MATHEMATICAL COMBINATORICS (INTERNATIONAL BOOK SERIES), Volume 1 / 2015

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Papers on Antidegree Equitable Sets in a Graph, One Modulo N Gracefulness of Some Arbitrary Supersubdivision and Removal Graphs, A New Approach to Natural Lift Curves of the Spherical Indicatrices of Timelike Bertrand Mate, On Signed Graphs Whose Two Path Signed Graphs are Switching Equivalent to Their Jump Signed Graphs, and other topics.Contributors: C. Adiga, K.N.S. Krishna, Mathew Varkey T.K, Sunoj B.S, V. Ramachandran, C. Sekar, W. Barbara, P. Sugirtha, R. Vasuki, J. Venkateswari, Yizhi Chen, Siyan Li, Wei Chen, and others.

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<ul><li><p>ISBN 978-1-59973-337-1</p><p>VOLUME 1, 2015</p><p>MATHEMATICAL COMBINATORICS</p><p>(INTERNATIONAL BOOK SERIES)</p><p>Edited By Linfan MAO</p><p>THE MADIS OF CHINESE ACADEMY OF SCIENCES AND</p><p>ACADEMY OF MATHEMATICAL COMBINATORICS &amp; APPLICATIONS</p><p>March, 2015</p></li><li><p>Vol.1, 2015 ISBN 978-1-59973-337-1</p><p>MATHEMATICAL COMBINATORICS</p><p>(INTERNATIONAL BOOK SERIES)</p><p>Edited By Linfan MAO</p><p>The Madis of Chinese Academy of Sciences and</p><p>Academy of Mathematical Combinatorics &amp; Applications</p><p>March, 2015</p></li><li><p>Aims and Scope: The Mathematical Combinatorics (International Book Series) is a</p><p>fully refereed international book series, quarterly comprising 100-150 pages approx. per volume,</p><p>which publishes original research papers and survey articles in all aspects of Smarandache</p><p>multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry</p><p>and topology and their applications to other sciences. Topics in detail 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 1.416),</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 (3nd)</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>S.Bhattacharya</p><p>Deakin University</p><p>Geelong Campus at Waurn Ponds</p><p>Australia</p><p>Email: Sukanto.Bhattacharya@Deakin.edu.au</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>Baizhou He</p><p>Beijing University of Civil Engineering and</p><p>Architecture, P.R.China</p><p>Email: hebaizhou@bucea.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>Nothing in life is to be feared. It is only to be understood.</p><p>By Marie Curie, a Polish and naturalized-French physicist and chemist.</p></li><li><p>Math.Combin.Book Ser. Vol.1(2015), 1-13</p><p>NC Smarandache Curves of Mannheim Curve CoupleAccording to Frenet Frame</p><p>Suleyman SENYURT and Abdussamet CALISKAN</p><p>(Faculty of Arts and Sciences, Department of Mathematics, Ordu University, 52100, Ordu/Turkey)</p><p>E-mail: senyurtsuleyman@hotmail.com</p><p>Abstract: In this paper, when the unit Darboux vector of the partner curve of Mannheim</p><p>curve are taken as the position vectors, the curvature and the torsion of Smarandache curve</p><p>are calculated. These values are expressed depending upon the Mannheim curve. Besides,</p><p>we illustrate example of our main results.</p><p>Key Words: Mannheim curve, Mannheim partner curve, Smarandache Curves, Frenet</p><p>invariants.</p><p>AMS(2010): 53A04</p><p>1. Introduction</p><p>A regular curve in Minkowski space-time, whose position vector is composed by Frenet frame</p><p>vectors on another regular curve, is called a Smarandache curve ([12]). Special Smarandache</p><p>curves have been studied by some authors .</p><p>Melih Turgut and Suha Ylmaz studied a special case of such curves and called it Smaran-</p><p>dache TB2 curves in the spaceE41 ([12]). Ahmad T.Ali studied some special Smarandache curves</p><p>in the Euclidean space. He studied Frenet-Serret invariants of a special case ([1]). Muhammed</p><p>Cetin , Ylmaz Tuncer and Kemal Karacan investigated special Smarandache curves according</p><p>to Bishop frame in Euclidean 3-Space and they gave some differential goematric properties of</p><p>Smarandache curves, also they found the centers of the osculating spheres and curvature spheres</p><p>of Smarandache curves ([5]). Senyurt and Calskan investigated special Smarandache curves in</p><p>terms of Sabban frame of spherical indicatrix curves and they gave some characterization of</p><p>Smarandache curves ([4]). Ozcan Bektas and Salim Yuce studied some special Smarandache</p><p>curves according to Darboux Frame in E3 ([2]). Nurten Bayrak, Ozcan Bektas and Salim Yuce</p><p>studied some special Smarandache curves in E31 [3]. Kemal Tas.kopru, Murat Tosun studied</p><p>special Smarandache curves according to Sabban frame on S2 ([11]).</p><p>In this paper, special Smarandache curve belonging to Mannheim partner curve suchas NC drawn by Frenet frame are defined and some related results are given.</p><p>1Received September 8, 2014, Accepted February 12, 2015.</p></li><li><p>2 Suleyman SENYURT and Abdussamet CALISKAN</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 ([6], [9])</p><p>T = N</p><p>N = T + BB = N.</p><p>(2.1)</p><p>For any unit speed : I E3, the vector W is called Darboux vector defined by</p><p>W = (s)T (s) + (s) +B(s).</p><p>If consider the normalization of the Darboux C =1</p><p>WW , we have</p><p>cos =(s)</p><p>W , sin =(s)</p><p>W ,</p><p>C = sinT (s) + cosB(s) (2.2)</p><p>where (W,B) = . 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 binormal vector B of the curve , then () is called a Mannheim curve and() a Mannheim partner curve of (). The pair (, ) is said to be Mannheim pair ([7], [8]).The relations between the Frenet frames {T (s), N(s), B(s)} and {T (s), N(s), B(s)} are asfollows: </p><p>T = cos T sin BN = sin T + cos B</p><p>B = N</p><p>(2.3)</p><p>cos =</p><p>dsds</p><p>sin = dsds</p><p>.(2.4)</p><p>where (T, T ) = ([8]).</p><p>Theorem 2.1([7]) The distance between corresponding points of the Mannheim partner curves</p><p>in E3 is constant.</p></li><li><p>NC Smarandache Curves of Mannheim Curve Couple According to Frenet Frame 3</p><p>Theorem 2.2 Let (, ) be a Mannheim pair curves in E3. For the curvatures and the torsionsof the Mannheim curve pair (, ) we have,</p><p> = sin dsds</p><p> = cos dsds</p><p>(2.5)</p><p>and =</p><p>d</p><p>ds = 2 + 2</p><p> = ( sin cos )dsds</p><p>(2.6)</p><p>Theorem 2.3 Let (, ) be a Mannheim pair curves in E3. For the torsions of theMannheim partner curve we have</p><p> =</p><p>Theorem 2.4([10]) Let (, ) be a Mannheim pair curves in E3. For the vector C is thedirection of the Mannheim partner curve we have</p><p>C =1</p><p>1 +( W</p><p>)2C +</p><p>W1 +</p><p>( W</p><p>)2N (2.7)</p><p>where the vector C is the direction of the Darboux vector W of the Mannheim curve .</p><p>3. NC Smarandache Curves of Mannheim Curve Couple According toFrenet Frame</p><p>Let (, ) be a Mannheim pair curves in E3 and {T NB} be the Frenet frame of theMannheim partner curve at (s). In this case, NC - Smarandache curve can be definedby</p><p>1(s) =12(N + C). (3.1)</p><p>Solving the above equation by substitution of N and C from (2.3) and (2.7), we obtain</p><p>1(s) =</p><p>(cos W+ sin </p><p>2 + W2)T + N + ( cos 2 + W2 sin W)B</p><p>2 + W2.</p><p>(3.2)</p></li><li><p>4 Suleyman SENYURT and Abdussamet CALISKAN</p><p>The derivative of this equation with respect to s is as follows,</p><p>T1(s) =</p><p>[(W</p><p>2+W2</p><p>)cos cos W</p><p>]T +</p><p>[ </p><p>(W</p><p>2+W2</p><p>) W</p><p>]N</p><p>[(W</p><p>2+W2</p><p>)2+W2</p><p>]2+ (</p><p>2+W2)W</p><p>[</p><p>W 2(</p><p>W2+W2</p><p>)1</p><p>]</p><p>+</p><p>[ sin W </p><p>(W</p><p>2+W2</p><p>)sin </p><p>]B [( W</p><p>2+W2</p><p>)2+W2</p><p>]2+ (</p><p>2+W2)W</p><p>[</p><p>W 2(</p><p>W2+W2</p><p>)1</p><p>](3.3)</p><p>In order to determine the first curvature and the principal normal of the curve 1(s), we</p><p>formalize</p><p>T 1(s) =</p><p>2[(r1 cos + r2 sin )T + r3N + (r1 sin + r2 cos )B</p><p>]([(</p><p>W2+W2</p><p>)2+W2</p><p>]2+ (</p><p>2+W2)W</p><p>[</p><p>W 2(</p><p>W2+W2</p><p>)1</p><p>])2</p><p>where</p><p>r1 = 2( </p><p>)2([( W2 + W2</p><p>)2 + W2</p><p>])( 2 + W2</p><p>)</p><p>( W</p><p>)[( W2 + W2</p><p>)2 + W2</p><p>][( W2 + W2</p><p>)2 + W2</p><p>]( </p><p>2 + W2)2( </p><p>)[( W2 + W2</p><p>)2 + W2</p><p>]([( W2 + W2</p><p>)2 + W2</p><p>])( W2 + W2</p><p>)( 2 + W2</p><p>)[( W</p><p>2 + W2)</p><p>2 + W2</p><p>]4( W2 + W2</p><p>)( </p><p>)2( W</p><p>)( </p><p>)2[( W2 + W2</p><p>)</p></li><li><p>NC Smarandache Curves of Mannheim Curve Couple According to Frenet Frame 5</p><p>2 + W2</p><p>]2( W2 + W2</p><p>)+ 2</p><p>( W</p><p>)[( W2 + W2</p><p>)2 + W2</p><p>]3</p><p>( W2 + W2</p><p>)( 2 + W2</p><p>)+ 2</p><p>( </p><p>)[( W2 + W2</p><p>)2 + W2</p><p>]3</p><p>( W2 + W2</p><p>)2( W</p><p>)2[( W2 + W2</p><p>)2 + W2</p><p>]2( W2 + W2</p><p>)</p><p>2( W</p><p>)[( W2 + W2</p><p>)2 + W2</p><p>]( 2 + W2</p><p>) 2</p><p>( W</p><p>)( </p><p>)[( W2 + W2</p><p>)2 + W2</p><p>]( W2 + W2</p><p>) </p><p>( </p><p>)[( W</p><p>2 + W2)2 + W2</p><p>]( 2 + W2</p><p>)+( W</p><p>)[( W2 + W2</p><p>)2 + W2</p><p>]( 2 + W2</p><p>)2( W</p><p>)[( W2 + W2</p><p>)2 + W2</p><p>]2</p><p>+( </p><p>)[( W2 + W2</p><p>)2 + W2</p><p>]2( W2 + W2</p><p>)( 2 + W2</p><p>)( W</p><p>)[( W2 + W2</p><p>)2 + W2</p><p>]([( W2 + W2</p><p>)2 + W2</p><p>])</p><p>+( W</p><p>)( </p><p>)( </p><p>)( W</p><p>)( </p><p>)([( W2 + W2</p><p>)2 + W2</p><p>])</p><p>( W2 + W2</p><p>)[( W</p><p>2 + W2)2 + W2</p><p>]( </p><p>)( W</p><p>)( W2 + W2</p><p>),</p><p>r2 =( W</p><p>)[( W2 + W2</p><p>)2 + W2</p><p>]3( 2 + W2</p><p>)+ 3</p><p>( W</p><p>)3[( W</p><p>2 + W2)2 + W2</p><p>]( 2 + W2</p><p>)+ 3</p><p>( </p><p>)2( W</p><p>)</p></li><li><p>6 Suleyman SENYURT and Abdussamet CALISKAN</p><p>[( W2 + W2</p><p>)2 + W2</p><p>]( 2 + W2</p><p>) 2</p><p>( W</p><p>)2[( W</p><p>2 + W2)2 + W2</p><p>]2( 2 + W2</p><p>)2( W</p><p>)2[( W</p><p>2 + W2)2 + W2</p><p>]2 ( </p><p>)2[( W2 + W2</p><p>)2 + W2</p><p>]2</p><p>+3( </p><p>)3[( W2 + W2</p><p>)2 + W2</p><p>]( W2 + W2</p><p>)+( </p><p>)[( W</p><p>2 + W2)2 + W2</p><p>]3( W2 + W2</p><p>) 2( </p><p>)2[( W2 + W2</p><p>)2 + W2</p><p>]2( W2 + W2</p><p>)2 4</p><p>( W</p><p>)( </p><p>)[( W2 + W2</p><p>)2 + W2</p><p>]2( W2 + W2</p><p>)( 2 + W2</p><p>)( W</p><p>)4 2</p><p>( W</p><p>)2( </p><p>)2+ 3</p><p>( W</p><p>)2( </p><p>)[( W2 + W2</p><p>)2 + W2</p><p>]( W2 + W2</p><p>)</p><p>r3 = 2( </p><p>)[( W2 + W2</p><p>)2 + W2</p><p>]2+( W</p><p>)2( </p><p>) 2( W</p><p>)</p><p>( </p><p>)[( W2 + W2</p><p>)2 + W2</p><p>]( 2 + W2</p><p>)( W</p><p>)2[( W</p><p>2 + W2)2 + W2</p><p>]( W2 + W2</p><p>)+( W</p><p>)[( W2 + W2</p><p>)2 + W2</p><p>][( W2 + W2</p><p>)2 + W2</p><p>]( W2 + W2</p><p>)( 2 + W2</p><p>)</p><p>+( </p><p>)[( W2 + W2</p><p>)2 + W2</p><p>][( W2 + W2</p><p>)2 + W2</p><p>]</p></li><li><p>NC Smarandache Curves of Mannheim Curve Couple According to Frenet Frame 7</p><p>( W</p><p>2 + W2)2[( W</p><p>2 + W2)2 + W2</p><p>]4( 2 + W2</p><p>)</p><p>( W</p><p>)2[( W2 + W2</p><p>)2 + W2</p><p>]2( 2 + W2</p><p>) ( </p><p>)[( W</p><p>2 + W2)2 + W2</p><p>]2( W2 + W2</p><p>)2 ( </p><p>)2[( W2 + W2</p><p>)2 + W2</p><p>]2( 2 + W2</p><p>)+ 2</p><p>( W</p><p>)[( W2 + W2</p><p>)2 + W2</p><p>]3( </p><p>2 + W2)2</p><p>+ 2( </p><p>)[( W2 + W2</p><p>)2 + W2</p><p>]3( W2 + W2</p><p>)( </p><p>2 + W2) ( </p><p>)[( W2 + W2</p><p>)2 + W2</p><p>][( W2 + W2</p><p>)2 + W2</p><p>] ( </p><p>)( W</p><p>)( W</p><p>)+( </p><p>)( W</p><p>)[( W2 + W2</p><p>)2 + W2</p><p>]( 2 + W2</p><p>)+( W</p><p>)( </p><p>)[( W2 + W2</p><p>)2 + W2</p><p>]( 2 + W2</p><p>)+( W</p><p>)( W</p><p>)[( W2 + W2</p><p>)2 + W2</p><p>]( W2 + W2</p><p>)( W</p><p>)[( W2 + W2</p><p>)2 + W2</p><p>]2( W</p><p>2 + W2)( </p><p>2 + W2)</p><p>The first curvature is</p><p>1 =</p><p>2(r12 + r22 + r32)([(</p><p>W2+W2</p><p>)2+W2</p><p>]2+ (</p><p>2+W2)W</p><p>[</p><p>W 2(</p><p>W2+W2</p><p>)1</p><p>])2 </p></li><li><p>8 Suleyman SENYURT and Abdussamet CALISKAN</p><p>The principal normal vector field and the binormal vector field are respectively given by</p><p>N1 =(r1 cos + r2 sin )T + r3N + (r1 sin + r2 cos )B</p><p>r12 + r22 + r32, (3.4)</p><p>B1(s) =14T +</p><p>24N +</p><p>34B, (3.5)</p><p>where</p><p>1 = r2 cos (</p><p>W2+W2</p><p>) W r2 cos </p><p>[r1</p><p> r1</p><p>(W</p><p>2+W2</p><p>) W</p><p>r3(</p><p>W2+W2</p><p>)+ r3</p><p>(</p><p>W)]</p><p>sin </p><p>2 = r1</p><p>[(W</p><p>2+W2</p><p>) W</p><p>]</p><p>3 = r2 sin </p><p>[(W</p><p>2+W2</p><p>) W </p><p>]+</p><p>[r1</p><p> r1</p><p>(W</p><p>2+W2</p><p>) W</p><p>r3(</p><p>W2+W2</p><p>)+ r3</p><p>W</p><p>]cos </p><p>4 =</p><p>((r1</p><p>2 + r22 + r3</p><p>2)</p><p>[(W</p><p>2+W2</p><p>)2+W2</p><p>]2+ (r1</p><p>2 + r22 + r3</p><p>2)(2+W2)W[</p><p>W 2</p><p>(W</p><p>2+W2</p><p>)1</p><p>]).</p><p>In order to calculate the torsion of the curve 1, we differentiate</p><p>1 =12</p><p>([cos </p><p>([( W2 + W2</p><p>)2 + W2</p><p>] 2 + W2</p><p>[( W</p><p>2 + W2)2 + W2</p><p>]2 W2 + W2</p><p>( W</p><p>))+</p><p>+sin </p><p>(( W</p><p>)[( W2 + W2</p><p>)2 + W2</p><p>]</p><p>2 + W2+( </p><p>)[( W</p><p>2 + W2)2 + W2</p><p>] W2 + W2</p><p>( W</p><p>)2 ( </p><p>)2)]T</p><p>+</p><p>[( </p><p>) [( W2 + W2</p><p>)2 + W2</p><p>] W2 + W2</p></li><li><p>NC Smarandache Curves of Mannheim Curve Couple According to Frenet Frame 9</p><p>[( W</p><p>2 + W2)2 + W2</p><p>]2</p><p>2 + W2</p><p>]N</p><p>[ sin </p><p>([( W2 + W2</p><p>)2 + W2</p><p>] 2 + W2</p><p>[( W</p><p>2 + W2)2 + W2</p><p>]2 W2 + W2</p><p>( W</p><p>))+</p><p>+cos </p><p>(( W</p><p>)[( W2 + W2</p><p>)2 + W2</p><p>]</p><p>2 + W2+( </p><p>)[( W</p><p>2 + W2)2 + W2</p><p>] W2 + W2</p><p>( W</p><p>)2 ( </p><p>)2)]B</p><p>).</p><p>and thus</p><p>1 =(t1 cos + t2 sin + t3)T + t3N + (t2 cos t1 sin + t3)T</p><p>2,</p><p>where</p><p>t1 =</p><p>[( W2 + W2</p><p>)2 + W2</p><p>]</p><p>2 + W2 3</p><p>[( W2 + W2</p><p>)2 + W2</p><p>][( W2 + W2</p><p>)2 + W2</p><p>] W2 + W2</p><p>[( W</p><p>2 + W2)2 + W2</p><p>]3</p><p>2 + W2( W</p><p>)</p><p>( W</p><p>)2[( W2 + W2</p><p>)2 + W2</p><p>]</p><p>2 + W2( W</p><p>)( </p><p>)[( W</p><p>2 + W2)2 + W2</p><p>] W2 + W2</p><p>+( W</p><p>)3+( W</p><p>)( </p><p>)2</p><p>t2 = 2( W</p><p>)[( W2 + W2</p><p>)2 + W2</p><p>]</p><p>2 + W2 2</p><p>( W</p><p>)[( W</p><p>2 + W2)2 + W2</p><p>]2 W2 + W2</p><p> 3( W</p><p>)( W</p><p>)</p></li><li><p>10 Suleyman SENYURT and Abdussamet CALISKAN</p><p>+( W</p><p>)[( W2 + W2</p><p>)2 + W2</p><p>]</p><p>2 + W2+( </p><p>)[( W</p><p>2 + W2)2 + W2</p><p>] W2 + W2</p><p>+ 2( </p><p>)[( W2 + W2</p><p>)2 + W2</p><p>] W2 + W2</p><p>+ 2</p><p>[( W2 + W2</p><p>)2 + W2</p><p>]2</p><p>2 + W2 3( </p><p>)( </p><p>)</p><p>t3 =</p><p>(( W</p><p>)( </p><p>)[( W2 + W2</p><p>)2 + W2</p><p>] 3</p><p>[( W2 + W2</p><p>)2 + W2</p><p>][( W2 + W2</p><p>)2 + W2</p><p>])</p><p>2 + W2+</p><p>(( </p><p>)2[( W</p><p>2 + W2)2 + W2</p><p>][( W</p><p>2 + W2)2 + W2</p><p>]</p><p>+</p><p>[( W2 + W2</p><p>)2 + W2</p><p>]3) W2 + W2</p><p>( W</p><p>)2( </p><p>)</p><p>( </p><p>)3+( </p><p>)The torsion is then given by</p><p>1 =det(1, </p><p>1 , </p><p>1 )</p><p>1 1 2,</p><p>1 =212</p><p>where</p><p>1 = 2t1( </p><p>)2( W2 + W2</p><p>) W</p><p>+ t1</p><p>[( W2 + W2</p><p>)]2 W22</p><p> t1( W</p><p>)2( W</p><p>2 + W2</p><p>) W</p><p>+( W</p><p>)t2( </p><p>)+</p><p>[( W2 + W2</p><p>)2 + W2</p><p>]3</p><p>2</p><p>2 + W2t2</p></li><li><p>NC Smarandache Curves of Mannheim Curve Couple According to Frenet Frame 11</p><p>+</p><p>[( W2 + W2</p><p>)]2t3</p><p>(</p><p>W</p><p>) 2</p><p>( W2 + W2</p><p>)t3</p><p>(</p><p>W</p><p>)2( W</p><p>2 + W2</p><p>)t3( </p><p>)2( W</p><p>2 + W2</p><p>)t2( </p><p>)(</p><p>W</p><p>)t2</p><p>[( W2 + W2</p><p>)2 + W2</p><p>]W</p><p>2 + W2(</p><p>W</p><p>)t2</p><p>[( W2 + W2</p><p>)2 + W2</p><p>]2</p><p>2 + W2+ t2</p><p>[( W2 + W2</p><p>)2 + W2</p><p>]</p><p>2 + W2</p><p>t2</p><p>[( W2 + W2</p><p>)2 + W2</p><p>]2W</p><p>2 + W2+</p><p>[( W2 + W2</p><p>)]2t3</p><p>W</p><p>+ t2</p><p>(</p><p>W</p><p>)( W</p><p>2 + W2</p><p>) W</p><p>(</p><p>W</p><p>)t3</p><p>( W2 + W2</p><p>) W</p><p> t1</p><p>(</p><p>W</p><p>)( W2 + W2</p><p>)+t1</p><p>[( W2 + W2</p><p>)]2+ t1</p><p>(</p><p>W</p><p>)2( </p><p>)+ t2</p><p>[( W2 + W2</p><p>)2 + W2</p><p>]3W2</p><p>2 + W2</p><p>t2(</p><p>W</p><p>)( </p><p>