prev

next

out of 144

View

8Category

## Documents

Embed Size (px)

DESCRIPTION

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.

ISBN 978-1-59973-337-1

VOLUME 1, 2015

MATHEMATICAL COMBINATORICS

(INTERNATIONAL BOOK SERIES)

Edited By Linfan MAO

THE MADIS OF CHINESE ACADEMY OF SCIENCES AND

ACADEMY OF MATHEMATICAL COMBINATORICS & APPLICATIONS

March, 2015

Vol.1, 2015 ISBN 978-1-59973-337-1

MATHEMATICAL COMBINATORICS

(INTERNATIONAL BOOK SERIES)

Edited By Linfan MAO

The Madis of Chinese Academy of Sciences and

Academy of Mathematical Combinatorics & Applications

March, 2015

Aims and Scope: The Mathematical Combinatorics (International Book Series) is a

fully refereed international book series, 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 1.416),

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 (3nd)

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

S.Bhattacharya

Deakin University

Geelong Campus at Waurn Ponds

Australia

Email: Sukanto.Bhattacharya@Deakin.edu.au

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

Baizhou He

Beijing University of Civil Engineering and

Architecture, P.R.China

Email: hebaizhou@bucea.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:

Nothing in life is to be feared. It is only to be understood.

By Marie Curie, a Polish and naturalized-French physicist and chemist.

Math.Combin.Book Ser. Vol.1(2015), 1-13

NC Smarandache Curves of Mannheim Curve CoupleAccording to Frenet Frame

Suleyman SENYURT and Abdussamet CALISKAN

(Faculty of Arts and Sciences, Department of Mathematics, Ordu University, 52100, Ordu/Turkey)

E-mail: senyurtsuleyman@hotmail.com

Abstract: In this paper, when the unit Darboux vector of the partner curve of Mannheim

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

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

we illustrate example of our main results.

Key Words: Mannheim curve, Mannheim partner curve, Smarandache Curves, Frenet

invariants.

AMS(2010): 53A04

1. Introduction

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 ([12]). Special Smarandache

curves have been studied by some authors .

Melih Turgut and Suha Ylmaz studied a special case of such curves and called it Smaran-

dache TB2 curves in the spaceE41 ([12]). Ahmad T.Ali studied some special Smarandache curves

in the Euclidean space. He studied Frenet-Serret invariants of a special case ([1]). Muhammed

Cetin , Ylmaz Tuncer and Kemal Karacan investigated special Smarandache curves according

to Bishop frame in Euclidean 3-Space and they gave some differential goematric properties of

Smarandache curves, also they found the centers of the osculating spheres and curvature spheres

of Smarandache curves ([5]). 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 ([2]). Nurten Bayrak, Ozcan Bektas and Salim Yuce

studied some special Smarandache curves in E31 [3]. Kemal Tas.kopru, Murat Tosun studied

special Smarandache curves according to Sabban frame on S2 ([11]).

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.

1Received September 8, 2014, Accepted February 12, 2015.

2 Suleyman SENYURT and Abdussamet CALISKAN

2. Preliminaries

The Euclidean 3-space E3 be inner product given by

, = x21 + x32 + x23

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])

T = N

N = T + BB = N.

(2.1)

For any unit speed : I E3, the vector W is called Darboux vector defined by

W = (s)T (s) + (s) +B(s).

If consider the normalization of the Darboux C =1

WW , we have

cos =(s)

W , sin =(s)

W ,

C = sinT (s) + cosB(s) (2.2)

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:

T = cos T sin BN = sin T + cos B

B = N

(2.3)

cos =

dsds

sin = dsds

.(2.4)

where (T, T ) = ([8]).

Theorem 2.1([7]) The distance between corresponding points of the Mannheim partner curves

in E3 is constant.

NC Smarandache Curves of Mannheim Curve Couple According to Frenet Frame 3

Theorem 2.2 Let (, ) be a Mannheim pair curves in E3. For the curvature