государственного университета. Серия: Математика. Механика. Физика

Alternative routs of games with rigid schedule

Статья научная

Corporative-competitive system, which is inside of corporations, can be determined as a “game”, step-by-step performing a certain type of activity. The system operates in real physical time, and the result of operation is the distance, which is divided into stages. The stages are passed by the team participants due to rigid schedule, which may be occasionally selected from the set of possible schedules. The abstraction “M-parallel semi-Markov process” is used for description of a system under consideration. In semi-Markov process degenerate distribution is used for description of time intervals between relay points. For analysis of relay-race evolution, recurrent method which takes into account rigidity of schedule and stochastic character of route selection is used. In accordance with the concept of distributed forfeit and proposed recurrent procedure, the method of calculation of summing forfeit, which one of competing teams receives from other teams, is proposed.

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Analysis of the boundary value problem for the Poisson equation

Статья научная

The mixed boundary value problem for the Poisson equation is considered in a bounded flat domain. The continuation of this problem through the boundary with the Dirichlet condition to a rectangular domain is carried out. Consideration of the continued problem in the operator form is proposed. To solve the continued problem, a method of iterative extensions is formulated in an operator form. The extended problem in operator form is considered on a finite-dimensional subspace. To solve the previous problem, an iterative extension method is formulated in operator form on a finite-dimensional subspace. The continued problem is presented in matrix form. To solve the continued problem in matrix form, the method of iterative extensions in matrix form is formulated. It is shown that in the proposed versions of the method of iterative extensions, the relative errors converge in a rate that is stronger than the energy norm of the extended problem with the rate of geometric progression. The iterative parameters in these methods are selected using the minimum residual method. Conditions are indicated that are sufficient for the convergence of the applied iterative processes. An algorithm is written that implements the method of iterative extensions in matrix form. In this algorithm, the iterative parameters are automatically selected and the stopping criterion is indicated when the estimate of the required accuracy is reached. Examples of application of the method of iterative extensions for solving problems on a computer are given.

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Asymptotic solution of the perturbed first boundary value problem with a non-smooth coefficient

Статья научная

In this paper, we consider the first boundary value problem, that is the Dirichlet problem in a ring for a linear inhomogeneous second-order elliptic equation with two independent variables containing a small parameter in front of the Laplacian. The equation potential is not a smooth function in the field under study. There exists a unique solution of the first boundary value problem under consideration. It is impossible to construct an obvious solution of the first boundary value problem. We are interested in the influence of the small parameter on the solution of the Dirichlet problem in the field under study when the small parameter tends to zero. That is why we need to construct an asymptotic solution of the first boundary value problem in a ring. The problem under consideration has two singularities (a bisingular problem): presence of a small parameter in front of the Laplacian, and solution of a relevant unperturbed equation is not a smooth function in the field under study. To construct an asymptotic solution, we use a modified method of boundary functions since it is impossible to use a classical method of boundary functions. To begin with, we construct a formal asymptotic solution as per the small parameter, and then we evaluate the remainder term of the asymptotic expansion. As a result, we have constructed complete uniform asymptotic expansion of the first boundary value problem in a ring as per the small parameter. The constructed series of the solution of the first boundary value problem is asymptotic in the sense of Erdey.

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Calculation of discrete semi-bounded operators’ eigenvalues with large numbers

Статья научная

In previous works of the article’s authors on development of the Galerkin method, linear formulas for calculating the approximate eigenvalues of discrete lower semi-bounded operators have been obtained. The formulas allow calculating the eigenvalues of the specified operators of any number, regardless of whether the eigenvalues of the previous numbers are known or not. At that, it is possible to calculate the eigenvalues with large numbers when application of the Galerkin method is becoming difficult. It is shown that eigenvalues of small numbers of various boundary-value problems, generated by discrete lower semi-bounded operators and calculated by linear formulas and by the Galerkin method, are in a good conformity. In this paper we use linear formulas to calculate approximate eigenvalues with large numbers of discrete lower semi-bounded operators. Results of calculation of eigenvalues by linear formulas and by known asymptotic formulas for two spectral problems are given. Comparison of the results of calculations of the approximate eigenvalues shows that they almost coincide for sufficiently large numbers. This proves the fact that linear formulas can be used for the considered spectral problems and sufficiently large numbers of eigenvalues.

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Статья научная

In this paper, we introduce a new sort of fractional derivative. For this, we consider the Cauchy's integral formula for derivatives and modify it by using Laplace transform. So, we obtain the fractional derivative formula F(α)(s) = L{(-1)(α)L-1{F(s)}}. Also, we find a relation between Weyl's fractional derivative and the formula above. Finally, we give some examples for fractional derivative of some elementary functions.

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Статья научная

We begin classification of prime knots in the thickened torus of genus 2 having diagrams with at most 4 crossings. To this end, it is enough to construct a table of prime knot projections with at most 4 crossings, and use the table to obtain table of prime diagrams, i. e. table of prime knots. In this paper, we present the result of the first step, i. e. we construct a table of prime projections of knots in the thickened torus of genus 2 having at most 4 crossings. First, we introduce definition of prime projection of a knot in the thickened torus of genus 2. Second, we construct a table of prime projections of knots in the thickened torus of genus 2 having at most 4 crossings. To this end, we enumerate graphs of special type and consider all possible embeddings of the graphs into the torus of genus 2 that lead to prime projections. In order to simplify enumeration of the embeddings, we prove some auxiliary statements. Finally, we prove that all obtained projections are inequivalent. Several known and new tricks allow us to keep the process within reasonable limits and rigorously theoretically prove the completeness of the constructed table.

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Correlations between elements and sequences in a numerical prism

Статья научная

A numerical prism, previously introduced by the author as an ordered set regarding the research of a three-parameter probability distribution of the hyperbolic-cosine type, which is a generalization of the known two-parameter Meixner distribution, is being considered. In geometry-related terminology, elements of a numerical prism are the coefficients of moment-forming polynomials for the specified distribution, which are obtained with the use of both differential and algebraic recurrence correlations. Each one of the infinite number of elements depends on three indices determining its position in a prism. Fixation of one or two indices results in cross-sections of the prism, which are numerical triangles or sequences. Among them, there are such well-known cross-sections as the Stirling number triangle, number triangle of coefficients in the Bessel polynomials, sequences of tangent and secant numbers, and others. However, the majority of numerical sets in the prism’s cross-sections have never been described in literature before. Considering the structure and construction algorithm, cross-sections of the numerical prism turn out to be interconnected not only by the general construction formula but also by certain correlations. As a result, formulas of connection between various groups of elements are presented in the article. In particular, expansion of secant numbers for the sum of products grouped by the number of tangent numbers’ cofactors with specification of corresponding coefficients in the expansion, representation (automatic expression) of elements of a sequence of alternating secant and tangent number through the previous ones, as well as a number of other correlations for the sequences and particular elements is determined.

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Description of some weighted exponential classes of subharmonic functions

Статья научная

The role of subharmonic functions in such sections of analysis as complex and real analysis is very significant. Such classes of functions are closely related to analytic harmonic functions and make an important contribution to the general theory of potential and mathematical physics. In the works of R. Nevanlinna and W. Heiman, parametric representations of subharmonic classes in the plane of functions, whose characteristic has a power growth at infinity, are obtained. The question of whether similar representations are true for weighted classes that admit a stronger growth at infinity (for example, the exponential growth) arises in the theory of entire and meromorphic functions. In this article, classes of subharmonic functions with Nevanlinna characteristic that is summable with exponential weight in a complex plane are introduced for consideration, and the representing measures of functions of such classes are studied. When proving the results, methods of complex and functional analysis are used. An important role in the study is played by potentials based on the factors of the modified Weierstrass product. The proof of the main result is based on the use of auxiliary assertions formulated in the form of lemmas.

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Exact solutions of the Hirota equation using the sine-cosine method

Статья научная

Nonlinear partial differential equations of mathematical physics are considered to be major subjects in physics. The study of exact solutions for nonlinear partial differential equations plays an important role in many phenomena in physics. Many effective and viable methods for finding accurate solutions have been established. In this work, the Hirota equation is examined. This equation is a nonlinear partial differential equation and is a combination of the nonlinear Schrödinger equation and the complex modified Korteweg-de Vries equation. The nonlinear Schrödinger equation is the physical model and occurs in various areas of physics, including nonlinear optics, plasma physics, superconductivity, and quantum mechanics. The complex modified Korteweg-de Vries equation has been applied as a model for the nonlinear evolution of plasma waves and represents the physical model that incorporates the propagation of transverse waves in a molecular chain model and in a generalized elastic solid. To find exact solutions of the Hirota equation, the sine-cosine method is applied. This method is an effective tool for searching exact solutions of nonlinear partial differential equations in mathematical physics. The obtained solutions can be applied when explaining some of the practical problems of physics.

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H-однородные lambda-пространства

Статья научная

Пусть Х - h-однородное сепарабельное метризуемое несчётное lambda-пространство. Тогда: 1) Х является CDH-пространством, 2) Х гомеоморфно Х \ А для любого счётного множества А из Х.

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Invariant spaces of Oskolkov stochastic linear equations on the manifold

Статья научная

The Oskolkov equation is obtained from the Oskolkov system of equations describing the dynamics of a viscoelastic fluid, after stopping one of the spatial variables and introducing a stream function. The article considers a stochastic analogue of the linear Oskolkov equation for plane-parallel flows in spaces of differential forms defined on a smooth compact oriented manifold without boundary. In these Hilbert spaces, spaces of random K-variables and K-“noises” are constructed, and the question of the stability of solutions of the Oskolkov linear equation in the constructed spaces is solved in terms of stable and unstable invariant spaces and exponential dichotomies of solutions. Oskolkov stochastic linear equation is considered as a special case of a stochastic linear Sobolev-type equation, where the Nelson-Glicklich derivative is taken as the derivative, and a random process acts as the unknown. The existence of stable and unstable invariant spaces is shown for different values of the parameters entering into the Oskolkov equation.

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Inverse problem for Sobolev type equation of the second order

Статья научная

The paper deals with the inverse problem for the Sobolev type equation of the second order in Banach spaces. The introduction contains a problem statement and the historiography of Sobolev type equations. The second part includes preliminary information based on the results of the theory of higher-order Sobolev type equations. In the third part the initial problem is reduced to the inverse regular and singular problems. A theorem of unique solvability of regular problem is formulated and proved. Using the results of the third part, the solution for the singular problem is obtained in the fourth part. The sum of regular and singular solutions is a solution to the original problem, thus a theorem on the unique solvability of the inverse problem for Sobolev type equation of the second order is stated and proved.

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Inverse problems of recovering the boundary data with integral over determination conditions

Статья научная

In the present article we examine an inverse problem of recovering unknown functions being part of the Dirichlet boundary condition together solving an initial boundary problem for a parabolic second order equation. Such problems on recovering the boundary data arise in various tasks of mathematical physics: control of heat exchange prosesses and design of thermal protection systems, diagnostics and identification of heat transfer in supersonic heterogeneous flows, identification and modeling of heat transfer in heat-shielding materials and coatings, modeling of properties and heat regimes of reusable heat protection of spacecrafts, study of composite materials, etc. As the overdetrermination conditions we take the integrals of a solution over the spatial domain with weights. The problem is reduced to an operator equation of the Volterra-type. The existence and uniqueness theorem for solutions to this inverse problem is established in Sobolev spaces. A solution is regular, i. e., all generalized derivatives occuring into the equation exists and are summable to some power. The proof relies on the fixed point theorem and bootstrap arguments. Stability estimates for solutions are also given. The solvability conditions are close to necessary conditions.

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Making investment decisions in an industrial enterprise under uncertainty

Статья научная

The article is devoted to the issue of selecting investment projects for the modernization of an industrial enterprise in the conditions of uncertainty. Uncertainty in this work is understood as the lack of data on the likelihood of the implementation of scenarios of alternative investment projects. As a methodological base, a set of generally accepted quantitative criteria for reducing uncertainty is proposed, which is characterized by different attitudes towards the probability of future events (pessimistic, optimistic, neutral, etc.). Additionally, a criterion is selected that combines expert and quantitative assessment of uncertainty. Practical calculations are made according to five criteria for three scenarios of five investment projects of an industrial enterprise. The aim of this work is to assess the difficulty of making unambiguous investment decisions under conditions of uncertainty. As a result, each of the four quantitative criteria has pointed to a different project. The fifth criterion has made it possible to limit the range of choice to two projects. This has confirmed the conclusions that making investment decisions requires the responsible person to use not only quantitative indicators, but also existing experience and intuition. The reliability of the findings has been confirmed by the use of generally accepted models and methods, as well as the practical implementation of the results. It is recommended to use these results in the scientific community when conducting subsequent methodological studies aimed at reducing uncertainty, as well as to business owners and investors when making strategic decisions.

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Статья научная

The work is the continuation of the authors' work on the simulation of the structure of the society by the method of random graphs generation. In their previous works, the authors experimentally proved that the social graph has a strict orientation of the information direction propagation from the shares of the graph with high bonds density to the graph parts having a lower bond density. The authors proposed a method for determining the orientation of the social graph by investigating the ratio of the number of outgoing links to the incoming ones. This method found its application in the analysis of postal mailing, but it turned out to be practically inapplicable in the analysis of social networks, since the counting of incoming and outgoing communications required the compilation of the announced social graphs which was a time-consuming and computational resources task that does not have a solution for polynomial time at present. Since the issue of the social networks analysis is not only of a strictly scientific, but also of a practical interest, the authors developed and tested a technique for determining the orientation of social graphs by the method of analyzing the numerical characteristics of a graph. The new method does not require a detailed analysis of the correspondence of users of social networks, but operates with open user information that is a list of friends (Friend List). The ultimate goal of this work is to develop simple and effective methods for analyzing social networks to identify “opinion leaders”, ways of disseminating information, including propagandizing deviant and dependent forms of behavior, identifying anti-systems and separate closed network communities, and general monitoring of the state of social systems.

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Methods of speech and text databases development for QA-systems

Статья научная

The paper is devoted to the problems of question-answer systems development (QA-systems). The subject of the study is discussion of approaches to the automatic filling of the database of the QA-system based on the analysis of the unstructured text sources currently available in the public domain of the Internet. The analysis reveals that the following ways of implementing QA-systems are distinguished: based on inference for ontologies, rules and syntax, using artificial neural networks. The methods for automatically search of question-answer pairs based on the structure of sentences and on the basis of associative-ontological analysis has been developed and tested in the research. The method based on the analysis of the structure of sentences is effective for texts such as lists of frequently asked questions (FAQ), as well as literature texts containing dialogs, direct speech, based on preliminary processing of the text, expressed in the form of a heuristic rule. The method based on associative-ontological analysis is focused to the class of reference and dictionary texts and is based on the assumption that in the descriptive text there is a sentence (or a group of sentences) containing the main idea of the text. In this case, the title of the text can be considered a question, and this sentence (or a group of sentences) is the answer. We need to make the selection of meaning-generating sentences due to the semantic reduction of the text automation. For this purpose, algorithms of self-referencing are applied based on the associative-ontological approach to the processing of texts in natural language. For the experimental verification of the possibility of creating an open QA-system based on the automatic collection of question-answer pairs from the Internet, a prototype of a collection module for the database of the QA-system has been developed.

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Статья научная

The article is devoted to the study of the morphology of the phase space of a degenerate two-component mathematical model of a nerve impulse propagation in the membrane shell. A mathematical model is studied in the case when the parameter at the time derivative of the component responsible for the dynamics of the membrane potential is equal to zero, and the theorem about the fact that the phase space is simple in this case is proved. A mathematical model is also considered in the case when the parameter at the time derivative of the component responsible for the ion currents is equal to zero, and the theorem on the presence of singularities of Whitney assemblies is proved. Based on the results obtained, the phase space of the mathematical model is constructed in the case when the parameters at the time derivative of both components of the system are equal to zero. The author gives examples of the construction of the phase space, illustrating the presence of features in the phase space of the studied problems based on the Galerkin method.

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Non-classical equations of mathematical physics. Linear Sobolev type equations of higher order

Статья научная

The article presents the review of authors' results in the field of non-classical equations of mathematical physics. The theory of Sobolev-type equations of higher order is introduced. The idea is based on generalization of degenerate operator semigroups theory in case of the following equations: decomposition of spaces, splitting of operators' actions, the construction of propagators and phase spaces for a homogeneous equation, as well as the set of valid initial values for the inhomogeneous equation. The author uses a proven phase space technology for solving Sobolev type equations consisting of reduction of a singular equation to a regular one defined on some subspace of initial space. However, unlike the first order equations, there is an extra condition that guarantees the existence of the phase space. There are some examples where the initial conditions should match together if the extra condition can't be fulfilled to solve the Cauchy problem. The reduction of nonclassical equations of mathematical physics to the initial problems for abstract Sobolev type equations of high order is conducted and justified.

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On a Q-boundary value problem with discontinuity conditions

Статья научная

In this paper, we studied q-analogue of Sturm-Liouville boundary value problem on a finite interval having a discontinuity in an interior point. We proved that the q-Sturm-Liouville problem is self-adjoint in a modified Hilbert space. We investigated spectral properties of the eigenvalues and the eigenfunctions of q-Sturm-Liouville boundary value problem. We shown that eigenfunctions of q-Sturm-Liouville boundary value problem are in the form of a complete system. Finally, we proved a sampling theorem for integral transforms whose kernels are basic functions and the integral is of Jackson’s type.

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On basis property of root functions for a class of the second order differential operators

Статья научная

It is well known that the Sturmian theory is an important tool in solving numerous problems of mathematical physics. Usually, eigenvalue parameter appears linearly only in the differential equation of the classic Sturm-Liouville problems. However, in mathematical physics there are also problems, which contain eigenvalue parameter not only in differential equation, but also in the boundary conditions. In this paper, we consider a Sturm-Liouville equation with the eigenparameter dependent boundary condition and with transmission conditions at two points of discontinuity. The aim of this paper is to investigate the completeness, minimality and basis properties of rootfunctions for the considered boundary value problem.

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