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.

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

  • ISBN 978-1-59973-464-4

    VOLUME 1, 2016

    MATHEMATICAL COMBINATORICS

    (INTERNATIONAL BOOK SERIES)

    Edited By Linfan MAO

    EDITED BY

    THE MADIS OF CHINESE ACADEMY OF SCIENCES AND

    ACADEMY OF MATHEMATICAL COMBINATORICS & APPLICATIONS, USA

    March, 2016

  • Vol.1, 2016 ISBN 978-1-59973-464-4

    MATHEMATICAL COMBINATORICS

    (INTERNATIONAL BOOK SERIES)

    Edited By Linfan MAO

    (www.mathcombin.com)

    Edited By

    The Madis of Chinese Academy of Sciences and

    Academy of Mathematical Combinatorics & Applications, USA

    March, 2016

  • Aims and Scope: 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. Topics in detail

    to be covered are:

    Smarandache multi-spaces with applications to other sciences, such as those of algebraic

    multi-systems, multi-metric spaces, , etc.. Smarandache geometries;Topological graphs; Algebraic graphs; Random graphs; Combinatorial maps; Graph and

    map enumeration; Combinatorial designs; Combinatorial enumeration;

    Differential Geometry; Geometry on manifolds; Low Dimensional Topology; Differential

    Topology; Topology of Manifolds; Geometrical aspects of Mathematical Physics and Relations

    with Manifold Topology;

    Applications of Smarandache multi-spaces to theoretical physics; Applications of Combi-

    natorics to mathematics and theoretical physics; Mathematical theory on gravitational fields;

    Mathematical theory on parallel universes; Other applications of Smarandache multi-space and

    combinatorics.

    Generally, papers on mathematics with its applications not including in above topics are

    also welcome.

    It is also available from the below international databases:

    Serials Group/Editorial Department of EBSCO Publishing

    10 Estes St. Ipswich, MA 01938-2106, USA

    Tel.: (978) 356-6500, Ext. 2262 Fax: (978) 356-9371

    http://www.ebsco.com/home/printsubs/priceproj.asp

    and

    Gale Directory of Publications and Broadcast Media, Gale, a part of Cengage Learning

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    Tel.: (248) 699-4253, ext. 1326; 1-800-347-GALE Fax: (248) 699-8075

    http://www.gale.com

    Indexing and Reviews: Mathematical Reviews (USA), Zentralblatt Math (Germany), Refer-

    ativnyi Zhurnal (Russia), Mathematika (Russia), Directory of Open Access (DoAJ), Interna-

    tional Statistical Institute (ISI), International Scientific Indexing (ISI, impact factor 1.659),

    Institute for Scientific Information (PA, USA), Library of Congress Subject Headings (USA).

    Subscription A subscription can be ordered by an email directly to

    Linfan Mao

    The Editor-in-Chief of International Journal of Mathematical Combinatorics

    Chinese Academy of Mathematics and System Science

    Beijing, 100190, P.R.China

    Email: maolinfan@163.com

    Price: US$48.00

  • Editorial Board (4th)

    Editor-in-Chief

    Linfan MAO

    Chinese Academy of Mathematics and System

    Science, P.R.China

    and

    Academy of Mathematical Combinatorics &

    Applications, USA

    Email: maolinfan@163.com

    Deputy Editor-in-Chief

    Guohua Song

    Beijing University of Civil Engineering and

    Architecture, P.R.China

    Email: songguohua@bucea.edu.cn

    Editors

    Arindam Bhattacharyya

    Jadavpur University, India

    Email: bhattachar1968@yahoo.co.in

    Said Broumi

    Hassan II University Mohammedia

    Hay El Baraka Ben Msik Casablanca

    B.P.7951 Morocco

    Junliang Cai

    Beijing Normal University, P.R.China

    Email: caijunliang@bnu.edu.cn

    Yanxun Chang

    Beijing Jiaotong University, P.R.China

    Email: yxchang@center.njtu.edu.cn

    Jingan Cui

    Beijing University of Civil Engineering and

    Architecture, P.R.China

    Email: cuijingan@bucea.edu.cn

    Shaofei Du

    Capital Normal University, P.R.China

    Email: dushf@mail.cnu.edu.cn

    Xiaodong Hu

    Chinese Academy of Mathematics and System

    Science, P.R.China

    Email: xdhu@amss.ac.cn

    Yuanqiu Huang

    Hunan Normal University, P.R.China

    Email: hyqq@public.cs.hn.cn

    H.Iseri

    Mansfield University, USA

    Email: hiseri@mnsfld.edu

    Xueliang Li

    Nankai University, P.R.China

    Email: lxl@nankai.edu.cn

    Guodong Liu

    Huizhou University

    Email: lgd@hzu.edu.cn

    W.B.Vasantha Kandasamy

    Indian Institute of Technology, India

    Email: vasantha@iitm.ac.in

    Ion Patrascu

    Fratii Buzesti National College

    Craiova Romania

    Han Ren

    East China Normal University, P.R.China

    Email: hren@math.ecnu.edu.cn

    Ovidiu-Ilie Sandru

    Politechnica University of Bucharest

    Romania

  • ii International Journal of Mathematical Combinatorics

    Mingyao Xu

    Peking University, P.R.China

    Email: xumy@math.pku.edu.cn

    Guiying Yan

    Chinese Academy of Mathematics and System

    Science, P.R.China

    Email: yanguiying@yahoo.com

    Y. Zhang

    Department of Computer Science

    Georgia State University, Atlanta, USA

    Famous Words:

    There is no royal road to science, and only those who do not dread the fatigu-

    ing climb of gaining its numinous summits.

    By Karl Marx, a German revolutionary .

  • Math.Combin.Book Ser. Vol.1(2016), 1-7

    NC Smarandache Curve ofBertrand Curves Pair According to Frenet Frame

    Suleyman Senyurt , Abdussamet Calskan and Unzile Celik

    (Faculty of Arts and Sciences, Department of Mathematics, Ordu University, Ordu, Turkey)

    E-mail: senyurtsuleyman@hotmail.com, abdussamet65@gmail.com, unzile.celik@hotmail.com

    Abstract: In this paper, let (, ) be Bertrand curve pair, when the unit Darboux vector

    of the curve are taken as the position vectors, the curvature and the torsion of Smaran-

    dache curve are calculated. These values are expressed depending upon the curve. Besides,

    we illustrate example of our main results.

    Key Words: Bertrand curves pair, Smarandache curves, Frenet invariants, Darboux vec-

    tor.

    AMS(2010): 53A04.

    1. Introduction

    It is well known that many studies related to the differential geometry of curves have been

    made. Especially, by establishing relations between the Frenet Frames in mutual points of two

    curves several theories have been obtained. The best known of the Bertrand curves discovered

    by J. Bertrand in 1850 are one of the important and interesting topics of classical special curve

    theory. A Bertrand curve is defined as a special curve which shares its principal normals with

    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

    Euclidean space. It shown that the distance between corresponding of Bertrand pair of curves

    and the angle between the tangent vector fields of these points are constant. Moreover Schell

    and Mannheim theorems are given in the Lorentzian space, [7]. The Bertrand curves are the

    Inclined curve pairs. On the other hand, it gave the notion of Bertrand Representation and

    found that the Bertrand Representation is spherical, [8]. Some characterizations for general

    helices in space forms were given, [11].

    A regular curve in Minkowski space-time, whose position vector is composed by Frenet

    frame vectors on another regular curve, is called a Smarandache curve [14]. Special Smarandache

    curves have been studied by some authors. Melih Turgut and Suha Ylmaz studied a special

    case of such curves and called it Smarandache TB2 curves in the space E41 ([14]). Ahmad T.Ali

    1Received August 9, 2015, Accepted February 2, 2016.

  • 2 Suleyman Senyurt , Abdussamet Calskan and Unzile Celik

    studied some special Smarandache curves in the Euclidean space. He studied Frenet-Serret

    invariants of a special case, [1]. Senyurt and Calskan investigated special Smarandache curves

    in terms of Sabban frame of spherical indicatrix curves and they gave some characterization

    of Smarandache curves, [4]Ozcan Bektas and Salim Yuce studied some special Smarandache

    curves according to Darboux Frame in E3, [3]. Kemal Tas.kopru andMurat Tosun studied special

    Smarandache curves according to Sabban frame on S2 ([2]). They defined NC-Smarandache

    curve, then they calculated the curvature and torsion of NB and TNB- Smarandache curves

    together with NC-Smarandache curve, [12]. It studied that the special Smarandache curve in

    terms of Sabban frame of Fixed Pole curve and they gave some characterization of Smarandache

    curves, [12]. When the unit Darboux vector of the partner curve of Mannheim curve were taken

    as the position vectors, the curvature and the torsion of Smarandache curve were calculated.

    These values were expressed depending upon the Mannheim curve, [6].

    In this paper, special Smarandache curve belonging to curve such as NC drawn byFrenet frame are defined and some related results are given.