## Mathematical structures and modeling ¹2(42)

 Mathematical structures and modeling. - Omsk : OmSU, 2017. ¹2(42), 145 p. ISSN (print): 2222-8772 ISSN (online): 2222-8799 For researchers, post-graduate students and senior students. Journal issue in one file

I.P. Bestsennyi
On the Reduction of Modalities in Deontic Calculi

Formal theories of deontic logic based on the Gensen-type propositional calculus are considered. Syntactic and semantic differences between the aletic and deontological calculi are analyzed in the aspect of reduction of sequences of successive modal signs
Keywords: modal logic, deontic calculi, Kripke semantic.

V.V. Varlamov
Mass Quantization and Lorentz Group

Mass spectrum of localized states of quantum micro-objects is studied in the framework of holistic (nonlocal) Heisenberg's scheme. As an object of a fundamental level (potential reality), the quantum micro-object exists outside of space-time. It is shown that state masses of lepton (except the neutrino) and hadron sectors of matter spectrum are proportional to the rest mass of electron with an accuracy of 0,41%.
Keywords: mass spectrum, Lorentz group, cyclic representations, mass formulae, mass quantization.

V.A. Shaptsev
About Dependence in the Bernoulli Trials

Analytical representation of scheme is discussed in independent Bernoulli trials by means of the ordered subsets of $$m$$ ($$m < n$$) numbers of successful and $$n - m$$ numbers of failed tests. The probabilistic description of $$P_{n}(m)$$ is determined by summing the binomial formula $${p^{m}(1-p)^{n-m}}$$ for all such subsets rather than a simple multiplication by the number of combinations $$C_{n}^{m}$$. Some appropriate expressions are given for the probability of success with $$k > 2$$ outcome of the experiments for the cases of equal and different probabilities of outcomes in different trials. In these symbolics, expressions for calculating the corresponding probabilities of the number of successful tests is specified in the presence of dependence of outcomes probabilities on a parameter varying in time (from test to test), regularly or accidentally. Thus, the basis is created for application of the scheme of dependent tests to estimate the probabilistic characteristics of complex stochastic systems, network architectures, in particular, characterized by some varying measure of quality. In conclusion, we present the expression for the mathematical expectation of the probability of m successful trials out of $n$ in the case of random parameter on which the probabilities of outcomes depend. Hypothesis about the possibility of using finite-dimensional characteristic function of the random parameter in the corresponding expression for the case of exponential dependence of the probability of the outcome from the random parameter is written.
Keywords: independent Bernoulli trials, ordered subsets of tests numbers, dependent tests.

The integral convolution-type equations of the first kind on the sphere are important for the geometric tomography. They have been studied by many researchers. In this paper, we consider the uniqueness and stability of solutions of such equations. We prove the uniqueness of the solution for the equation with the kernel of convolution type and obtain a formula for the average value of a function on a subsphere. The latter is used for the deriving of the inversion formula of Radon spherical transformation on sphere. For the Blaschke-Levy equation and for the convolution type singular integral equations of the linear transfer theory, the uniqueness theorems are proved and find estimates for the stability of solutions are found. In all the cases we use the expansion of a function into series of spherical harmonics.
Keywords: geometric tomography, convolution-type equations of the first kind, uniqueness, stability, Radon spherical transform, inversion formula, Blaschke-Levy equation, equation with a singular kernel.

Applied Mathematics and Modeling

S.V. Belim, T.B. Smirnova, A.N. Mironenko
Application of an Associative Rules Method to the Activities Analysis of Public Organizations

In the article the activities indicators analysis method of public organizations is developed. The method of the associative rules is used. Transactions are created of polling questionnaires. From coded responses of questionnaires the associative rules are obtained. The rules possessing high support and confidence are selected. On the basis of the associative rules the oriented weighted correlations graph is constructed. On this graph search of communities is executed. The selected community allowed to reveal activities factors of public organizations the most tightly connected with each other.
Keywords: associative rules, public organizations, community, Data Mining.

V.P. Golubyatnikov
The Existence of a Stable Cycle in a Model of a Molecular Repressor

We consider nonlinear 6-dimensional dynamical system which describes a model of functioning of one simple molecular repressilator. We find sufficient conditions of existence of a stable cycle in the phase portrait of this system.
Keywords: nonlinear dynamical system, gene network models, hyperbolic stationary points, cycles, Brouwer's fixed-point theorem, stability.

L.A. Volodchenkova, A.K. Guts
Equilibrium Dynamics of Forest Ecosystems Based on the Relationship ”Vegetation-Soil”

In the article the Nash equilibrium state for the forest ecosystems based on the relationship ”vegetation-soil” and the theory of differential games are investigated.
Keywords: The Nash equilibrium, forest ecosystem, soil, vegetation, differential games.

V.A. Shovin, V.V. Goltyapin
Latency Analysis Based on Penalty Method for Multidimensional Binary Indicators

A comparison of following classification algorithms with learning is carried out: FORDIASIMPT, latency analysis based on penalty method, KORA, naive Bayes classifier, k-nearest neighbors algorithm, decision tree based on information growth and reduction of the average entropy in an example of multidimensional binary indicators.
Keywords: classification with learning, FORDIASIMPT, latency analysis, penalty method, KORA, naive Bayes classifier, k-nearest neighbors algorithm, decision tree.

The compactness of groups of patients with hypertension before and after physiotherapy calculates by FRiS function. The ”defenses” of groups of patients is determined by counting the number of nearest neighbors of each group of objects that alternate FRiS function. The article offers a special FRiS function that improves the accuracy of statistical estimates of the compactness of classes. As the result of the calculations, it turned out that from the group of the patients ”after physical therapy” a compact group is allocated, while the objects of the third part were not protected by objects of the own group. From a medical point of view, it seems to be interpreted as a positive effect of treatment for some patients.
Keywords: FRiS compact function, hypertension.

Abstract. The results of analytical and numerical simulations of one-dimensional ideal gas shockfree strong compression problem while compression piston’s radius is not decrease are presented. The analytic formula for generalized compression Riemann wave’s sound characteristics is the main result. Comparison of the formula with calculation of the problem by characteristic method in time decreasing case is also presented. Obtained formulas for sound characteristics was used to calculate gasdynamic characteristic (velocity, density, etc.) of ideal gas layer while time increase. The computational domain is the area between compression piston and the last sound characteristic of generalized Riemann wave. The main results of numerical simulations are shown in graphs and tables.
Keywords: gas strong compression, the characteristic series method, solution feature.

D.N. Lavrov, A.A. Laptev, M.A Mamontova
Identification of a Parametric Model of Changes in the Number of Students Enrolled on a Small Sample

A parametric model of changes in the number of students of the University with a small sample is built. The number of high school graduates in the region is taken as an input parameter. With the help of information criteria, the order of autoregressive moving-average is selected. Applying the identification system feature of Scilab, the forecast of the number of applicants for admission campaign in 2017 was made. A similar result was obtained through the identification of the model in the form of the transfer function.
Keywords: parametric identification, autoregressive-moving-average model, forecast of the number of high school students, Akaike information criterion.

When we need to add several integers, computers add them one by one, while we usually add them digit by digit: first, we add all the lowest digits, then we add all next lowest digits, etc. Which way is faster? Should we learn from computers or should we teach computers to add several integers our way? In this paper, we show that the computer way is faster. This adds one more example to the list of cases when computer-based arithmetic algorithms are much more efficient than the algorithms that we humans normally use.

Computer Science

The main reasons for the need for a preliminary analysis of the initial data for the calculation of teaching loads are presented. We confirmed the presence of factors that affect the optimal distribution of the teaching load, such as a classroom university fund, threading of groups and other. The study is based on the analysis and subsequent optimization of the university curriculum. The curricula for the current academic year of Kazakh University of Economy, Finance and International Trade (Astana, Kazakhstan) were used as the initial data, as well as academic calendars and rules for the organization of educational process on credit technology in the Republic of Kazakhstan.
Keywords: optimization, modelling, curriculum, teaching load, classroom fund.

G. Muela, C. Servin, V. Kreinovich
How to Make Machine Learning Robust Against Adversarial Inputs

It has been recently shown that it is possible to "cheat" many machine learning algorithms - i.e., to perform minor modifications of the inputs that would lead to a wrong classification. This feature can be used by adversaries to avoid spam detection, to create a wrong identification allowing access to classified information, etc. In this paper, we propose a solution to this problem: namely, instead of applying the original machine learning algorithm to the original inputs, we should first perform a random modification of these inputs. Since machine learning algorithms perform well on random data, such a random modification ensures us that the algorithm will, with a high probability, work correctly on the modified inputs. An additional advantage of this idea is that it also provides an additional privacy protection.
Keywords: machine learning, adversarial inputs, robust learning

Information Security

D.M. Brechka, A.A. Litvinenko
Ðàçðàáîòêà ñèñòåìû ìàíäàòíîãî óïðàâëåíèÿ äîñòóïîì äëÿ îïåðàöèîííûõ ñèñòåì ñåìåéñòâà Windows

Abstract. The article describes a development of Mandatory Access Control System for Windows operating system. It is proposed the approach of creating the minifilter driver for the file system to intercept application calls. The interface of the Mandatory Access Control system, and the results of testing are described.
Keywords: Mandatory Access Control, operating system, Windows, driver, access control, subjects, objects.