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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.

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

27500 Drake Rd. Farmington Hills, MI 48331-3535, USA

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.