{"id":39723,"date":"2015-04-04T20:59:00","date_gmt":"2015-04-04T17:59:00","guid":{"rendered":"https:\/\/new.azbuki.bg\/matematics\/matharticles\/mathinfocontent022015\/"},"modified":"2019-06-24T17:33:15","modified_gmt":"2019-06-24T14:33:15","slug":"mathinfocontent022015","status":"publish","type":"post","link":"https:\/\/science.azbuki.bg\/en\/matematics\/matharticles\/mathinfocontent022015\/","title":{"rendered":"Mathematics and Informatics, Number 2\/2015, Volume 58"},"content":{"rendered":"<p><strong><a name=\"top\">&nbsp;<\/a>CONTENTS \/ \u0421\u042a\u0414\u042a\u0420\u0416\u0410\u041d\u0418\u0415&nbsp;<\/strong><\/p>\n<p><strong>EDUCATIONAL TECHNOLOGY \/ \u041e\u0411\u0420\u0410\u0417\u041e\u0412\u0410\u0422\u0415\u041b\u041d\u0418 \u0422\u0415\u0425\u041d\u041e\u041b\u041e\u0413\u0418\u0418<\/strong><\/p>\n<p><a href=\"#art01\"> Spreadsheets as Tools for Constructing Mathematical Concepts <\/a>\/ <br \/><i>Erich Neuwirth &#8211; \u0441\u0442\u0440. 119<\/i><\/p>\n<p><i><a href=\"https:\/\/science.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/i><\/p>\n<p>&nbsp;<\/p>\n<p><a href=\"#art02\"> Education of a Mathematician \u2013 Experimentalist, or Soft Manifesto of Experimental Mathematics <\/a>\/<i> <br \/>Alexander Yastrebov, Maria Shabanova &#8211; \u0441\u0442\u0440. 129<\/i><\/p>\n<p><i><a href=\"https:\/\/science.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/i><\/p>\n<p>&nbsp;<\/p>\n<p><a href=\"#art03\"> Computer-discovered Mathematics: Lalesco Products <\/a>\/ <br \/><i>Sava Grozdev, Deko Dekov &#8211; \u0441\u0442\u0440. 143<\/i><\/p>\n<p><i><a href=\"https:\/\/science.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/i><\/p>\n<p>&nbsp;<\/p>\n<p><a href=\"#art04\"> \u0421\u0430\u043c\u043e\u0441\u0442\u043e\u044f\u0442\u0435\u043b\u043d\u043e \u0440\u0435\u0448\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u0437\u0430\u0447\u0430\u0447\u0438 \u0441 Excel <br \/>[Independent Problem Solving with Excel]<\/a> \/ <br \/><i>\u041f\u043b\u0430\u043c\u0435\u043d \u041f\u0435\u043d\u0435\u0432, \u0414\u0438\u0430\u043d\u0430 \u0421\u0442\u0435\u0444\u0430\u043d\u043e\u0432\u0430 \/ Plamen Penev, Diana Stefanova &#8211; \u0441\u0442\u0440. 149<\/i><\/p>\n<p><i><a href=\"https:\/\/science.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/i><\/p>\n<p>&nbsp;<\/p>\n<p><strong>EDUCATIONAL RESEARCH \/ \u041d\u0410\u0423\u0427\u041d\u041e\u2013\u041c\u0415\u0422\u041e\u0414\u0418\u0427\u0415\u0421\u041a\u0418 \u0421\u0422\u0410\u0422\u0418\u0418<\/strong><\/p>\n<p><a href=\"#art05\"> \u041c\u043e\u0434\u0438\u0444\u0438\u043a\u0430\u0446\u0438\u044f \u043c\u0435\u0442\u043e\u0434\u0430 \u043f\u0440\u043e\u0435\u043a\u0446\u0438\u0439 \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u044f \u0440\u0430\u0441\u0441\u0442\u043e\u044f\u043d\u0438\u044f \u043c\u0435\u0436\u0434\u0443 \u0441\u043a\u0440\u0435\u0449\u0438\u0432\u0430\u044e\u0442\u0449\u0438\u043c\u0438\u0441\u044f \u043f\u0440\u044f\u043c\u044b\u043c\u0438 <br \/>[Modification of the Projection Method to Find the Distrance Between Skew Lines] <\/a>\/<br \/> <i>\u0412\u043b\u0430\u0434\u0438\u043c\u0438\u0440 \u0416\u0443\u043a \/ Vladimir Zhuk &#8211; \u0441\u0442\u0440. 157<\/i><\/p>\n<p><i><a href=\"https:\/\/science.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/i><\/p>\n<p>&nbsp;<\/p>\n<p><a href=\"#art06\"> \u041e\u0431\u043e\u0431\u0449\u0435\u043d\u0438\u044f \u0437\u0430 \u043d\u044f\u043a\u043e\u0438 \u043a\u043b\u0430\u0441\u043e\u0432\u0435 \u043d\u0435\u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438 \u0438\u043d\u0442\u0435\u0433\u0440\u0430\u043b\u0438, \u0441\u0432\u0435\u0436\u0434\u0430\u0449\u0438 \u0441\u0435 \u0434\u043e \u0440\u0435\u043a\u0443\u0440\u0435\u043d\u0442\u043d\u0430 \u0437\u0430\u0432\u0438\u0441\u0438\u043c\u043e\u0441\u0442 <br \/>[Generalization of Some Types of Indefinite Integrals, Which Can Be Solved by Recurrence] <\/a>\/ <br \/><i>\u0420\u043e\u0441\u0435\u043d \u041d\u0438\u043a\u043e\u043b\u0430\u0435\u0432 \/ Rosen Nikolaev &#8211; \u0441\u0442\u0440. 170<\/i><\/p>\n<p><a href=\"https:\/\/science.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/em><\/p>\n<p>&nbsp;<\/p>\n<p><a href=\"#art08\"> \u041f\u043e\u0434\u0445\u043e\u0434\u0438 \u043f\u0440\u0438 \u0440\u0430\u0437\u0440\u0430\u0431\u043e\u0442\u0432\u0430\u043d\u0435 \u043d\u0430 \u0433\u0440\u0430\u0444\u0438\u0447\u0435\u043d \u0438\u043d\u0442\u0435\u0440\u0444\u0435\u0439\u0441 \u0437\u0430 \u0440\u0430\u0431\u043e\u0442\u0430 \u0441 \u043d\u044f\u043a\u043e\u043b\u043a\u043e \u0434\u043e\u043a\u0443\u043c\u0435\u043d\u0442\u0430 <br \/>[Approaches in Developing GUI and Working With Several Applications] <\/a>\/ <br \/><i>\u0422\u043e\u0434\u043e\u0440\u043a\u0430 \u0422\u0435\u0440\u0437\u0438\u0435\u0432\u0430 \/ Todorka Terzieva &#8211; \u0441\u0442\u0440. 195<\/i><\/p>\n<p><i><a href=\"https:\/\/science.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/i><\/p>\n<p>&nbsp;<\/p>\n<p><strong>CONTEST PROBLEMS \/ \u041a\u041e\u041d\u041a\u0423\u0420\u0421\u041d\u0418 \u0417\u0410\u0414\u0410\u0427\u0418<\/strong><\/p>\n<p>Contest Problems of this Issue <br \/>[Contest Problems of the Issue] &#8211;&nbsp;<i>&nbsp;p.&nbsp;<\/i>213<\/p>\n<p><em><a href=\"https:\/\/science.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/em><\/p>\n<p>&nbsp;<\/p>\n<p>\u0420\u0435\u0448\u0435\u043d\u0438\u044f \u043d\u0430 \u043a\u043e\u043d\u043a\u0443\u0440\u0441\u043d\u0438\u0442\u0435 \u0437\u0430\u0434\u0430\u0447\u0438 \u043e\u0442 \u0431\u0440\u043e\u0439 3, 2014 <br \/>[Solutions of the Contest Problems from issue 3, 2014] &#8211;&nbsp;<i>&nbsp;p.&nbsp;<\/i>214<\/p>\n<p><a class=\"doclink\" href=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2014\/04\/azbuki.bg_dmdocuments_MathInfo-02-2014_resheniq.pdf\"><img decoding=\"async\" loading=\"lazy\" class=\"size-full wp-image-38022\" src=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2012\/08\/components_com_docman_themes_default_images_icons_16x16_pdf.png\" alt=\"icon\" border=\"0\" width=\"16\" height=\"16\" \/><em>&nbsp;Open the full text<\/em><\/a><\/p>\n<p>&nbsp;<\/p>\n<p><strong>READ IN THE LATEST ISSUES OF \u201eAZ BUKI\u201c JOURNALS \/ <\/strong><br \/><strong>\u0412 \u041d\u041e\u0412\u0418\u0422\u0415 \u0411\u0420\u041e\u0415\u0412\u0415 \u041d\u0410 \u0421\u041f\u0418\u0421\u0410\u041d\u0418\u042f\u0422\u0410 \u041d\u0410 \u0418\u0417\u0414\u0410\u0422\u0415\u041b\u0421\u0422\u0412\u041e \u201e\u0410\u0417 \u0411\u0423\u041a\u0418\u201c \u0427\u0415\u0422\u0415\u0422\u0415<\/strong> &#8211;&nbsp;<i>&nbsp;p.&nbsp;<\/i>217<\/p>\n<p><strong>GUIDE FOR AUTHORS \/ \u0423\u041a\u0410\u0417\u0410\u041d\u0418\u0415 \u0417\u0410 \u0410\u0412\u0422\u041e\u0420\u0418\u0422\u0415<\/strong> &#8211;&nbsp;<i>&nbsp;p.&nbsp;<\/i>219<\/p>\n<p style=\"text-align: right;\"><a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p><a name=\"art01\">&nbsp;<\/a><strong>SPREADSHEETS AS TOOLS FOR CONSTRUCTING MATHEMATICAL CONCEPTS<\/strong><\/p>\n<p><strong>Erich Neuwirth<\/strong><\/p>\n<p><i>University of Vienna<\/i><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Abstract. <\/strong>The formal representation of mathematical relations in spreadsheets is different from the usual algebraic representation and computer programs doing mathematics. We will demonstrate how this alternative representation can help understanding mathematical concepts in a new way that can be very helpful for learners not yet fluent in the \u201calgebraic language\u201d of mathematics.<\/p>\n<p><i>Keywords: <\/i>spreadsheet tools, spreadsheet, mathematics, programming<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Prof. Dr. Erich Neuwirth<\/strong><\/p>\n<p>Faculty of Computer Science<\/p>\n<p>University of Vienna<\/p>\n<p>Vienna, Austria<\/p>\n<p style=\"text-align: right;\"><a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p><a name=\"art02\">&nbsp;<\/a><strong>EDUCATION OF A MATHEMATICIAN \u2013 EXPERIMENTALIST, OR SOFT MANIFESTO OF EXPERIMENTAL MATHEMATICS<\/strong><\/p>\n<p><strong><sup>1)<\/sup>Alexander Yastrebov, <sup>2)<\/sup>Maria&nbsp; Shabanova<\/strong><\/p>\n<p><i><sup>1)<\/sup>Yaroslavl State Pedagogical University<\/i><\/p>\n<p><i><sup>2)<\/sup>Institute of Mathematics, Information and Space Technologies<\/i><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Abstract. <\/strong>This article is written under the influence of V. I. Arnold\u2019s works, in which he suggested the possibility of considering mathematics as an experimental science, and proved the existence of phenomena in the description of which \u201csoft\u201d models have an advantage over the \u201chard\u201d. The article is a kind of response to the manifesto, which was delivered by R. V. Shamin, head of the group of experimental mathematics. By this manifesto the group declares the emergence of a new branch in mathematics, which is characterized by the extensive use of computer experiments. Due to the fact that the means and methods of experimental mathematics have lately become increasingly involved in teaching mathematics, the authors of this article have considered it necessary to have their own manifesto published. The article aims to remind readers what place was occupied and is occupied by experimental methods in mathematics and mathematics education; to show positive and negative aspects of the impact which computer experiments involved in the process of learning mathematics may have on the formation of the style of pupils\u2019 scientific thinking; to call for reasonable and prudent use of computer experiments both in mathematics and&nbsp;mathematics education.<\/p>\n<p><i>Keywords: <\/i>experimental mathematics, mathematics education, dynamic geometry software, computer experiment, experimental-theoretical gap<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Prof. Alexander Yastrebov, DSc.<\/strong><\/p>\n<p>Yaroslavl State Pedagogical University<\/p>\n<p>Yaroslavl, Russia<\/p>\n<p><strong>Prof. Maria Shabanova, DSc.<\/strong><\/p>\n<p>Institute of Mathematics, Information and Space Technologies<\/p>\n<p>Arkhangelsk, Russia<\/p>\n<p style=\"text-align: right;\"><a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p><a name=\"art03\">&nbsp;<\/a><strong>COMPUTER-DISCOVERED MATHEMATICS: LALESCO PRODUCTS<\/strong><\/p>\n<p><strong>Sava Grozdev, Deko Dekov<\/strong><\/p>\n<p><i>Institute of Mathematics and Informatics<\/i><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Abstract. <\/strong>The computer program \u201cDiscoverer\u201d, created by the authors, is the first computer program, able easily to discover new theorems in mathematics, and possibly, the first computer program, able easily to discover new knowledge in science. In this paper we use \u201cDiscoverer\u201d in order to investigate the Lalesco products in the geometry of the triangle.<\/p>\n<p><i>Keywords: <\/i>geometry of the triangle, computer-discovered mathematics, Discoverer<\/p>\n<p>&nbsp;<\/p>\n<p><strong>\u041a\u041e\u041c\u041f\u042e\u0422\u042a\u0420\u041d\u041e \u041e\u0422\u041a\u0420\u0418\u0422\u0410\u0422\u0410 \u041c\u0410\u0422\u0415\u041c\u0410\u0422\u0418\u041a\u0410: \u041f\u0420\u041e\u0418\u0417\u0412\u0415\u0414\u0415\u041d\u0418\u042f \u041d\u0410 \u041b\u0410\u041b\u0415\u0421\u041a\u041e<\/strong><\/p>\n<p><strong>Abstract.<\/strong> \u041a\u043e\u043c\u043f\u044e\u0442\u044a\u0440\u043d\u0430\u0442\u0430 \u043f\u0440\u043e\u0433\u0440\u0430\u043c\u0430 \u201e\u041e\u0442\u043a\u0440\u0438\u0432\u0430\u0442\u0435\u043b\u201c, \u0441\u044a\u0437\u0434\u0430\u0434\u0435\u043d\u0430 \u043e\u0442 \u0430\u0432\u0442\u043e\u0440\u0438\u0442\u0435, \u0435 \u043f\u044a\u0440\u0432\u0430\u0442\u0430 \u043a\u043e\u043c\u043f\u044e\u0442\u044a\u0440\u043d\u0430 \u043f\u0440\u043e\u0433\u0440\u0430\u043c\u0430, \u043a\u043e\u044f\u0442\u043e \u043b\u0435\u0441\u043d\u043e \u043c\u043e\u0436\u0435 \u0434\u0430 \u043e\u0442\u043a\u0440\u0438\u0432\u0430 \u043d\u043e\u0432\u0438 \u0442\u0435\u043e\u0440\u0435\u043c\u0438 \u0432 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430\u0442\u0430, \u0438 \u043c\u043e\u0436\u0435 \u0431\u0438 \u043f\u044a\u0440\u0432\u0430\u0442\u0430 \u043a\u043e\u043c\u043f\u044e\u0442\u044a\u0440\u043d\u0430 \u043f\u0440\u043e\u0433\u0440\u0430\u043c\u0430, \u043a\u043e\u044f\u0442\u043e \u043b\u0435\u0441\u043d\u043e \u043c\u043e\u0436\u0435 \u0434\u0430 \u043e\u0442\u043a\u0440\u0438\u0432\u0430 \u043d\u043e\u0432\u043e \u0437\u043d\u0430\u043d\u0438\u0435 \u0432 \u043d\u0430\u0443\u043a\u0430\u0442\u0430. \u0412 \u0442\u0430\u0437\u0438 \u0441\u0442\u0430\u0442\u0438\u044f \u043d\u0438\u0435 \u0438\u0437\u043f\u043e\u043b\u0437\u0432\u0430\u043c\u0435 \u201e\u041e\u0442\u043a\u0440\u0438\u0432\u0430\u0442\u0435\u043b\u201c, \u0437\u0430 \u0434\u0430 \u0438\u0437\u0443\u0447\u0438\u043c \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044f\u0442\u0430 \u043d\u0430 Lalesco \u0432 \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u044f\u0442\u0430 \u043d\u0430 \u0442\u0440\u0438\u044a\u0433\u044a\u043b\u043d\u0438\u043a\u0430.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Prof. Sava Grozdev, DSc<\/strong><\/p>\n<p>Institute of Mathematics and Informatics \u2013 BAS<\/p>\n<p>Sofia, Bulgaria<\/p>\n<p><strong>&nbsp;Dr. Deko Dekov, Assoc. Professor<\/strong><\/p>\n<p>Stara Zagora, Bulgaria<\/p>\n<p style=\"text-align: right;\"><a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p><a name=\"art04\">&nbsp;<\/a><strong>\u0421\u0410\u041c\u041e\u0421\u0422\u041e\u042f\u0422\u0415\u041b\u041d\u041e \u0420\u0415\u0428\u0410\u0412\u0410\u041d\u0415 \u041d\u0410 \u0417\u0410\u0414\u0410\u0427\u0418 \u0421 EXCEL<\/strong><\/p>\n<p><strong><sup>1)<\/sup>\u041f\u043b\u0430\u043c\u0435\u043d&nbsp; \u041f\u0435\u043d\u0435\u0432, <sup>2)<\/sup>Diana Stefanova<\/strong><\/p>\n<p><sup>1)<\/sup>\u0421\u041e\u0423 \u201e\u041f\u0430\u043d\u0430\u0439\u043e\u0442 \u0412\u043e\u043b\u043e\u0432\u201c \u2013 \u0428\u0443\u043c\u0435\u043d<\/p>\n<p><sup>2)<\/sup>\u041e\u0423 \u201e\u041d\u0438\u043a\u043e\u043b\u0430 \u0419\u043e\u043d\u043a\u043e\u0432 \u0412\u0430\u043f\u0446\u0430\u0440\u043e\u0432\u201c \u2013 \u0410\u0441\u0435\u043d\u043e\u0432\u0433\u0440\u0430\u0434<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Abstract. <\/strong>\u0412 \u0441\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0435\u043a\u0441\u043f\u0435\u0440\u0438\u043c\u0435\u043d\u0442\u0430\u043b\u043d\u0430\u0442\u0430 \u0441\u0440\u0435\u0434\u0430 \u043d\u0430 Excel \u0441\u0435 \u0438\u0437\u043f\u043e\u043b\u0437\u0432\u0430 \u0437\u0430 \u043f\u0440\u043e\u0432\u0435\u0440\u043a\u0430 \u043d\u0430 \u0438\u0434\u0435\u0438, \u0445\u0440\u0443\u043c\u0432\u0430\u043d\u0438\u044f, \u0434\u043e\u0433\u0430\u0434\u043a\u0438, \u043e\u0431\u0443\u0441\u043b\u0430\u0432\u044f\u0449\u0438 \u043f\u0440\u043e\u0446\u0435\u0441\u0430 \u043d\u0430 \u0440\u0435\u0448\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u043f\u043e-\u0442\u0440\u0443\u0434\u043d\u0438 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u0438 \u0437\u0430\u0434\u0430\u0447\u0438. \u041e\u043f\u0438\u0441\u0430\u043d\u0430 \u0435 \u043c\u0435\u0442\u043e\u0434\u0438\u043a\u0430, \u043f\u043e \u043a\u043e\u044f\u0442\u043e \u0443\u0447\u0435\u043d\u0438\u043a\u044a\u0442, \u0440\u0430\u0437\u0447\u0438\u0442\u0430\u0439\u043a\u0438 \u043d\u0430 \u201e\u0438\u043d\u0442\u0435\u043b\u0438\u0433\u0435\u043d\u0442\u043d\u0430\u0442\u0430\u201c \u043f\u043e\u043c\u043e\u0449 \u043d\u0430 Excel, \u0441\u0430\u043c\u043e\u0441\u0442\u043e\u044f\u0442\u0435\u043b\u043d\u043e \u043c\u043e\u0436\u0435 \u0434\u0430 \u0441\u0442\u0438\u0433\u043d\u0435 \u0434\u043e \u0440\u0435\u0448\u0435\u043d\u0438\u0435\u0442\u043e \u043d\u0430 \u043d\u0435\u0441\u0442\u0430\u043d\u0434\u0430\u0440\u0442\u043d\u0430 \u0437\u0430\u0434\u0430\u0447\u0430.<\/p>\n<p><i>Keywords:<\/i> problem solving, equation, numbers, tables, macros<\/p>\n<p>&nbsp;<\/p>\n<p><strong>INDEPENDENT PROBLEM SOLVING WITH EXCEL<\/strong><\/p>\n<p><strong>Abstract.<\/strong> In this paper, the experimental medium of Excel is used for a verification of ideas, thoughts, and guesses, which arise in the process of solving harder mathematical problems. A description of methodology, which can help the student to independently solve non-standard problem with the intelligent assistance of Excel, is given.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Mr. Plamen Penev<\/strong><\/p>\n<p>Informatics Teacher<\/p>\n<p>Secondary School \u201ePanayot Volov\u201c \u2013 Shumen<\/p>\n<p>Shumen, Bulgaria<\/p>\n<p><strong>Dr. Diana Stefanova<\/strong><\/p>\n<p>Mathematics Teacher<\/p>\n<p>Primary School \u201eNikola Yonkov Vaptzarov\u201c \u2013&nbsp; Asenovgrad<\/p>\n<p>Asenovgrad, Bulgaria<\/p>\n<p style=\"text-align: right;\"><a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p><a name=\"art05\">&nbsp;<\/a><strong>\u041c\u041e\u0414\u0418\u0424\u0418\u041a\u0410\u0426\u0418\u042f \u041c\u0415\u0422\u041e\u0414\u0410 \u041f\u0420\u041e\u0415\u041a\u0426\u0418\u0419 \u0412\u042cI\u0427\u0418\u0421\u041b\u0415\u041d\u0418\u042f \u0420\u0410\u0421\u0421\u0422\u041e\u042f\u041d\u0418\u042f \u041c\u0415\u0416\u0414\u0423 \u0421\u041a\u0420\u0415\u0429\u0418\u0412\u0410\u042e\u0429\u0418\u041c\u0418\u0421\u042f \u041f\u0420\u042f\u041c\u042cI\u041c\u0418<\/strong><\/p>\n<p><strong>\u0412\u043b\u0430\u0434\u0438\u043c\u0438\u0440 \u0416\u0443\u043a<\/strong><\/p>\n<p><i>\u0420\u0435\u0441\u043f\u0443\u0431\u043b\u0438\u043a\u0430\u043d\u0441\u043a\u0430\u044f \u0441\u043f\u0435\u0446\u0438\u0430\u043b\u0438\u0437\u0438\u0440\u043e\u0432\u0430\u043d\u043d\u0430\u044f \u0444\u0438\u0437\u0438\u043a\u043e-\u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u0430\u044f&nbsp;<\/i><i>\u0441\u0440\u0435\u0434\u043d\u044f\u044f \u0448\u043a\u043e\u043b\u0430-\u0438\u043d\u0442\u0435\u0440\u043d\u0430\u0442 \u0438\u043c\u0435\u043d\u0438 \u041e. \u0416\u0430\u0443\u0442\u044b\u043a\u043e\u0432\u0430<\/i><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Abstract. <\/strong>\u0412 \u0434\u0430\u043d\u043d\u043e\u0439 \u0441\u0442\u0430\u0442\u044c\u0435 \u0440\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0435\u043d\u044b \u043c\u0435\u0442\u043e\u0434\u0438\u0447\u0435\u0441\u043a\u0438\u0435 \u043f\u043e\u0434\u0445\u043e\u0434\u044b \u043a \u0437\u0430\u0434\u0430\u0447\u0435 \u043d\u0430\u0445\u043e\u0436\u0434\u0435\u043d\u0438\u044f \u0440\u0430\u0441\u0441\u0442\u043e\u044f\u043d\u0438\u044f \u043c\u0435\u0436\u0434\u0443 \u0441\u043a\u0440\u0435\u0449\u0438\u0432\u0430\u044e\u0449\u0438\u043c\u0438\u0441\u044f \u043f\u0440\u044f\u043c\u044b\u043c\u0438.<\/p>\n<p><i>Keywords: <\/i>skew lines, distance, projection, angle, solid geometry, methodology<\/p>\n<p>&nbsp;<\/p>\n<p><strong>MODIFICATION OF THE PROJECTION METHOD TO FIND THE DISTANCE BETWEEN SKEW LINES<\/strong><\/p>\n<p><strong>Abstract. <\/strong>The present paper considers methodological approaches to the problem of finding the distance between skew lines.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Dr. Vladimir Zhuk<\/strong><\/p>\n<p>Teacher in Mathematics<\/p>\n<p>Secondary School for Talented Students<\/p>\n<p>Almaty, Kazhakhstan<\/p>\n<p style=\"text-align: right;\"><a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p><a name=\"art06\">&nbsp;<\/a><strong>\u041e\u0411\u041e\u0411\u0429\u0415\u041d\u0418\u042f \u0417\u0410 \u041d\u042f\u041a\u041e\u0418 \u041a\u041b\u0410\u0421\u041e\u0412\u0415 \u041d\u0415\u041e\u041f\u0420\u0415\u0414\u0415\u041b\u0415\u041d\u0418 \u0418\u041d\u0422\u0415\u0413\u0420\u0410\u041b\u0418, \u0421\u0412\u0415\u0416\u0414\u0410\u0429\u0418 \u0421\u0415 \u0414\u041e \u0420\u0415\u041a\u0423\u0420\u0415\u041d\u0422\u041d\u0410 \u0417\u0410\u0412\u0418\u0421\u0418\u041c\u041e\u0421\u0422<\/strong><\/p>\n<p><strong>\u0420\u043e\u0441\u0435\u043d \u041d\u0438\u043a\u043e\u043b\u0430\u0435\u0432<\/strong><\/p>\n<p><i>University of Economics \u2013 Varna<\/i><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Abstract.<\/strong> \u0412 \u0441\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0441\u0435 \u0434\u043e\u0440\u0430\u0437\u0432\u0438\u0432\u0430 \u0440\u0435\u043a\u0443\u0440\u0435\u043d\u0442\u043d\u0430\u0442\u0430 \u0432\u0440\u044a\u0437\u043a\u0430 \u0437\u0430 \u0438\u0437\u0432\u0435\u0441\u0442\u043d\u0438\u044f \u043a\u043b\u0430\u0441 \u0438\u043d\u0442\u0435\u0433\u0440\u0430\u043b\u0438 \u043e\u0442\u0432\u0438\u0434\u0430 \u222b dx\u2044(a<sup>2<\/sup>+x<sup>2<\/sup>)<sup>n<\/sup>&nbsp;\u0438 \u0441\u0435 \u0438\u0437\u0432\u0435\u0436\u0434\u0430 \u0444\u043e\u0440\u043c\u0443\u043b\u0430 \u0437\u0430 \u0442\u044f\u0445\u043d\u043e\u0442\u043e \u043f\u0440\u0435\u0441\u043c\u044f\u0442\u0430\u043d\u0435.<\/p>\n<p>\u041f\u0440\u0435\u0434\u043b\u0430\u0433\u0430 \u0441\u0435 \u043d\u043e\u0432\u0430 \u043c\u0435\u0442\u043e\u0434\u0438\u043a\u0430\u0438 \u0441\u0435 \u0438\u0437\u0432\u0435\u0436\u0434\u0430 \u0444\u043e\u0440\u043c\u0443\u043b\u0430 \u0437\u0430 \u043f\u0440\u0435\u0441\u043c\u044f\u0442\u0430\u043d\u0435 \u043d\u0430 \u0435\u0434\u0438\u043d \u043a\u043b\u0430\u0441 \u0438\u043d\u0442\u0435\u0433\u0440\u0430\u043b\u0438 \u043e\u0442 \u0432\u0438\u0434\u0430 \u222b (x<sup>k\u03b1+\u03b1\u20442-1<\/sup>\u2044\u221a1-x<sup>\u03b1 <\/sup>)dx, \u043a\u044a\u0434\u0435\u0442\u043e k<span style=\"font-size: 12pt;\">\u220a<\/span>\u00a5 , \u03b1<span style=\"font-size: 16px;\">\u220a<\/span>\u00a1{0}.<\/p>\n<p><em>Keywords:<\/em> indefinite integral, recurrence, differential binomial<\/p>\n<p>&nbsp;<\/p>\n<p><strong>GENERALIZATION OF SOME TYPES OF INDEFINITE INTEGRALS, WHICH CAN BE SOLVED BY RECURRENCE<\/strong><\/p>\n<p><strong>Abstract.<\/strong>&nbsp;In this article we develop a recurrence for the indefinite integrals of the famoustype \u222b dx\u2044(a<sup>2<\/sup>+x<sup>2<\/sup>)<sup>n<\/sup>&nbsp;and a formula for their calculation. A new methodology and a formula forsolving integrals of the type \u222b (x<sup>k\u03b1+\u03b1\u20442-1<\/sup>\u2044\u221a1-x<sup>\u03b1&nbsp;<\/sup>)dx, where&nbsp;k<span style=\"font-size: 12pt;\">\u220a<\/span>\u00a5 , \u03b1<span style=\"font-size: 16px;\">\u220a<\/span>\u00a1{0} are also described.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Dr. Rosen Nikolaev, Assoc. Prof.<\/strong><\/p>\n<p>University of Economics \u2013 Varna<\/p>\n<p>&nbsp;Varna, Bulgaria<\/p>\n<p style=\"text-align: right;\"><a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p><a name=\"art07\">&nbsp;<\/a><strong>APPLYING CLOUD COMPUTING&nbsp;<\/strong><strong>FOR A MORE EFFICIENT EVALUATION OF PRACTICAL PROJECTS<\/strong><\/p>\n<p><strong>Emil Delinov<\/strong><\/p>\n<p><strong>Abstract. <\/strong>Significant competitive advantage both for businesses and for the other&nbsp;entities are the well prepared, adaptive, and constantly improving staff. One approach&nbsp;of the education system for the creation and development of adequate to the needs staff, is the development of projects with topics from real economy. Important preconditions for successful realization of the project are the close collaboration between the trainer (teacher, lecturer, mentor, supervisor\/consultant, etc.) and the learner and the motivation of the latter to create a product of high quality. The accurate and objective evaluation, set and announced on time, could play a significant role for the positive motivation and respectively for the successful outcome of the project and the acquisition and consolidation of knowledge. This paper considers some features of Cloud Computing used to increase the efficiency in the evaluation of projects with topics from real economy.<\/p>\n<p><i>Keywords: <\/i>Cloud Computing; Cloud Computing in education; usage of Cloud technologies in education; project work; evaluation of project work; road map of practical projects; Cloud Computing for evaluation<\/p>\n<p>&nbsp;<\/p>\n<p><strong>\u041f\u0420\u0418\u041b\u041e\u0416\u0415\u041d\u0418\u0415 \u041d\u0410 \u201e\u041e\u0411\u041b\u0410\u0427\u041d\u0418 \u0422\u0415\u0425\u041d\u041e\u041b\u041e\u0413\u0418\u0418\u201c \u0417\u0410 \u041f\u041e-\u0415\u0424\u0415\u041a\u0422\u0418\u0412\u041d\u041e \u041e\u0426\u0415\u041d\u042f\u0412\u0410\u041d\u0415 \u041d\u0410 \u041f\u0420\u0410\u041a\u0422\u0418\u0427\u0415\u0421\u041a\u0418 \u041f\u0420\u041e\u0415\u041a\u0422\u0418<\/strong><\/p>\n<p><strong>Emil Delinov<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Abstract.<\/strong> \u0421\u044a\u0449\u0435\u0441\u0442\u0432\u0435\u043d\u043e \u043a\u043e\u043d\u043a\u0443\u0440\u0435\u043d\u0442\u043d\u043e \u043f\u0440\u0435\u0434\u0438\u043c\u0441\u0442\u0432\u043e \u0437\u0430 \u0441\u0442\u043e\u043f\u0430\u043d\u0441\u043a\u0438\u0442\u0435 \u0438 \u0437\u0430 \u0434\u0440\u0443\u0433\u0438 \u0441\u0443\u0431\u0435\u043a\u0442\u0438 \u0441\u0430 \u043f\u043e\u0434\u0433\u043e\u0442\u0432\u0435\u043d\u0438\u0442\u0435, \u0430\u0434\u0430\u043f\u0442\u0438\u0432\u043d\u0438 \u0438 \u043f\u043e\u0441\u0442\u043e\u044f\u043d\u043d\u043e \u0443\u0441\u044a\u0432\u044a\u0440\u0448\u0435\u043d\u0441\u0442\u0432\u0430\u0449\u0438 \u0441\u0435 \u043a\u0430\u0434\u0440\u0438. \u0415\u0434\u0438\u043d \u043e\u0442 \u043f\u043e\u0434\u0445\u043e\u0434\u0438\u0442\u0435 \u043d\u0430 \u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u0442\u0435\u043b\u043d\u0430\u0442\u0430 \u0441\u0438\u0441\u0442\u0435\u043c\u0430 \u0437\u0430 \u0441\u044a\u0437\u0434\u0430\u0432\u0430\u043d\u0435\u0442\u043e \u0438 \u0440\u0430\u0437\u0432\u0438\u0442\u0438\u0435\u0442\u043e \u043d\u0430 \u0430\u0434\u0435\u043a\u0432\u0430\u0442\u043d\u0438 \u043d\u0430 \u043d\u0443\u0436\u0434\u0438\u0442\u0435 \u043a\u0430\u0434\u0440\u0438 \u0435 \u0440\u0430\u0437\u0440\u0430\u0431\u043e\u0442\u043a\u0430\u0442\u0430 \u043d\u0430 \u043f\u0440\u043e\u0435\u043a\u0442\u0438 \u0441 \u0442\u0435\u043c\u0438 \u043e\u0442 \u0440\u0435\u0430\u043b\u043d\u0430\u0442\u0430 \u0438\u043a\u043e\u043d\u043e\u043c\u0438\u043a\u0430. \u0412\u0430\u0436\u043d\u0438 \u0443\u0441\u043b\u043e\u0432\u0438\u044f \u0437\u0430 \u0443\u0441\u043f\u0435\u0448\u043d\u0430 \u0440\u0435\u0430\u043b\u0438\u0437\u0430\u0446\u0438\u044f \u043d\u0430 \u043f\u0440\u043e\u0435\u043a\u0442 \u0441\u0430 \u0431\u043b\u0438\u0437\u043a\u043e\u0442\u043e \u0441\u044a\u0442\u0440\u0443\u0434\u043d\u0438\u0447\u0435\u0441\u0442\u0432\u043e \u043c\u0435\u0436\u0434\u0443 \u043e\u0431\u0443\u0447\u0430\u0432\u0430\u0449\u0438\u044f (\u0443\u0447\u0438\u0442\u0435\u043b, \u043f\u0440\u0435\u043f\u043e\u0434\u0430\u0432\u0430\u0442\u0435\u043b, \u043d\u0430\u0441\u0442\u0430\u0432\u043d\u0438\u043a, \u043d\u0430\u0443\u0447\u0435\u043d \u0440\u044a\u043a\u043e\u0432\u043e\u0434\u0438\u0442\u0435\u043b\/\u043a\u043e\u043d\u0441\u0443\u043b\u0442\u0430\u043d\u0442 \u0438 \u0434\u0440.) \u0438 \u043e\u0431\u0443\u0447\u0430\u0435\u043c\u0438\u044f \u0438 \u043c\u043e\u0442\u0438\u0432\u0430\u0446\u0438\u044f\u0442\u0430 \u043d\u0430 \u043f\u043e\u0441\u043b\u0435\u0434\u043d\u0438\u044f \u0437\u0430 \u0441\u044a\u0437\u0434\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u043a\u0430\u0447\u0435\u0441\u0442\u0432\u0435\u043d \u043f\u0440\u043e\u0434\u0443\u043a\u0442. \u0422\u043e\u0447\u043d\u0430\u0442\u0430 \u0438 \u043e\u0431\u0435\u043a\u0442\u0438\u0432\u043d\u0430 \u043e\u0446\u0435\u043d\u043a\u0430, \u043f\u043e\u0441\u0442\u0430\u0432\u0435\u043d\u0430 \u0438 \u043e\u043f\u043e\u0432\u0435\u0441\u0442\u0435\u043d\u0430 \u043d\u0430\u0432\u0440\u0435\u043c\u0435, \u0431\u0438 \u043c\u043e\u0433\u043b\u0430 \u0434\u0430 \u0438\u0437\u0438\u0433\u0440\u0430\u0435 \u0437\u043d\u0430\u0447\u0438\u0442\u0435\u043b\u043d\u0430 \u0440\u043e\u043b\u044f \u0437\u0430 \u043f\u043e\u043b\u043e\u0436\u0438\u0442\u0435\u043b\u043d\u0430 \u043c\u043e\u0442\u0438\u0432\u0430\u0446\u0438\u044f \u0438 \u0440\u0435\u0441\u043f\u0435\u043a\u0442\u0438\u0432\u043d\u043e \u0437\u0430 \u0443\u0441\u043f\u0435\u0448\u043d\u0438 \u0440\u0435\u0437\u0443\u043b\u0442\u0430\u0442\u0438 \u043e\u0442 \u0440\u0430\u0431\u043e\u0442\u0430\u0442\u0430 \u043f\u043e \u043f\u0440\u043e\u0435\u043a\u0442, \u043f\u0440\u0438\u0434\u043e\u0431\u0438\u0432\u0430\u043d\u0435 \u0438 \u0437\u0430\u0442\u0432\u044a\u0440\u0436\u0434\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u0437\u043d\u0430\u043d\u0438\u044f. \u0412 \u043d\u0430\u0441\u0442\u043e\u044f\u0449\u0438\u044f \u043c\u0430\u0442\u0435\u0440\u0438\u0430\u043b \u0441\u0435 \u0440\u0430\u0437\u0433\u043b\u0435\u0436\u0434\u0430\u0442 \u043d\u044f\u043a\u043e\u0438 \u0432\u044a\u0437\u043c\u043e\u0436\u043d\u043e\u0441\u0442\u0438 \u043d\u0430 \u201e\u041e\u0431\u043b\u0430\u0447\u043d\u0438\u0442\u0435 \u0442\u0435\u0445\u043d\u043e\u043b\u043e\u0433\u0438\u0438\u201c (Cloud Computing) \u0437\u0430 \u043f\u043e\u0432\u0438\u0448\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u0435\u0444\u0435\u043a\u0442\u0438\u0432\u043d\u043e\u0441\u0442\u0442\u0430 \u043f\u0440\u0438 \u043e\u0446\u0435\u043d\u044f\u0432\u0430\u043d\u0435 \u043d\u0430 \u043f\u0440\u043e\u0435\u043a\u0442\u0438 \u0441 \u0442\u0435\u043c\u0438 \u043e\u0442 \u0440\u0435\u0430\u043b\u043d\u0430\u0442\u0430 \u0438\u043a\u043e\u043d\u043e\u043c\u0438\u043a\u0430.&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Mr. Emil Delinov<\/strong><\/p>\n<p>Sofia, Bulgaria<\/p>\n<p style=\"text-align: right;\"><a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p><a name=\"art08\">&nbsp;<\/a><strong>\u041f\u041e\u0414\u0425\u041e\u0414\u0418 \u041f\u0420\u0418 \u0420\u0410\u0417\u0420\u0410\u0411\u041e\u0422\u0412\u0410\u041d\u0415 \u041d\u0410 \u0413\u0420\u0410\u0424\u0418\u0427\u0415\u041d \u0418\u041d\u0422\u0415\u0420\u0424\u0415\u0419\u0421 \u0417\u0410 \u0420\u0410\u0411\u041e\u0422\u0410 \u0421 \u041d\u042f\u041a\u041e\u041b\u041a\u041e \u0414\u041e\u041a\u0423\u041c\u0415\u041d\u0422\u0410<\/strong><\/p>\n<p><strong>Todorka Terzieva<\/strong><\/p>\n<p><i>University of Plovdiv \u201cPaisii Hilendarski\u201d<\/i><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Abstract.<\/strong> \u0418\u0437\u0432\u0435\u0441\u0442\u043d\u0430 \u0435 \u043e\u0441\u043e\u0431\u0435\u043d\u0430\u0442\u0430 \u0440\u043e\u043b\u044f \u043d\u0430 \u0437\u0430\u0434\u0430\u0447\u0438\u0442\u0435 \u0432 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435\u0442\u043e \u043f\u043e \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430&nbsp;\u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u043e\u043d\u043d\u0438 \u0442\u0435\u0445\u043d\u043e\u043b\u043e\u0433\u0438\u0438 \u2013 \u0432 \u0447\u0430\u0441\u0442\u043d\u043e\u0441\u0442, \u0442\u0435 \u0441\u0430\u043c\u0438\u0442\u0435 \u043c\u043e\u0433\u0430\u0442 \u0434\u0430 \u0431\u044a\u0434\u0430\u0442 \u0441\u0440\u0435\u0434\u0441\u0442\u0432\u043e \u0437\u0430&nbsp;\u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435. \u0412 \u043f\u0440\u043e\u0446\u0435\u0441\u0430 \u043d\u0430 \u0440\u0435\u0448\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u0437\u0430\u0434\u0430\u0447\u0438 \u043e\u0431\u0443\u0447\u0430\u0435\u043c\u0438\u0442\u0435 \u0441\u0430\u043c\u043e\u0441\u0442\u043e\u044f\u0442\u0435\u043b\u043d\u043e \u0437\u0430\u0442\u0432\u044a\u0440\u0434\u044f\u0432\u0430\u0442 \u043d\u0430\u0443\u0447\u0435\u043d\u043e\u0442\u043e, \u0430 \u0441\u044a\u0449\u043e \u0442\u0430\u043a\u0430 \u043e\u0442\u043a\u0440\u0438\u0432\u0430\u0442 \u043d\u043e\u0432\u0438 \u0445\u0430\u0440\u0430\u043a\u0442\u0435\u0440\u0438\u0441\u0442\u0438\u043a\u0438 \u043d\u0430 \u0438\u0437\u0443\u0447\u0430\u0432\u0430\u043d\u0438\u0442\u0435 \u0435\u043b\u0435\u043c\u0435\u043d\u0442\u0438 \u0447\u0440\u0435\u0437 \u0446\u0435\u043b\u0435\u043d\u0430\u0441\u043e\u0447\u0435\u043d\u0430 \u0430\u043a\u0442\u0438\u0432\u043d\u043e\u0441\u0442 \u043e\u0442 \u0441\u0432\u043e\u044f \u0441\u0442\u0440\u0430\u043d\u0430. \u0412 \u0441\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0441\u0435 \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u044f \u043f\u0440\u043e\u0431\u043b\u0435\u043c\u043d\u043e \u043e\u0440\u0438\u0435\u043d\u0442\u0438\u0440\u0430\u043d \u043f\u043e\u0434\u0445\u043e\u0434 \u0437\u0430 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435 \u043f\u043e \u0443\u0447\u0435\u0431\u043d\u0430\u0442\u0430 \u0434\u0438\u0441\u0446\u0438\u043f\u043b\u0438\u043d\u0430 \u201e\u0421\u044a\u0437\u0434\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u0433\u0440\u0430\u0444\u0438\u0447\u0435\u043d \u043f\u043e\u0442\u0440\u0435\u0431\u0438\u0442\u0435\u043b\u0441\u043a\u0438 \u0438\u043d\u0442\u0435\u0440\u0444\u0435\u0439\u0441 C#\u201c \u043d\u0430 \u0441\u0442\u0443\u0434\u0435\u043d\u0442\u0438 \u043e\u0442 \u043f\u044a\u0440\u0432\u0438 \u043a\u0443\u0440\u0441, \u0441\u043f\u0435\u0446\u0438\u0430\u043b\u043d\u043e\u0441\u0442 \u201e\u0421\u043e\u0444\u0442\u0443\u0435\u0440\u043d\u0438 \u0442\u0435\u0445\u043d\u043e\u043b\u043e\u0433\u0438\u0438 \u0438 \u0434\u0438\u0437\u0430\u0439\u043d\u201c. \u0422\u0435\u043c\u0430\u0442\u0430, \u043a\u043e\u044f\u0442\u043e \u0441\u0435 \u0440\u0430\u0437\u0433\u043b\u0435\u0436\u0434\u0430, \u0435 \u0441\u0432\u044a\u0440\u0437\u0430\u043d\u0430 \u0441\u044a\u0441 \u0441\u0442\u0438\u043b\u043e\u0432\u0435 \u0437\u0430 \u0434\u0438\u0437\u0430\u0439\u043d \u043d\u0430 \u0438\u043d\u0442\u0435\u0440\u0444\u0435\u0439\u0441 \u0438 \u0440\u0430\u0431\u043e\u0442\u0430 \u0441 \u043d\u044f\u043a\u043e\u043b\u043a\u043e \u0434\u043e\u043a\u0443\u043c\u0435\u043d\u0442\u0430 \u0435\u0434\u043d\u043e\u0432\u0440\u0435\u043c\u0435\u043d\u043d\u043e.<\/p>\n<p><i>Keywords: <\/i>graphical user interface, SDI and MDI applications<\/p>\n<p>&nbsp;<\/p>\n<p><strong>APPROACHES IN DEVELOPING GUI AND WORKING WITH&nbsp;<\/strong><strong>SEVERAL APPLICATIONS<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Abstract. <\/strong>Tasks have a very specific role in the learning of informatics and information technologies. In particular, they can be educational tools within themselves. In the process of solving a problem, students consolidate their knowledge and also discover new characteristics of previously studied elements through activity on their part. The paper presents problem-oriented approach for teaching the course \u201cCreating a GUI C #\u201d for first-year students, specialty \u201cSoftware technologies and design\u201d. This article emphasis on a methodological approach for teaching styles of interface design and working with several applications at the same time.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Dr. Todorka Terzieva, Assist. Prof.<\/strong><\/p>\n<p>Faculty of Mathematics and Informatics<\/p>\n<p>Plovdiv University \u201cPaisii Hilendarski\u201d<\/p>\n<p>Plovdiv, Bulgaria<\/p>\n<p><strong>&nbsp;<\/strong><\/p>\n<p style=\"text-align: right;\"><a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>","protected":false},"excerpt":{"rendered":"<p>&nbsp;CONTENTS \/ \u0421\u042a\u0414\u042a\u0420\u0416\u0410\u041d\u0418\u0415&nbsp; EDUCATIONAL TECHNOLOGY \/ \u041e\u0411\u0420\u0410\u0417\u041e\u0412\u0410\u0422\u0415\u041b\u041d\u0418 \u0422\u0415\u0425\u041d\u041e\u041b\u041e\u0413\u0418\u0418 Spreadsheets as Tools for Constructing Mathematical Concepts \/ Erich Neuwirth &#8211; \u0441\u0442\u0440. 119 &nbsp; Education of a Mathematician \u2013 Experimentalist, or Soft Manifesto of Experimental Mathematics \/ Alexander Yastrebov, Maria Shabanova &#8211; \u0441\u0442\u0440. 129 &nbsp; Computer-discovered Mathematics: Lalesco Products \/ Sava Grozdev, Deko Dekov &#8211; \u0441\u0442\u0440. 143 [&hellip;]<\/p>","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jnews-multi-image_gallery":[],"jnews_single_post":[],"jnews_primary_category":[]},"categories":[2882],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.7 - 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