## Mathematical Structures and Modeling №1 (41) Mathematical structures and modeling . - Omsk: OmSU, 2017. No.1(41), 138  p. ISSN (print): 2222-8772 ISSN (online): 2222-8799 For researchers, post-graduate students and senior students. Journal issue in one file

Fundamental mathematics and physics

The necessary and sufficient conditions for convergence of distributions of symmetric functions of random variables to the normal law are obtained in this article. These conditions include the so-called minimal conditions of the weak dependence.
Keywords: Symmetric functions of random variables, central limit theorem, minimal conditions of the weak dependence.

I.A. Eganova, W. Kallies
Time Structure of the Complex Systems: Methodical Review

The structure of time series that describes the dynamics of the key characteristic of the complex system internal state is discussed as time structure corresponding to that system and assigning the mode of its existence. Based on the notion about the information included in the time structure, its mathematical description is proposed using the well-known means: the mean value of the key characteristic and its instantaneous deviation from the mean value. From this point of view, we discovered the meaning of the function used by H. Hurst in his analysis of time series that describe the dynamics of natural processes and phenomena: it assigns time structure, and its range is the size of the time structure in the period of time covered by observation. The authors define the size of the elements that compose the structure and propose an interpretation of the Hurst empirical law as a ratio that describes the quantity of structural elements in the given period of time. This interpretation allowed to propose an essentially new approach to the explanation of the so-called Hurst phenomenon (the values of the Hurst exponent are bigger than 1/2), as well as its properties observed in various factual evidence. In conclusion, the accessory of complex organized systems to harmonic systems (Yu.G. Kosarev, 1988) is discussed in brief.
Keywords: time series, time structure, R/S analysis, Hurst rescaled range, Hurst exponent, R/S statistics, Hurst statistics, Hurst phenomenon, Hurst empirical law.

A.N. Romanov

The article discusses the relationship between the behavior of the causal structure of space-time and its topology, namely, attention is paid to the study of the causes of the presence or absence of a causal closure sets of the past and the future depending on the conditions of compactness sets associated with past and future causal space-time points. An example, when the presence of closed non-compact sets of the space-time associated with the cause of future or past of any point entails the fact of not closed causal past and the future of some points, is given.
Keywords: space-time, compactness, causality.

Applied Mathematics and Modeling

Abstract. For the first time the two-dimensional disordered Ising model with deformed Tsallis statistics, with spin concentrations of 0.95 and 0.8 is investigated. The values of critical temperatures and the critical exponents are obtained. For disordered model with deformed statistics the emergence of a new type of critical behavior depending on the concentration of impurities is revealed.
Keywords: phase transitions, deformed statistics, Ising model.

L.A. Volodchenkova, A.K. Guts
Climax Forest as the Nash Equilibrium of Forest Ecosystems

To find the possible equilibrium states of forest ecosystems one are encouraged to use the theory of differential games. Within the 4-tier model of mosaic forest communities the existence in such ecosystems of the Nash equilibrium states is established.
Keywords: forest ecosystem, the equilibrium of the ecosystem, differential game, Nash equilibrium, climax forest.

S.L. Deryabin, A.S. Kiryanova
Generalization of a Centered Riemann Wave Taking into Account the Forces of Gravity

The paper examines two-dimensional isentropic flow of a polytropic gas under the action of gravity. As a mathematical model a system of equations of gas dynamics is used. To put the problem of decay of a special break the degenerate change of variables is made in the system, namely: dependent and independent variables change roles. In the new variables for the system initial-boundary value problem with data on the characteristics of the sound and the additional condition is put. This condition describes the instantaneous destruction of the impermeable wall separating the gas from the vacuum at the initial time. We prove the existence and uniqueness of the initial-boundary value problem in the vicinity of the sound characteristics. Next, the solution is constructed in the form of a power series. To determine the coefficients of the series systems of ordinary differential equations are written and integrated. Coefficients of the series structure analysis has proved the existence of the built solution in the range from the sound characteristics to the boundary of the gas-vacuum inclusive. To determine the law of motion of gas-vacuum boundary quasi-linear system of partial differential equations is written, which by means of a characteristic parameter is reduced to a system of ordinary differential equations. After integration of the latter system in parametric form the law of motion of gas-vacuum boundary values and parameters of the gas on it are obtained.
Keywords: polytropic gas, vacuum, force of gravity, the gas dynamics equations, gas-vacuum boundary, initial-boundary value problem, Riemann problem, centered wave.

A.V. Eremeev, C.R. Reeves
On Confidence Intervals for the Number of Local Optima

Abstract. The number of local optima is an important indicator of optimization problem difficulty for local search algorithms. Here we will discuss some methods of finding the confidence intervals for this parameter in problems where the large cardinality of the search space does not allow exhaustive investigation of solutions. Computational results are reported that were obtained by using these methods for $${\mathcal NK}$$ landscapes model of S.A. Kauffman, for the low autocorrelation binary sequence, for buffer allocation problems in production line, and vertex cover problems.
Keywords: local search, combinatorial optimization problem, Schnabel census, conservative confidence interval.

The paper presents the construction and study of the dynamical system describing the trajectory of the observer by measuring accelerations, which in turn is an estimate of the unknown control. The problem arises in the construction of detection systems of installed unauthorized wireless access points indoors. We studied the observability of the system, investigated the work of Kalman filter and optimal smoothing filter. Heuristic algorithms of trajectory recovery are proposed. A computer simulation of the proposed algorithms is conducted.
Keywords: Kalman filter, smoothing, positioning, wireless access points.

A.V. Lisin, K.S. Yakovenko
Hybrid Methods for Solving Constrained Optimization Problems Using Metaheuristics

Abstract. In the article hybrid methods of numerical solving constrained optimization problems based on classical approaches such as penalty functions method, Lagrange multipliers theory and metaheuristics are discussed. The example of hybrid method based on particle swarm optimization and augmented Lagrangian method is given. Numerical experiment results are provided.
Keywords: constrained optimization, metaheuristics, penalty functions, Lagrange multipliers.

Abstract. A method for solving the one-dimensional ideal gas shock-free strong compression problem in R. Mises configuration is proposed. The method combines finite-difference method ”ROMB” and tracking feature method. The method allows to calculate gas-dynamic characteristic (velocity, density, etc.) of ideal gas layer while time increases and provides better accuracy in comparison with other finite-difference methods. The accuracy of the proposed method was demonstrated in calculations of test plane-symmetry problem. Exact solution and numerical one agree quite well. Numerical results of solving one-dimensional problems with different symmetry and gas characteristic are also shown. The main results of numerical simulations are shown in graphs and tables.
Keywords: gas strong compression, finite difference method ”Romb”, discontinuity tracking method.

Abstract. A scheme for analysis of linear dynamical systems described by stochastic integro-differential equations with nondifference kernels is considered. Such equations are mathematical models of a significant number of phenomena in various scientific and technological fields including the theory of oscillations for objects with lumped and distributed parameters taking into account aeroautoelasticity, heredity, (thermo)viscoelasticity and aging of materials (asphalt, concrete, biopolymers, rocks, colloidal solutions, composites, natural and synthetic polymers, suspensions, glass, cellulose, etc.) and others. The calculation scheme proposed is based on a modification of the iterative method for approximation of the matrix Green’s function and is designed to compute the first moment functions of the state vector of the system including functions of mathematical expectation and covariance functions. The example shows an application of our scheme for an analysis of a model system with two degrees of freedom.
Keywords: stochastic analysis, linear dynamic system, state vector, integro-differential equation, matrix Green’s function, moment function.

O. Kosheleva, V. Kreinovich
Why Most Bright Stars Are Binary But Most Dim Stars Are Single: A Simple Qualitative Explanation

It is known that most visible stars are binary: they have a nearby companion star, and these two stars orbit around each other. Based on this fact, until recently, astronomers believed that, in general, most stars are binary. A few years ago, a surprising paper showed that while most bright stars are indeed binary, most dim stars are single. In this paper, we provide a simple qualitative explanation for this empirical fact.
Keywords: binary stars, single stars, statistics

In general, malignant tumors are known to grow fast, cancer cells that form these tumors divide and spread around. Tumors also experience the process of metastasis, when cancer cells invade neighboring organs. A recent experiment has shown that, contrary to the previous assumptions, when cancer cells are invading, they stop dividing. In this paper, we provide a geometric explanation for this empirical phenomenon.
Keywords: cancer, metastasis, symmetry

O. Kosheleva, V. Kreinovich
Yes- and No-Gestures Explained by Symmetry

In most cultures, “yes” is indicated by a vertical head movement (nod), while “no” is indicated by a left-right movement (shake). In this paper, we show that basic symmetries can explain this cultural phenomenon.
Keywords: yes-gestures, no-gestures, nodding vs.~shaking, symmetry

Сomputer Science

F. Zapata and V. Kreinovich
Why Pairwise Testing Works So Well: A Possible Theoretical Explanation of an Empirical Phenomenon

Some software defects can be detected only if we consider all possible combinations of three, four, or more inputs. However, empirical data shows that the overwhelming majority of software defects are detected during pairwise testing, when we only test the software on combinations of pairs of different inputs. In this paper, we provide a possible theoretical explanation for the corresponding empirical data.
Keywords: software testing, pairwise testing, empirical data, theoretical explanation