MATHEMATICAL COMBINATORICS (INTERNATIONAL BOOK SERIES), Volume 1 / 2016

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The Mathematical Combinatorics (International Book Series) is a fully refereed international book series with ISBN number on each issue, sponsored by the MADIS of Chinese Academy of Sciences and published in USA quarterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences.

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<ul><li><p>ISBN 978-1-59973-464-4</p><p>VOLUME 1, 2016</p><p>MATHEMATICAL COMBINATORICS</p><p>(INTERNATIONAL BOOK SERIES)</p><p>Edited By Linfan MAO</p><p>EDITED BY</p><p>THE MADIS OF CHINESE ACADEMY OF SCIENCES AND</p><p>ACADEMY OF MATHEMATICAL COMBINATORICS &amp; APPLICATIONS, USA</p><p>March, 2016</p></li><li><p>Vol.1, 2016 ISBN 978-1-59973-464-4</p><p>MATHEMATICAL COMBINATORICS</p><p>(INTERNATIONAL BOOK SERIES)</p><p>Edited By Linfan MAO</p><p>(www.mathcombin.com)</p><p>Edited By</p><p>The Madis of Chinese Academy of Sciences and</p><p>Academy of Mathematical Combinatorics &amp; Applications, USA</p><p>March, 2016</p></li><li><p>Aims and Scope: The Mathematical Combinatorics (International Book Series) is</p><p>a fully refereed international book series with ISBN number on each issue, sponsored by the</p><p>MADIS of Chinese Academy of Sciences and published in USA quarterly comprising 100-150</p><p>pages approx. per volume, which publishes original research papers and survey articles in all</p><p>aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics,</p><p>non-euclidean geometry and topology and their applications to other sciences. Topics in detail</p><p>to be covered are:</p><p>Smarandache multi-spaces with applications to other sciences, such as those of algebraic</p><p>multi-systems, multi-metric spaces, , etc.. Smarandache geometries;Topological graphs; Algebraic graphs; Random graphs; Combinatorial maps; Graph and</p><p>map enumeration; Combinatorial designs; Combinatorial enumeration;</p><p>Differential Geometry; Geometry on manifolds; Low Dimensional Topology; Differential</p><p>Topology; Topology of Manifolds; Geometrical aspects of Mathematical Physics and Relations</p><p>with Manifold Topology;</p><p>Applications of Smarandache multi-spaces to theoretical physics; Applications of Combi-</p><p>natorics to mathematics and theoretical physics; Mathematical theory on gravitational fields;</p><p>Mathematical theory on parallel universes; Other applications of Smarandache multi-space and</p><p>combinatorics.</p><p>Generally, papers on mathematics with its applications not including in above topics are</p><p>also welcome.</p><p>It is also available from the below international databases:</p><p>Serials Group/Editorial Department of EBSCO Publishing</p><p>10 Estes St. Ipswich, MA 01938-2106, USA</p><p>Tel.: (978) 356-6500, Ext. 2262 Fax: (978) 356-9371</p><p>http://www.ebsco.com/home/printsubs/priceproj.asp</p><p>and</p><p>Gale Directory of Publications and Broadcast Media, Gale, a part of Cengage Learning</p><p>27500 Drake Rd. Farmington Hills, MI 48331-3535, USA</p><p>Tel.: (248) 699-4253, ext. 1326; 1-800-347-GALE Fax: (248) 699-8075</p><p>http://www.gale.com</p><p>Indexing and Reviews: Mathematical Reviews (USA), Zentralblatt Math (Germany), Refer-</p><p>ativnyi Zhurnal (Russia), Mathematika (Russia), Directory of Open Access (DoAJ), Interna-</p><p>tional Statistical Institute (ISI), International Scientific Indexing (ISI, impact factor 1.659),</p><p>Institute for Scientific Information (PA, USA), Library of Congress Subject Headings (USA).</p><p>Subscription A subscription can be ordered by an email directly to</p><p>Linfan Mao</p><p>The Editor-in-Chief of International Journal of Mathematical Combinatorics</p><p>Chinese Academy of Mathematics and System Science</p><p>Beijing, 100190, P.R.China</p><p>Email: maolinfan@163.com</p><p>Price: US$48.00</p></li><li><p>Editorial Board (4th)</p><p>Editor-in-Chief</p><p>Linfan MAO</p><p>Chinese Academy of Mathematics and System</p><p>Science, P.R.China</p><p>and</p><p>Academy of Mathematical Combinatorics &amp;</p><p>Applications, USA</p><p>Email: maolinfan@163.com</p><p>Deputy Editor-in-Chief</p><p>Guohua Song</p><p>Beijing University of Civil Engineering and</p><p>Architecture, P.R.China</p><p>Email: songguohua@bucea.edu.cn</p><p>Editors</p><p>Arindam Bhattacharyya</p><p>Jadavpur University, India</p><p>Email: bhattachar1968@yahoo.co.in</p><p>Said Broumi</p><p>Hassan II University Mohammedia</p><p>Hay El Baraka Ben Msik Casablanca</p><p>B.P.7951 Morocco</p><p>Junliang Cai</p><p>Beijing Normal University, P.R.China</p><p>Email: caijunliang@bnu.edu.cn</p><p>Yanxun Chang</p><p>Beijing Jiaotong University, P.R.China</p><p>Email: yxchang@center.njtu.edu.cn</p><p>Jingan Cui</p><p>Beijing University of Civil Engineering and</p><p>Architecture, P.R.China</p><p>Email: cuijingan@bucea.edu.cn</p><p>Shaofei Du</p><p>Capital Normal University, P.R.China</p><p>Email: dushf@mail.cnu.edu.cn</p><p>Xiaodong Hu</p><p>Chinese Academy of Mathematics and System</p><p>Science, P.R.China</p><p>Email: xdhu@amss.ac.cn</p><p>Yuanqiu Huang</p><p>Hunan Normal University, P.R.China</p><p>Email: hyqq@public.cs.hn.cn</p><p>H.Iseri</p><p>Mansfield University, USA</p><p>Email: hiseri@mnsfld.edu</p><p>Xueliang Li</p><p>Nankai University, P.R.China</p><p>Email: lxl@nankai.edu.cn</p><p>Guodong Liu</p><p>Huizhou University</p><p>Email: lgd@hzu.edu.cn</p><p>W.B.Vasantha Kandasamy</p><p>Indian Institute of Technology, India</p><p>Email: vasantha@iitm.ac.in</p><p>Ion Patrascu</p><p>Fratii Buzesti National College</p><p>Craiova Romania</p><p>Han Ren</p><p>East China Normal University, P.R.China</p><p>Email: hren@math.ecnu.edu.cn</p><p>Ovidiu-Ilie Sandru</p><p>Politechnica University of Bucharest</p><p>Romania</p></li><li><p>ii International Journal of Mathematical Combinatorics</p><p>Mingyao Xu</p><p>Peking University, P.R.China</p><p>Email: xumy@math.pku.edu.cn</p><p>Guiying Yan</p><p>Chinese Academy of Mathematics and System</p><p>Science, P.R.China</p><p>Email: yanguiying@yahoo.com</p><p>Y. Zhang</p><p>Department of Computer Science</p><p>Georgia State University, Atlanta, USA</p><p>Famous Words:</p><p>There is no royal road to science, and only those who do not dread the fatigu-</p><p>ing climb of gaining its numinous summits.</p><p>By Karl Marx, a German revolutionary .</p></li><li><p>Math.Combin.Book Ser. Vol.1(2016), 1-7</p><p>NC Smarandache Curve ofBertrand Curves Pair According to Frenet Frame</p><p>Suleyman Senyurt , Abdussamet Calskan and Unzile Celik</p><p>(Faculty of Arts and Sciences, Department of Mathematics, Ordu University, Ordu, Turkey)</p><p>E-mail: senyurtsuleyman@hotmail.com, abdussamet65@gmail.com, unzile.celik@hotmail.com</p><p>Abstract: In this paper, let (, ) be Bertrand curve pair, when the unit Darboux vector</p><p>of the curve are taken as the position vectors, the curvature and the torsion of Smaran-</p><p>dache curve are calculated. These values are expressed depending upon the curve. Besides,</p><p>we illustrate example of our main results.</p><p>Key Words: Bertrand curves pair, Smarandache curves, Frenet invariants, Darboux vec-</p><p>tor.</p><p>AMS(2010): 53A04.</p><p>1. Introduction</p><p>It is well known that many studies related to the differential geometry of curves have been</p><p>made. Especially, by establishing relations between the Frenet Frames in mutual points of two</p><p>curves several theories have been obtained. The best known of the Bertrand curves discovered</p><p>by J. Bertrand in 1850 are one of the important and interesting topics of classical special curve</p><p>theory. A Bertrand curve is defined as a special curve which shares its principal normals with</p><p>another special curve, called Bertrand mate or Bertrand curve Partner. If = + N, = const., then (, ) are called Bertrand curves pair. If and Bertrand curves pair,then T, T = cos = constant, [9], [10]. The definition of n-dimensional Bertrand curves inLorentzian space is given by comparing a well-known Bertrand pair of curves in n- dimensional</p><p>Euclidean space. It shown that the distance between corresponding of Bertrand pair of curves</p><p>and the angle between the tangent vector fields of these points are constant. Moreover Schell</p><p>and Mannheim theorems are given in the Lorentzian space, [7]. The Bertrand curves are the</p><p>Inclined curve pairs. On the other hand, it gave the notion of Bertrand Representation and</p><p>found that the Bertrand Representation is spherical, [8]. Some characterizations for general</p><p>helices in space forms were given, [11].</p><p>A regular curve in Minkowski space-time, whose position vector is composed by Frenet</p><p>frame vectors on another regular curve, is called a Smarandache curve [14]. Special Smarandache</p><p>curves have been studied by some authors. Melih Turgut and Suha Ylmaz studied a special</p><p>case of such curves and called it Smarandache TB2 curves in the space E41 ([14]). Ahmad T.Ali</p><p>1Received August 9, 2015, Accepted February 2, 2016.</p></li><li><p>2 Suleyman Senyurt , Abdussamet Calskan and Unzile Celik</p><p>studied some special Smarandache curves in the Euclidean space. He studied Frenet-Serret</p><p>invariants of a special case, [1]. Senyurt and Calskan investigated special Smarandache curves</p><p>in terms of Sabban frame of spherical indicatrix curves and they gave some characterization</p><p>of Smarandache curves, [4]Ozcan Bektas and Salim Yuce studied some special Smarandache</p><p>curves according to Darboux Frame in E3, [3]. Kemal Tas.kopru andMurat Tosun studied special</p><p>Smarandache curves according to Sabban frame on S2 ([2]). They defined NC-Smarandache</p><p>curve, then they calculated the curvature and torsion of NB and TNB- Smarandache curves</p><p>together with NC-Smarandache curve, [12]. It studied that the special Smarandache curve in</p><p>terms of Sabban frame of Fixed Pole curve and they gave some characterization of Smarandache</p><p>curves, [12]. When the unit Darboux vector of the partner curve of Mannheim curve were taken</p><p>as the position vectors, the curvature and the torsion of Smarandache curve were calculated.</p><p>These values were expressed depending upon the Mannheim curve, [6].</p><p>In this paper, special Smarandache curve belonging to curve such as NC drawn byFrenet frame are defined and some related results are given.</p><p>2. Preliminaries</p><p>The Euclidean 3-space E3 be inner product given by</p><p> , = x21 + x32 + x23</p><p>where (x1, x2, x3) E3. Let : I E3 be a unit speed curve denote by {T,N,B} themoving Frenet frame . For an arbitrary curve E3, with first and second curvature, and respectively, the Frenet formulae is given by [9], [10].</p><p>T = N</p><p>N = T + BB = N.</p><p>(2.1)</p><p>Figure 1 Darboux vector</p><p>For any unit speed curve : I E3, the vector W is called Darboux vector defined by</p><p>W = T + B. (2.2)</p></li><li><p>NC Smarandache Curve of Bertrand Curves Pair According to Frenet Frame 3</p><p>If we consider the normalization of the Darboux, we have</p><p>sin =</p><p>W , cos =</p><p>W (2.3)</p><p>and</p><p>C = sinT + cosB, (2.4)</p><p>where (W,B) = .</p><p>Definition 2.1([9]) Let : I E3 and : I E3 be the C2 class differentiable unitspeed two curves and let {T (s), N(s), B(s)} and {T (s), N(s), B(s)} be the Frenet frames ofthe curves and , respectively. If the principal normal vector N of the curve is linearlydependent on the principal vector N of the curve , then the pair (, ) is said to be Bertrandcurves pair.</p><p>The relations between the Frenet frames {T (s), N(s), B(s)} and {T (s), N(s), B(s)} areas follows: </p><p>T = cos T + sin B</p><p>N = N</p><p>B = sin T + cos B.(2.5)</p><p>where (T, T ) = </p><p>Theorem 2.2([9], [10]) The distance between corresponding points of the Bertrand curves pair</p><p>in E3 is constant.</p><p>Theorem 2.3([10]) Let (, ) be a Bertrand curves pair in E3. For the curvatures and thetorsions of the Bertrand curves pair (, ) we have</p><p> = sin2 (1 ) , = constant</p><p> =sin2 </p><p>2</p><p>(2.6)</p><p>Theorem 2.4([9]) Let (, ) be a Bertrand curves pair in E3. For the curvatures and thetorsions of the Bertrand curves pair (, ) we have</p><p> ds</p><p>ds= cos sin ,</p><p> ds</p><p>ds= sin + cos .</p><p>(2.7)</p><p>By using equation (2.2), we can write Darboux vector belonging to Bertrand mate .</p><p>W = T + B. (2.8)</p></li><li><p>4 Suleyman Senyurt , Abdussamet Calskan and Unzile Celik</p><p>If we consider the normalization of the Darboux vector, we have</p><p>C = sinT + cosB. (2.9)</p><p>From the equation (2.3) and (2.7), we can write</p><p>sin =</p><p>W = sin + cos </p><p>W = sin(+ ), (2.10)</p><p>cos =</p><p>W = cos sin </p><p>W = cos(+ ),</p><p>where W =2 + 2 = W and (W , B) = . By the using (2.5) and (2.10),</p><p>the final version of the equation (2.9) is as follows:</p><p>C = sinT + cosB. (2.11)</p><p>3. NC Smarandache Curve of Bertrand Curves Pair According to Frenet Frame</p><p>Let (, ) be a Bertrand curves pair in E3 and {T , N, B} be the Frenet frame of the curve at (s). In this case, NC - Smarandache curve can be defined by</p><p>(s) =12(N + C). (3.1)</p><p>Solving the above equation by substitution of N and C from (2.5) and (2.11), we obtain</p><p>(s) =sinT +N + cosB</p><p>2 (3.2)</p><p>The derivative of this equation with respect to s is as follows,</p><p> = Tdsds</p><p>=(+ cos)T + ( sin)B</p><p>2(3.3)</p><p>and by substitution, we get</p><p>T =(+ cos)T + ( sin)B</p><p>W2 2W+ 2, (3.4)</p><p>where</p><p>dsds</p><p>=</p><p>W2 2W+ 2</p><p>2 (3.5)</p><p>In order to determine the first curvature and the principal normal of the curve (s), we</p></li><li><p>NC Smarandache Curve of Bertrand Curves Pair According to Frenet Frame 5</p><p>formalize</p><p>T (s) =</p><p>2[(1 cos + 3 sin )T + 2N + (1 sin + 3 cos )B</p><p>][W2 2W+ 2]2 , (3.6)where</p><p>1 =( cos + sin + cos(+ ))(W2 2W+ 2) ( cos + sin + cos(+ )</p><p>)(WW W W + )2 =</p><p>( W2 + W)(W2 2W+ 2)3 =</p><p>( sin + cos sin(+ ))(W2 2W+ 2) ( sin + cos sin(+ ))(WW W W + )</p><p>The first curvature is</p><p> = T , =2(12 + 22 + 32)[W2 2W+ 2]2 </p><p>The principal normal vector field and the binormal vector field are respectively given by</p><p>N =</p><p>[(1 cos + 3 sin )T + 2N + (1 sin + 3 cos )B</p><p>]12 + 22 + 32</p><p>, (3.7)</p><p>B =</p><p>2[ 2 sin cos + (sin2 cos2 ) + sin]T</p><p>+1[ sin + cos sin(+ )]N + 2[2 sin cos </p><p>+(sin2 cos2 ) + cos]B(W2 2W+ 2)(12 + 22 + 32) (3.8)The torsion is then given by</p><p> =det(, , ) 2 ,</p><p> =</p><p>2( + + )</p><p>2 + 2 + 2</p><p>where </p><p> =( cos( + ) cos + sin ) + ( cos sin )W2( cos sin )W</p><p> = ( cos sin )( cos(+ ) cos + sin ) + (W2+W) ( sin + cos )( sin + cos sin(+ ))</p><p> = ( sin cos )W2 + ( sin + cos )W+ ( sin + cos sin(+ )),</p></li><li><p>6 Suleyman Senyurt , Abdussamet Calskan and Unzile Celik</p><p> = ( W2 + W)( sin + cos sin(+ )) = </p><p>[( cos(+ ) cos + sin )( sin + cos sin(+ ))</p><p>+( cos(+ ) cos + sin )( sin + cos sin(+ ))]</p><p> =( cos(+ ) cos + sin )( W2 + W).</p><p>Example 3.1 Let us consider the unit speed curve and curve:</p><p>(s) =12</p><p>( cos s, sin s, s) and (s) = 12(cos s, sin s, s)</p><p>The Frenet invariants of the curve, (s) are given as following:T (s) =</p><p>12( sin s, cos s, 1), N(s) = ( cos s, sin s, 0)</p><p>B(s) =12(sin s, cos s, 1), C(s) = (0, 0, 1)</p><p>(s) =12, (s) =</p><p>12</p><p>In terms of definitions, we obtain special Smarandache curve, see Figure 1.</p><p>Figure 2 NC-Smarandache Curve</p><p>References</p><p>[1] Ali A. T., Special Smarandache curves in the Euclidean space, International J.Math.Combin.,</p><p>2(2010), 30-36.</p><p>[2] Bektas O. and Yuce S., Special Smarandache curves according to Darboux frame in Eu-</p><p>clidean 3-space, Romanian Journal of Mathematics and Computer Science, 3(1)(2013),</p><p>48-59.</p><p>[3] Calskan A. and Senyurt S., Smarandache curves in terms of Sabban frame of spherical</p></li><li><p>NC Smarandache Curve of Bertrand Curves Pair According to Frenet Frame 7</p><p>indicatrix curves, Gen. Math. Notes, 31(2)(2015), 1-15.</p><p>[4] Calskan A. and Senyurt S., Smarandache curves in terms of Sabban frame of fixed pole</p><p>curve, Boletim da Sociedade parananse de Mathemtica, 34(2)(2016), 5362.</p><p>[5] Calskan A. and Senyurt S., NC- Smarandache curves of Mannheim curve couple ac-cording to Frenet frame, International J.Math. Combin., 1(2015), 113.</p><p>[6] Ekmekci N. and Ilarslan K., On Bertrand curves and their characterization, Differential</p><p>Geometry-Dynamical Systems, 3(2)(2001), 17-24.</p><p>[7] Gorgulu A. and Ozdamar E., A generalizations of the Bertrand curves as general inclined</p><p>curve in En, Commun. Fac. Sci. Uni. Ankara, Series A1, 35(1986), 53-60.</p><p>[8] Hacsalihoglu H.H., Differential Geometry, Inonu University, Malatya, Mat. No.7, 1983</p><p>[9] Kasap E. and Kuruoglu N., Integral invariants of the pairs of the Bertrand ruled surface,</p><p>Bulletin of Pure and Applied Sciences, 21(2002), 37-44.</p><p>[10] Sabuncuoglu A., Differential Geometry, Nobel Publications, Ankara, 2006</p><p>[11] Senol A., Zplar E. and Yayl Y., General helices and Bertrand curves in Riemannian space</p><p>form, Mathematica Aeterna, 2(2)(2012), 155-161.</p><p>[12] Senyurt S. and Sivas S., An application of Sma...</p></li></ul>