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Austrian Journal of Technical and Natural Sciences Austrian Journal of Technical and Natural Sciences № 7–8 2016 July–August «East West» Association for Advanced Studies and Higher Education GmbH Vienna 2016 Austrian Journal of Technical and Natural Sciences Scientific journal № 7–8 2016 (July–August) ISSN 2310-5607 Editor-in-chief Hong Han, China, Doctor of Engineering Sciences International editorial board Andronov Vladimir Anatolyevitch, Ukraine, Doctor of Engineering Sciences Bestugin Alexander Roaldovich, Russia, Doctor of Engineering Sciences Frolova Tatiana Vladimirovna, Ukraine, Doctor of Medicine Inoyatova Flora Ilyasovna, Uzbekistan, Doctor of Medicine Kushaliyev Kaisar Zhalitovich, Kazakhstan, Doctor of Veterinary Medicine Mambetullaeva Svetlana Mirzamuratovna, Uzbekistan, Doctor of Biological Sciences Nagiyev Polad Yusif, Azerbaijan, Ph.D. of Agricultural Sciences Nemikin Alexey Andreevich, Russia, Ph.D. of Agricultural Sciences Nenko Nataliya Ivanovna, Russia, Doctor of Agricultural Sciences Skopin Pavel Igorevich, Russia, Doctor of Medicine Suleymanov Suleyman Fayzullaevich, Uzbekistan, Ph.D. of Medicine Zhanadilov Shaizinda, Uzbekistan, Doctor of Medicine Proofreading Kristin Theissen Cover design Andreas Vogel Additional design Stephan Friedman Editorial office European Science Review “East West” Association for Advanced Studies and Higher Education GmbH, Am Gestade 1 1010 Vienna, Austria Email: [email protected] Homepage: www.ew-a.org Austrian Journal of Technical and Natural Sciences is an international, German/English/Russian language, peer-reviewed journal. It is published bimonthly with circulation of 1000 copies. The decisive criterion for accepting a manuscript for publication is scientific quality. All research articles published in this journal have undergone a rigorous peer review. Based on initial screening by the editors, each paper is anonymized and reviewed by at least two anonymous referees. Recommending the articles for publishing, the reviewers confirm that in their opinion the submitted article contains important or new scientific results. East West Association GmbH is not responsible for the stylistic content of the article. The responsibility for the stylistic content lies on an author of an article. Instructions for authors Full instructions for manuscript preparation and submission can be found through the “East West” Association GmbH home page at: http://www.ew-a.org. Material disclaimer The opinions expressed in the conference proceedings do not necessarily reflect those of the «East West» Association for Advanced Studies and Higher Education GmbH, the editor, the editorial board, or the organization to which the authors are affiliated. East West Association GmbH is not responsible for the stylistic content of the article. The responsibility for the stylistic content lies on an author of an article. © «East West» Association for Advanced Studies and Higher Education GmbH All rights reserved; no part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior written permission of the Publisher. Typeset in Berling by Ziegler Buchdruckerei, Linz, Austria. Printed by «East West» Association for Advanced Studies and Higher Education GmbH, Vienna, Austria on acid-free paper. The construction of the sums of a geometric progression to a degree Section 1. Mathematics Druzhinin Victor Vladimirovich, Alekseev Vladimir Vasilevich National research University “MEPHI” Sarov physical-technical Institute E mail: [email protected] The construction of the sums of a geometric progression to a degree Abstract: The analytical expressions for the erection of sums of geometric progressions to a degree. The results are used for the summation of series, complex finite sums, solving algebraic equations. Keywords: geometric progression, complex finite and infinite sums. In analytical mathematics there are many examples decrease symmetrically, and cfkn= mk− � and �bekn= mk− . We of finite sums and series, with a relatively simple formula have identified the coefficients of the different letters as the result. These include the binomial theorem, sums they are considered by different formulas. The basic of arithmetic and geometric progressions, sums of factors are calculated using binomial coefficient ()n ≥ 2 Bernoulli — ordered amount of the one power, whole and ()mk+−1 ! k cnkm(),m = =C +−k 1. (4) fractional numbers, arithmetic-geometric progressions ()mk−1 !! and other subjects [1; 2; 3; 4]. In this article we focus For the coefficients bk and ds we have not received on the construction in the natural degree m amounts of General formula, but found them to specific m = 234,,�� . a geometric progression If nm is an odd number, then in the middle of the right Ga(,nm,) = row (3) there is not one greatest factor, and two of the m n m n+1 k 23 n m a −1 same factor. If nt=+21,�mp=+21 that are coefficients = �� ∑aa =+()1 ++aa+…+a �= . (1) k=0 a −1 dd22pt ++tp= pt ++tp+1. Subsequently the coefficients of mirror In reference books and educational literature we have symmetry occurs. not found this problem. Meanwhile here there is a large If m = 2 then n 2 n 2n set of new deemed according to the formula amounts and k k k Ga(),,na21= ∑∑ =+� ()ka++∑ ()21nk− a. (5) ranks as well as taking the integrals and the solution of k==0 k 01kn=+ For example, algebraic equations. In (1) the final result is the right side, 2 but as reveals the amount of degree, what its symmetry ()11++aa2 =+23aa++232aa+ 4 , (6) 2 and structure are we show below. First we write the ()11++aa23+aa=+23++aa2343++aa452 +a 6 . (7) polynomial theorem From identities (6 –7) a symmetry of the right and n m m! tt12 tn left parts and an algorithm for the construction of any ∑∑xk =� ()xx12()...()xn ,, (2) k=0 ()tt12!!()...()t n ! geometric progression is in the square. where ()tt12++…+tmn = and the sum goes through all In the construction of Ga(,n,)3 in the cube we obtain combinations of these numbers. With the special the following result. If n is even, nt= 2 , if xx= k 2 structure of numbers k � (our case) the left part is 21tk+≤ ≤−31tb,;��k =+13tt()+1 −−()mt k nm +1 much easier. Number of terms is (), the highest dt3t =+13()t +1 . Examples a nm nm 23456 degree is . General record in even has a view Ga(),,23 =+13aa++67aa++63aa+ . (8) n s −1 21sn−− mn 234 k k s k k Ga(),,43� =+13aa++6101aa++5 Ga(),,nm =+∑∑cak bak ++das ∑ eak + ∑ fak . (3) k==01kn+ ks=+1 k=2ssn− 56712 (9) k ++18aaa 19++ 18 …+a . Coefficients with increasing monotonically increase 567 d d Ga(),,63� =+12…+12aa83++3a up to including s . The central coefficient s has the (10) greatest value. The following ratios monotonically ++36aa89 37+… 36aa10+ 18 . 3 Section 1. Mathematics 456 If n is odd, nt=+21, then 22tk+≤≤=31tb,;��31tq+− dq 31t + −+()q Ga(),,44� =+13…+ 55aa++268a + (15) 22tk+≤≤=31tb,;dq−+q 11≤≤qt− 78916 ��31tq+− 31t + () . The highest coefficients ++ 80aa 85+… 80aa+ . dd==31t + 2 . 31tt++32 () Examples It is also possible to obtain other degrees Ga(),,nm� . Ga(),,33� =+13aa++6123401aa++2 (11) Consider the possible applications of these structures. ++12aa56 10 ++63aa78+a 9 .� 1. The sum of a convergent series of generalized Ga(),,53 =+13aa++612301aa++5 4 geometric progression with 01<<a species arises (12) 56718 ∞ ++21aa 25++ 27aa…+ .� ()mk+−1 ! k 1 ∑ a = m .� � � (16) These amounts are calculated according to right side k=0 ()mk−1 !! ()1−a (1). In the construction of Ga(,n,)4 in the fourth degree a we get such formulas, regardless of the evenness or 2. If we take the derivative at , we get the new calculated amount. For example, ∂Ga(),,23� / ∂a gives unevenness of n . If kn==21,/�dn2n ++52nn()−14()n + 3 kn==21,/dn++52nn−14n + 3 3 2 2 � 2n ()(), the maximum value of the ()aa−12()−−a 1 ++23++45+= kn=−21� 14aa78aa52a 3 .(17)� coefficient. If have the greatest ()a −1 bd21nn− =−2 ()n +1 . The other coefficients b2nl− , with 12≤≤ln− are calculated by recurrent relations 3. There appears the possibility of analytical solutions 2 bb21nl−− =−2 nl− ()21ln+ +−()32ll− /.2 � of algebraic equations of high order. The equation Examples 65432 23 aa++36aa++76aa+−3260= � is reduced to the Ga(),,24� =+14aa++10 16a + 2 (13) equation aa+−20= � which has two aa12==12,� − . ++45+…+ 8 19aa 16 a .� � 4. Combinations of sums and products of Ga(),,nm� Ga(),,34� =+14aa++1023 20aa++ 31 4 (14) form a new hope of final amount and series and also give ++40aa56 44++ 40aa71.. + 2 . the opportunity to find the integral analytically. References: 1. Sizyi S. V. Lectures on the theory of numbers, FIZMATGIZ, – M., 2007. 2. Korn. G., Korn. T. Handbook of mathematics, Science, GHML, – M., 1974. P. 31, 135. 3. Gradshteyn I. S., Ryzhik I. – M. Tables of integrals, sums, series and products. GIFML, – M, 1962. P. 15–16. 4. Druzhinin V. V., Sirotkina A. G. NTVP, No. 4, 2016, P. 15–16. 4 The construction of the sums of a geometric progression to a degree Section 2. Medical science Anoshina Tatiana Nikolaevna, PhD, Assistant of the Department of Obstetrics, Gynecology and Reproduction, Shupik National Medical Academy of Postgraduate Education, Kyiv, Ukraine E-mail: [email protected] Clinical course of genital herpes in HIV-infected pregnant women Abstract: HIV-infected pregnant women have severe manifestations of the clinical course of genital herpes, which was accompanied by a characteristic colpo- and vulvoscopic picture (inflammatory background, herpes vesicles, ulcers, erosion, hyperemia, apparent edema, serous-inflammatory discharge, multi-vessels, severe reaction on acetic acid, rough epithelial relief with micropapillary needled like outgrowths of the connective tissue). It was demonstrated the expediency of express diagnostics (Tzanck sample, Schiller test). There were specific for HSV cytologic manifestations: increased sizes of nuclei in epithelial cells, form of watch-glass, multinucleated cells, muddy unstructured chromatin.
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