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VOLUME EDITORS

Alexander N. Prokopenya (Warsaw University of Life Sciences – SGGW, Poland)

Miroslaw Jakubiak (University of Natural Sciences and Humanities in Siedlce, Poland)

SCIENTIFIC BOARD

Sergei Abramov (Moscow, Russia) Viktor I. Korzyuk (Minsk, Belarus)

Michael V. Alania (Siedlce, Poland) Robert Kragler (Weingarten, Germany)

Michail Barbosu (New York, USA) Pavel S. Krasilnikov (Moscow, Russia)

Andrzej Barczak (Siedlce, Poland) Anatoly P. Markeev (Moscow, Russia)

Moulay Barkatou (Limoges, France) Arkadiusz Orłowski (Warsaw, Poland)

Wiesława Barszczewska (Siedlce, Poland) Alexander Prokopenya (Warsaw, Poland)

Carlo Cattani (Salerno, Italy) Agnieszka Prusińska (Siedlce, Poland)

Alexander V. Chichurin (Brest, Belarus) Bogusław Radziszewski (Siedlce, Poland)

Vladimir P. Gerdt (Dubna, Russia) Valeriy Hr. Samoylenko (Kiev, Ukraine)

Vasile Glavan (Siedlce, Poland) Marek Siłuszyk (Siedlce, Poland)

Valeriu Gutu (Chishinau, Moldova) Doru Stefanescu (Bucharest, Romania)

Katica R. (Stevanovic) Hedrih (Serbia) Aleksander Strasburger (Warsaw, Poland)

Krzysztof Iskra (Siedlce, Poland) Alexey Tret’yakov (Siedlce, Poland)

Miroslaw Jakubiak (Siedlce, Poland) Andrzej Walendziak (Siedlce, Poland)

LIST OF REVIEWERS

Carlo Cattani (Salerno, Italy) Robert Kragler (Weingarten, Germany)

Alexander V. Chichurin (Brest, Belarus) Renata Modzelewska-Lagodzin (Siedlce)

Agnieszka Gil (Siedlce, Poland) Alexander Prokopenya (Warsaw, Poland)

Vasile Glavan (Siedlce, Poland) Agnieszka Siluszyk (Siedlce, Poland)

Valeriu Gutu (Chishinau, Moldova) Marek Siłuszyk (Siedlce, Poland)

Katica R. (Stevanovic) Hedrih (Serbia) Aleksander Strasburger (Warsaw, Poland)

Krzysztof Iskra (Siedlce, Poland) Alexey Tikhonov (St. Petersburg, Russia)

Cover designed by Paweł Trojanowski

© Copyright Uniwersytet Przyrodniczo-Humanistyczny w Siedlcach, Siedlce 2015

ISSN 2300-7397

ISBN 978-83-7051-779-3

Uniwersytet Przyrodniczo-Humanistyczny w Siedlcach

Instytut Matematyki i Fizyki

08-110 Siedlce, ul. 3-Maja 54,

tel. +48 25 643 1003, e-mail: imif@uph.edu.pl

3

Contents

I. Mathematical Modeling and Differential Equations

M.V. Alania, G.G. Didebulidze, R. Modzelewska, M. Todua, A. Wawrzynczak.

Annual variations of the galactic cosmic ray intensity and seasonal distribution

of the cloudless days and cloudless nights in Abastumani (41.75º N, 42.82 º E;

Georgia): (1) experimental study and (2) theoretical modeling . . . . . . . . . . . . . .

5

N.N. Aprausheva, V.V. Dikusar, S.V. Sorokin. Gradient-statistical algorithm for

calculating critical points of density probability of Gaussian mixture . . . . . . . .

15

Z. Binderman, B. Borkowski, A. Prokopenya, W. Szczesny. Application of

dissimilarity measures to objects odering and concentration measurement . . . .

23

A. Chichurin, E. Ovsiyuk, A.Red’ko, V. Red’kov. Spin 1 particle in the Coulomb

field on the background of Lobachevsky geometry: general Heun functions,

analytical and numerical study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

A. Chichurin, A. Shvychkina. Simulating the population dynamics of the

bacterial plasmids with the equal half-saturation constants . . . . . . . . . . . . . . . . .

55

V. Dikusar, N. Olenev. Parallel programming in MATLAB for modeling an

economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

V. Dikusar, M. Wojtowicz, E. Zasukhina. Solving optimal control problems

with control-state constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

G. Filipuk. A Remark on the Bӓcklund transformation of the fifth

Painlevé equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

G. Filipuk, Yu. Bibilo. A remark on non-Schlesinger deformations . . . . . . . . . . 87

G. Filipuk, S. Hilger. A remark on the tensor product of two (q,h)-Weyl

algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

A. Gil. Modulation of galactic cosmic rays during the unusually prolonged solar

minimum of 2007 – 2009 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

V. Glavan, V. Gutu. Eventually contracting affine IFS . . . . . . . . . . . . . . . . . . . . 111

K. Iskra, R. Modzelewska, M. Siluszyk, M. Alania, W. Wozniak, P. Wolinski.

The estimation of the parameters characterizing the galactic cosmic ray

modulation based on the measurements of the anisotropy in different sectors of

the interplanetary magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

122

M. Jakubiak, D. Kozak-Superson, A. Prusińska. Mankiw-Romer-Weil model of

economic growth dynamics with Cobb - Douglas production function in

Mathematica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

132

A. Jatsko, S. Zasukhin. Optimal control problems with control-state constrains . 143

S. Keska. On Hausdorff moment sequences under permutation . . . . . . . . . . . . . 153

V. I. Korzyuk, I.I. Staliarchuk. Mixed problem for Klein-Gordon-Fock equation

with curve derivatives in boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . .

166

R. Kragler. Using the method of inverse differential operators (MIDO) . . . . . . . 182

D. Pylak, P. Karczmarek. The full proof of the error estimates of approximate

4

solutions of a singular integral equation on the quarter plane . . . . . . . . . . . . . . . 198

Z.Z. Rzeszotko. Time series analysis for some market indices . . . . . . . . . . . . . . . 209

V. Samoylenko, Yu. Samoylenko. Asymptotic many phase soliton type solutions

to Cauchy problem for Korteweg-de Vries equation with variable coefficients

and a small parameter of the first degree at the highest derivative . . . . . . . . . . .

219

M. Siluszyk. Theoretical and experimental study of the 11-year variations of

galactic cosmic ray intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

227

M. Siluszyk, K. Iskra, M. Alania, S. Miernicki. Properties of the interplane-tary

magnetic field turbulence in different cycles of solar activity . . . . . . . . . . . . . .

237

V.R. Sultanov, V.V. Dikusar. Optimal selection of task team . . . . . . . . . . . . . . . 250

E. Szczepanik, A. Tret’yakov. Algorithm of the method for solving degenerate

sub-definite nonlinear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

254

A. Wawrzynczak, R. Modzelewska, A. Gil, M. Kluczek. Solving the stochastic

differential equations equivalent to Parker’s transport equation by the various

numerical methods. Model of the 27-day variation of cosmic rays . . . . . . . . . . .

269

P. Wolinski, P. Woyciechowski. Mathematical model of carbonization concrete

with calcareous fly ash . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

282

II. Problems of Classical Mechanics

K.R. (Stevanovic) Hedrih. Rolling heavy disk along rotating circle with

constant angular velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

293

I.I. Kharlamova. An example of the constructor of topological invariants for

integrable Hamiltonian systems with parameters . . . . . . . . . . . . . . . . . . . . . . . .

305

M.P. Kharlamov, P.E. Ryabov. Phase topology of the Kowalevski –Komarov –

Sokolov top . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

320

M. Maksimenko. Numerical algorithm for calculation of the stress state of a

rock massif including the mine in MATLAB . . . . . . . . . . . . . . . . . . . . . . . .

343

A.F. Mselati, S. Bosiakov. Conditions for translational movement of the

composite paraboloid with one plane of symmetry . . . . . . . . . . . . . . . . . . . .

355

A. Siluszyk. The finiteness in the circular restricted five-body problem . . . . . 365

III. Education and Didactics

T. Botchorishvili. On some properties of the rapidly converging series for

computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

376

V. Taranchuk, V. Kulinkovich. On the preparation and distribution of interactive

graphics applications using Mathematica . . . . . . . . . . . . . . . . . . . .

380

V. Taranchuk, V. Kulinkovich. On programming interactive graphics

applications in Mathematica system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

388

Author index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396

On the Preparation and Distribution of InteractiveGraphics Applications Using Mathematica

Valery Taranchuk 1), Viktoria Kulinkovich 2)

1,2) Belarusian State University4, Nezavisimosti Ave. 220030 Minsk, Belarus

taranchuk@bsu.by

kulinkovichva@gmail.com

Abstract. The paper presents new possibilities of preparation and distributionbased on Mathematica interactive graphics applications

1 Introduction

Nowadays, computer hardware and software provides a variety of options for creatingand using dynamic and interactive documents. Such documents have several advan-tages over printed materials. A vital task is to define requirements for the contentof electronic documents, processes for their preparation, information visualizationtools. That requires solving some technical and organizational problems.

This work covers recommendations for creating, maintaining and freely distribut-ing interactive electronic educational resources using Wolfram technologies; and inparticular, computer algebra system Mathematica, computable document format(CDF), the Wolfram Demonstrations Project. The article contains practical exam-ples originated from developing training materials for “Computer Graphics” course.The distinguishing feature of this discipline is that each theoretical topic requiresillustrations.

This course provides introduction to basic computer graphics algorithms, so it isimportant to be able to do the calculations and conversions in mathematical nota-tion, on personal computer. One of the necessary requirements for a correct under-standing of the models and their descriptions is clear presentation of the material.It can be achieved by constructing imaginary models based on their mathematicaldescription. Developing interactive software applications requires special program-ming skills, and is also very time consuming. Wolfram Mathematica provides asolution to this problem, in particular, through the software modules in WolframDemonstrations Project.

2 The Basic Tools

About Mathematica. Mathematica is one of the most powerful computational soft-ware systems [1]. The system allows working with numbers, formulas, functions,graphics, sound, databases, documents, and supports almost all types of analytical

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and numerical calculations. Mathematica provides following capabilities to the user:numeric and symbolic tools for discrete and continuous calculus; 2D and 3D visu-alization and animation tools; image importing, processing, and exporting [1, 2].Mathematica is a great aid in educational process and scientific research.

Computable Document Format (CDF). CDF format features and objec-tives ([3]), using CDF format to create interactive software applications, documents,that can be launched in freely distributed CDF player application on most comput-ers and operating systems, are described in [4].

Wolfram Demonstrations Project. The Wolfram Demonstrations Project ishosted by Wolfram Research [5]. It currently contains over 10190 freely distributedinteractive demonstrations (on July, 2015), and its systematic catalog is updatedregularly. Its goal is to demonstrate the possibilities of Wolfram Mathematica andto increase the number of Wolfram users. Small interactive programs, includedin Wolfram Demonstrations Project, illustrate concepts from a number of fields:science, technology, mathematics, art, finance, etc. The collection covers a variety oflevels; from elementary school mathematics to much more advanced topics, includingquantum mechanics and models of biological organisms.

Most of the demonstration modules have a straightforward user interface that re-computes plot or visualization dynamically in response to user actions like moving aslider, clicking a button or dragging one of the graphic elements. Each demonstrationalso includes a brief description about the concept being shown. All demonstrationsare available for download in Mathematica NB format and computable documentformat (CDF). Open access allows users to create their own interactive applicationswith minimal effort.

The present paper recites methodical aspects, lists new options of creation;through examples illustrates recommended steps to adapt existing educational mate-rials, as well as developing and utilizing interactive educational resources. Program-ming specific questions, steps necessary to prepare interactive educational modulesfor free distribution, special items that should be considered when using Mathe-matica to develop applications distributed widely (including embedding them inweb pages that can be displayed in different browsers) are presented in paper “OnProgramming Interactive Graphic Applications in Mathematica System”.

3 Examples and Implementation Stages

Main components, tools used to create and support interactive educational systemsare illustrated using educational materials from “Computer Graphics” course as anexample.

The software modules used in Computer Graphics. Training processof “Computer Graphics” course (“Applied Computer Science” speciality) taughtin Belarusian State University on faculty of Applied Mathematics and Computer

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Science utilizes interactive demonstrations (software applications-projects) availablein collection [5] on following topics:

Color in computer graphics; additive color systems; subtractive color systems;color space; color cube; intuitive color model; color space conversion.

Mathematical foundations of computer graphics. Point, vector, distance on aplane and in three-dimensional space. Equation of a line segment, equation ofa ray in 2D and 3D: parametric, with direction vector. Normal. Distance toa point. Angle between straight lines. Transformation of coordinates. Homo-geneous coordinates. Geometric transformation in 2D and 3D. Matrix repre-sentation of affine transformations (translate, reflect, rotate, scale). Rotationabout an arbitrary axis, about a point. Composite 3D transformations, trans-formation commutativity. Geometry transformation pipeline. Projections,projection matrices.

Digital image processing: linear and nonlinear filtering, mathematical mor-phology, image binarization. Gradient based edge detection, Laplacian.

Image-based rendering and lighting. Illumination models in computer graph-ics. Transparency modelling. Construction of shades. Texture. Voxel graphicsconcept and examples.

Now we consider the examples of three topics in Computer Graphics and cor-responding software applications from the collections [5] recommended to use. Thefirst topic is devoted mainly technical issues, and the foundations of the theory of col-ors, the second and third – mathematical and algorithmic foundations of computergraphics.

One of the main topics in “Computer Graphics” course is color and color mod-els. We recommend to use following interactive visualization modules, in particularapplications: Colors of the Visible Spectrum; Overlapping Light Colors; ColoredLights; Named Colors; Select, View, and Compare Named Colors; Analogous andComplementary Colors; Newton’s Color Wheel; Color Cube; Color Triangles; ColorSpace; Cartesian Color Coordinate Spaces; RGB and CMYK Colors; RGB Explorer;Orthogonal Views of Named RGB Colors; HSV Colors; HSV Loci in the RGB ColorSpace; CIE Chromaticity Diagram.

Modules used in the topic “Mathematical foundations of computer graphics”:Understanding 2D Translation; Understanding 2D Shearing; Understanding 2D Ro-tation; Understanding 2D Reflection; Understanding 2D Rescaling; 3D graphicsmodules: Understanding 3D Rotation; Understanding 3D Scaling; Understanding3D Reflection; Understanding 3D Shearing; Two Models of Projective Geometry;Orthographic Projection of Parallelepipeds; Stereographic Projection of PlatonicSolids; Cutoff Parallelepipeds.

Modules used in the topic “Digital image processing”: Playing with Image Chan-nels; Image Color Analyzer; Processing Various Parts of an Image Differently; ImageProcessing on Partitions; Interactive Color Posterization; Posterization of Grayscale

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Images; Morphological Processing; Morphological Operations; Image Sharpening;Image Quality Adjustment; Histogram Equalization; Image Smoothing Using Sta-tionary Wavelet Transform; Image Compression via the Fourier Transform; UsingDifferent Types of Filters; Image Kernels and Convolution (Linear Filtering); Non-linear Image Filtering; Convolution Linear Filtering; Filtering Using Common Value;Filtering Images in the Frequency Domain; Nonlinear Image Filtering; Image JitterFilter.

The source codes of these projects in NB or CDF format can be downloadedfrom [5]. Search box available on the site to can be used to find the project; allmatching applications will be displayed with hyperlinks and names. All moduleshave slider control on the panel (geometric parameters, color, transparency of dis-played objects); user can also change scale and viewing angle. We provide studentswith modified applications that use Russian terminology and formatting commonlyused in training materials and textbooks.

Example of content and organizational structure of electronic mate-rials for one of the topics. Interactive electronic resources are used during allstages of learning process: lectures, practical classes, individual work, intermediatecontrol, final exams (computer-based testing). The following components of elec-tronic materials are on topic “Geometric transformations in 2D and 3D. Matrixrepresentation; composition of 3D transformations”. Let’s mark components of theelectronic methodical complex on an example of studying by that.

During classes (lectures, practical classes) students use one or more softwaremodules in CDF format. Figure 1 shows an illustration of one of them; 3 fragmentsof sections are shown: main section, section “Homogeneous Coordinates”, subsection“Where to apply”. We can expand / collapse group of sections by click on bordering]-key or index ∨/∧ on left of the title. Students can work with the module inMathematica or using CDF Player.

Modules include sections with basic theory, links to recommended textbooks andmanuals, tasks for students. All texts where there are formulas, are written in themathematical notation, and can be export to rtf, pdf. In addition to sections withexplanations there are executable sections. In executable sections (section “In [7]:=” in Figure 2) students can do different calculations and transformations, matrixoperations, visualizations. It is possible to work with graphics interactively.

The functions (system commands) are written in sections in the form of exercises,and students can change values, receive and view the results, combine operations,construct expressions of the proposed templates, combining standard arithmeticoperations and elementary and more complex functions. The results can be copiedto the clipboard and exported.

Figures 2 and 3 show the subsections “Affine transformation matrices”, “Com-position of 3D transformations”.

Figure 2 illustrates Mathematica functions: Inverse – matrix inverse, Simplify –simplify expressions, MatrixForm – output elements of array in a matrix format.

Figure 3 shows block “geometric transformation pipeline” of subsection “Com-position of 3D transformations”.

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Figure 1: View of software module window; titles of the sections.

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Figure 2: Fragment of subsection “Affine transformation matrices”.

We consider classical problem of obtaining the final transformation matrix. Thesolution includes 4 steps: translation and 3 rotations around the coordinate axes(figure with fragments from subsection “Steps to process standard transformation”).The transformations are implemented by application of appropriate standard ma-trices. Graphics for each step and calculated, coordinate points are displayed in themodule. For example, figure 3 shows the results of calculating the coordinates ofpoint Q3 in section “Coordinates control after transformations”. After this trans-formation point must be on the axis 0Z (the first two coordinates are zero).

We provide students document with theory and software module Understand-ing3DRotation+.cdf that is adapted according to the original from the catalog [5](we use Russian terminology, change initial camera angles and zoom viewing, addcomments). Appearance of tools of handle and the explanation of constituents ofthe scene of the mentioned demonstration project are explained in paper “On Pro-gramming Interactive Graphic Applications in Mathematica System”.

4 Conclusion

Features and recommended application of Mathematica system, computable doc-ument format CDF, modules from Wolfram demonstration project collection de-scribed in the present paper simplify creation and extend boundaries of free distri-bution of electronic interactive educational resources. Using program modules listedin this paper during educational process drastically increases comprehension levelof the material.

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Figure 3: View of fragments from “Composition of 3D transformation” subsectionthat illustrate transformation pipeline stages.

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References

[1] V.B. Taranchuk. Basic functions of computer algebra systems: manual for studentsof fac. Applied mathematics and computer science. Minsk, BSU (2013).

[2] Wolfram Mathematica. The world’s definitive system for modern technical comput-ing. [Electronic resource] / Access mode: http://www.wolfram.com/mathematica/ -Access date: 24.06.2015

[3] Computable Document Format CDF. [Electronic resource] / Access mode:http://www.wolfram.com/cdf/ - Access date: 24.06.2015

[4] V.B Taranchuk. About creation of interactive educational resources with usage of Wol-fram technologies. Information of Education, 1, pp. 78.–89 (2014).

[5] Wolfram Demonstrations Project. [Electronic resource] / Access mode:http://demonstrations.wolfram.com. - Access date: 24.06.2015

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