## Mathematical Structures and Modeling N2(46)

 Mathematical Structures and Modeling. - Omsk : OmSU, 2018. N2(46), 170 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

An axiomatic realization of the group theoretical description is considered for periodic system of elements. The periodic system is represented as a single quantum system of structureless states. Masses of elements of the superactinide group are calculated. A relationship between an algebraic formulation of the single quantum system and twistor theory is established.
Keywords: Mendeleev periodic system, single quantum system, conformal group, Rumer-Fet group, twistor structure.

A.K. Guts, G.B. Goldina, A.N. Kabanov.
Affine Representations as Lie Groups of Transformations of Three-Dimensional Solvable Lie Groups.

Using the Yamaguchi method, we give the representations as Lie groups of affine transformations of all three-dimensional solvable Lie groups.
Keywords: affine structures, affine representation, three-dimensional solvable Lie groups.

Applied Mathematics and Modeling

We study the problem of three-layered nonhomogeneous rectilinear rods on a nonlinear elastic foundation under the pressure of compressive loads in this article. It is assumed that the rod is in the uneven temperature field and the elasticity modules of the material layers depend on temperature. For the elastic foundation of nonlinear model is accepted and it is assumed that the hypothesis of plane sections is valid for the entire thickness of the element of the rod. In general, we achieve the steadiness equation of the considered rod, and a formula is found for determining the critical load in the certain case.
Keywords: nonhomogeneous three-layered rod, temperature, stability, critical load.

V.V. Goltyapin, V.A. Shovin, E.V. Nadey, V.I. Sovalkin, G.I. Nechaeva.
Building of Dispersion Complexes for Estimation of Efficiency of Immunotherapy of Allergic Bronchial Asthma.

In this article, the efficacy of allergen-specific immunotherapy (ASIT) of allergic bronchial asthma (BA), comorbid with allergic rhinitis (AP) and atopic dermatitis (ATD) over a three-year period by means of a single-factor variance analysis of multigradation features was evaluated.
Keywords: correlation relation, organized factors, discrete values, variance analysis, bronchial asthma, allergy, allergen-specific immunotherapy.

Keywords: hypertension, factor analysis, structural equations.

A.A. Kondyurina, D.N. Lavrov.
Results of the Wireless Access Point Detection Experiment by Modified Trilateration Method.

The research paper presents the results of an experiment to detect wireless access point in the open space based on the early suggested model.
Keywords: positioning, wireless access point detection, trilateration, Kolmogorov-Smirnov test.

Francisco Zapata, Olga Kosheleva, Vladik Kreinovich.
Why Under Stress Positive Reinforcement is More Effective? Why Optimists Study Better? Why People Become Restless? Simple Utility-Based Explanations.

In this paper, we use the utility-based approach to decision making to provide simple answers to the following three questions: Why under stress positive reinforcement is more effective? Why optimists study better? Why people become restless?
Keywords: utility theory, positive vs. negative reinforcement, optimists vs. pessimists.

Vladik Kreinovich, Olga Kosheleva, Mahdokht Afravi, Genesis Bejarano, Marisol Chacon.
Economics of Commitment: Why Giving Away Some Freedom Makes Sense.

In general, the more freedom we have, the better choices we can make, and thus, the better possible economic outcomes. However, in practice, people often artificially restrict their future options by making a commitment. At first glance, commitments make no economic sense, and so their ubiquity seems puzzling. Our more detailed analysis shows that commitment often makes perfect economic sense: namely, it is related to the way we take future gains and losses into account. With the traditionally assumed exponential discounting, commitment indeed makes no economic sense, but with the practically observed hyperbolic discounting, commitment is indeed often economically beneficial.
Keywords: commitment, economics, discounting, exponential discounting, hyperbolic discounting.

Computer Sciences

S.V. Belim, S.B. Larionov.
Algorithm of a Learning Set Formation for Image Segmentation Artificial Neural Network.

An algorithm for forming a learning set for an artificial neural network in the problem of image segmentation is proposed in the article. A distinctive feature is the use of only one segmented image. Segmentation is performed using a three-layer perceptron. The method of growing areas is used. The neural network is used to decide whether the pixel belongs to the segment being formed. Impulse noise is used to form the learning set. Pixels modified by noise do not belong to any segment. A computer experiment was conducted in an automatic and interactive mode.
Keywords: image segmentation, neural network, impulse noise.

S.V. Lejhter, S.N. Chukanov.
The Matching of Images Based on the Construction of the Hamilton Equations.

The problem of comparing the template and the terminal images is considered. To analyze deformations of a image, the group of diffeomorphisms is considered. The problem is solved on the basis of the method for constructing a minimized functional characterizing the evolution of the diffeomorphic image transformation from the template to the terminal image, and the penalty for deviating the image path from the required trajectory. The formulation of the problem based on the construction and solution of Hamilton's equations for the group of diffeomorphisms of particles --- points of landmark of the image is given. An algorithm for solving Hamilton's equations for a diffeomorphic transformation is developed based on the stochastic gradient descent method.
Keywords: pattern recognition, stochastic gradient descent, Hamilton equation, diffeomorphic transformation.

S.V. Guss, D.N. Lavrov.
Approaches to Implementing a Secure Delivery Network Protocol for Multipath Data Transfer.

The paper considers the problem of increasing the reliability of data transmission in a network with flickering nodes. Some comments are given on the development of algorithms for dividing the data flow and recovering lost data. We propose general considerations related to the implementation of the protocol for multipath routing in self-organizing dynamic networks.
Keywords: routing, secret sharing, data recovery, bandwidth, network fault tolerance.

How to Store Tensors in Computer Memory: An Observation.

In this paper, after explaining the need to use tensors in computing, we analyze the question of how to best store tensors in computer memory. Somewhat surprisingly, with respect to a natural optimality criterion, the standard way of storing tensors turns out to be one of the optimal ones.
Keywords: tensors, computing, computer memory.

Why Deep Learning Methods Use KL Divergence Instead of Least Squares: A Possible Pedagogical Explanation.

In most applications of data processing, we select the parameters that minimize the mean square approximation error. The same Least Squares approach has been used in the traditional neural networks. However, for deep learning, it turns out that an alternative idea works better --- namely, minimizing the Kullback-Leibler (KL) divergence. The use of KL divergence is justified if we predict probabilities, but the use of this divergence has been successful in other situations as well. In this paper, we provide a possible explanation for this empirical success. Namely, the Least Square approach is optimal when the approximation error is normally distributed --- and can lead to wrong results when the actual distribution is different from normal. The need to have a robust criterion, i.e., a criterion that does not depend on the corresponding distribution, naturally leads to the KL divergence.
Keywords: deep learning, Kullback-Leibler divergence.

In many practical applications, we are interested in the values of the quantities $$y_1,\ldots,y_m$$ which are difficult (or even impossible) to measure directly. A natural idea to estimate these values is to find easier-to-measure related quantities $$x_1,\ldots,x_n$$ and to use the known relation to estimate the desired values $$y_j$$. Measurements come with uncertainty, and often, the only thing we know about the actual value of each auxiliary quantity $$x_i$$ is that it belongs to the interval $$[\underline x_i,\overline x_i]=[\widetilde x_i-\Delta_i,\widetilde x_i+\Delta_i]$$, where $$\widetilde x_i$$ is the measurement result, and $$\Delta_i$$ is the upper bound on the absolute value of the measurement error $$\widetilde x_i-x_i$$. In such situations, instead of a single value of a tuple $$y=(y_1,\ldots,y_m)$$, we have a range of possible values. In this paper, we provide calculus-based algorithms for computing this range.
Keywords: data processing, interval uncertainty, indirect measurements, calculus.

Information Security

The work is devoted to the development of the method of information concealment in a raster image using steganography together with cryptography. The suggested method allows to solve the problem of message recovery if the image containing the hidden data was damaged. The basic idea is that the data with the help of steganography is not placed in the image entirely, but using the $$(t, n)$$ --- threshold scheme, the insertion of each part occurs independently. A software has been developed to test the proposed method. A series of experiments confirming the possibility of applying the proposed method is carried out.