The Smarandache Curves on

Page 1

Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 193586, 12 pages http://dx.doi.org/10.1155/2014/193586

Research Article The Smarandache Curves on đ?‘†12 and Its Duality on đ??ť02 Atakan TuLkan Yakut,1 Murat SavaG,2,3 and TuLba Tamirci1 1

Department of Mathematics, Arts and Sciences Faculty, Ni˘gde University, 51240 Ni˘gde, Turkey Department of Mathematics, Columbia University, New York, NY 10027, USA 3 Department of Mathematics, Gazi University, 06500 Ankara, Turkey 2

Correspondence should be addressed to Murat Savas¸; murat@math.columbia.edu Received 20 March 2014; Revised 18 July 2014; Accepted 22 July 2014; Published 21 August 2014 Academic Editor: Fernando SimËœoes Copyright Š 2014 Atakan Tu˘gkan Yakut et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We introduce special Smarandache curves based on Sabban frame on đ?‘†12 and we investigate geodesic curvatures of Smarandache curves on de Sitter and hyperbolic spaces. The existence of duality between Smarandache curves on de Sitter space and Smarandache curves on hyperbolic space is shown. Furthermore, we give examples of our main results.

1. Introduction Curves as a subject of differential geometry have been intriguing for researchers throughout mathematical history and so they have been one of the interesting research fields. Regular curves play a central role in the theory of curves in differential geometry. In the theory of curves, there are some special curves such as Bertrand curves, Mannheim curves, involute and evolute curves, and pedal curves in which differential geometers are interested. A common approach to characterization of curves is to consider the relationship between the corresponding Frenet vectors of two curves. Bertrand and Mannheim curves are excellent examples for such cases. In the study of fundamental theory and the characterizations of space curves, the corresponding relations between the curves are a very fascinating problem. Recently, a new special curve is named according to the Sabban frame in the Euclidean unit sphere; Smarandache curve has been defined by Turgut and Yilmaz in Minkowski space-time [1]. Ali studied Smarandache curves with respect to the Sabban frame in Euclidean 3-space [2]. Then Tas¸k¨opr¨u and Tosun studied Smarandache curves on đ?‘†2 [3]. Smarandache curves have also been studied by many researchers [1, 4– 7]. Smarandache curves are one of the most important tools in Smarandache geometry. Smarandache geometry has an important role in the theory of relativity and parallel universes. There are many results related to Smarandache curves

in Euclidean and Minkowski spaces, but Smarandache curves are getting more tedious and complicated when de Sitter space is concerned. A regular curve in Minkowski space-time, whose position vector is associated with Frenet frame vectors on another regular curve, is called a Smarandache curve [1]. In this paper, we define Smarandache curves on de Sitter surface according to the Sabban frame {đ?›ź, đ?‘Ą, đ?œ‚} in Minkowski 3-space. We obtain the geodesic curvatures and the expressions for the Sabban frame’s vectors of special Smarandache curves on de Sitter surface. Furthermore, we give some examples of special de Sitter and hyperbolic Smarandache curves in Minkowski 3-space.

2. Preliminaries In this section, we prepare some definitions and basic facts. For basic concepts and details of properties, see [8, 9]. Consider R3 as a three-dimensional vector space. For any vectors đ?‘Ľâƒ— = (đ?‘Ľ0 , đ?‘Ľ1 , đ?‘Ľ2 ) and đ?‘Śâƒ— = (đ?‘Ś0 , đ?‘Ś1 , đ?‘Ś2 ) in R3 the pseudoscalar product of đ?‘Ľâƒ— and đ?‘Śâƒ— is defined by â&#x;¨â‹…, â‹…â&#x;Šđ??ż = −đ?‘Ľ0 đ?‘Ś0 + đ?‘Ľ1 đ?‘Ś1 + đ?‘Ľ2 đ?‘Ś2 . It is called đ??¸13 = (R3 , â&#x;¨â‹…, â‹…â&#x;Šđ??ż ) Minkowski 3-space. Recall that a nonzero vector đ?‘Ľâƒ— ∈ đ??¸13 is spacelike if â&#x;¨đ?‘Ľ,⃗ đ?‘Ľâ&#x;Šâƒ— đ??ż > 0, timelike if â&#x;¨đ?‘Ľ,⃗ đ?‘Ľâ&#x;Šâƒ— đ??ż < 0, and null (lightlike) if â&#x;¨đ?‘Ľ,⃗ đ?‘Ľâ&#x;Šâƒ— đ??ż = 0. The norm (length) of a vector đ?‘Ľâƒ— ∈ đ??¸13 is given by ‖đ?‘Ľâ€–⃗ đ??ż = √|â&#x;¨đ?‘Ľ,⃗ đ?‘Ľâ&#x;Šâƒ— đ??ż | and two vectors đ?‘Ľâƒ— and đ?‘Śâƒ— are said to be orthogonal if â&#x;¨đ?‘Ľ,⃗ đ?‘Śâ&#x;Šâƒ— đ??ż = 0. Next, we say that an arbitrary curve đ?›ź = đ?›ź(đ?‘ ) in

2

Journal of Applied Mathematics

đ??¸13 can locally be spacelike, timelike, or null (lightlike) if all of its velocity vectors đ?›źó¸€ (đ?‘ ) are, respectively, spacelike, timelike, or null (lightlike) for all đ?‘ ∈ đ??ź. If ‖đ?›źó¸€ (đ?‘ )‖đ??ż ≠ 0 for every đ?‘ ∈ đ??ź, then đ?›ź is a regular curve in đ??¸13 . A spacelike (timelike) regular curve đ?›ź is parameterized by a pseudoarclength parameter đ?‘ which is given by đ?›ź : đ??ź ⊂ R → đ??¸13 , and then the tangent vector đ?›źó¸€ (đ?‘ ) along đ?›ź has unit length; that is, â&#x;¨đ?›źó¸€ (đ?‘ ), đ?›źó¸€ (đ?‘ )â&#x;Šđ??ż = 1 (â&#x;¨đ?›źó¸€ (đ?‘ ), đ?›źó¸€ (đ?‘ )â&#x;Šđ??ż = −1) for all đ?‘ ∈ đ??ź. Let đ?‘Ľâƒ— = (đ?‘Ľ0 , đ?‘Ľ1 , đ?‘Ľ2 ), đ?‘Śâƒ— = (đ?‘Ś0 , đ?‘Ś1 , đ?‘Ś2 ) ∈ đ??¸13 . The Lorentzian vector cross-product is defined as follows: óľ„¨óľ„¨âˆ’đ?‘’ đ?‘’ đ?‘’ 󾄨󾄨 󾄨󾄨 0 1 2 󾄨󾄨 󾄨 󾄨 đ?‘Ľâƒ— ∧ đ?‘Śâƒ— = 󾄨󾄨󾄨 đ?‘Ľ0 đ?‘Ľ1 đ?‘Ľ2 󾄨󾄨󾄨 , 󾄨 󾄨󾄨 󾄨󾄨 đ?‘Ś0 đ?‘Ś1 đ?‘Ś2 󾄨󾄨󾄨

(1)

(2)

and hyperbolic space in Minkowski 3-space by (3)

We can express a new frame different from the Frenet frame for a regular curve. Let đ?›ź : đ??ź ⊂ R → đ?‘†12 be a regular unit speed curve lying fully on đ?‘†12 . Then its position vector đ?›ź is spacelike, which implies that the tangent vector đ?›źó¸€ = đ?‘Ą is the unit timelike, spacelike, or null vector for all đ?‘ ∈ đ??ź. In our work, we are concerned with the vector đ?›źó¸€ = đ?‘Ą which may be the unit timelike or spacelike. Let đ?›ź : đ??ź ⊂ R → đ?‘†12 be a regular unit speed curve lying fully on đ?‘†12 for all đ?‘ ∈ đ??ź and its position vector đ?›ź a unit spacelike vector; then đ?›źó¸€ = đ?‘Ą is a unit timelike and so đ?œ‚ is a unit spacelike vector. In this case, the curve đ?›ź is called a timelike curve. If đ?›źó¸€ = đ?‘Ą is a unit spacelike vector, then đ?œ‚ is a unit timelike vector. In this case, the curve đ?›ź is called a spacelike curve and we have an orthonormal Sabban frame {đ?›ź(đ?‘ ), đ?‘Ą(đ?‘ ), đ?œ‚(đ?‘ )} along the curve đ?›ź, where đ?œ‚(đ?‘ ) = đ?›ź(đ?‘ ) ∧ đ?‘Ą(đ?‘ ) is the unit spacelike or timelike vector. Then Frenet formulas of đ?›ź are given by

đ?‘Ą = đ?›ź ∧ đ?œ‚,

đ?œ‚ = đ?›ź ∧ đ?‘Ą.

3. De Sitter and Hyperbolic Smarandache Curves for Timelike Curves In this section we give different Smarandache curves on de Sitter and hyperbolic spaces in Minkowski-space. Let đ?›ź be a timelike curve on đ?‘†12 ; then the Smarandache partner curve of đ?›ź is either timelike/spacelike or hyperbolic curve. We refer to the hyperbolic Smarandache curve of a timelike curve đ?›ź as the hyperbolic duality of đ?›ź. To avoid repetition we use đ?œ€ = Âą1 in the following theorems in this section. If we take đ?œ€ = 1, then the Smarandache curve đ?›˝ is timelike or spacelike, and if we take đ?œ€ = −1, then đ?›˝ is hyperbolic. Definition 2. Let đ?›ź = đ?›ź(đ?‘ ) be a unit speed regular timelike curve lying fully on đ?‘†12 . The curve đ?›˝ : đ??ź ⊂ R → đ?‘†12 (đ?›˝ : đ??ź ⊂ R → đ??ť02 ) of đ?›ź defined by 1 (đ?‘? đ?›ź (đ?‘ ) + đ?‘?2 đ?œ‚ (đ?‘ )) √2 1

(6)

(4)

where đ?œ€ = Âą1, the curve đ?›ź is timelike for đ?œ€ = 1 and spacelike for đ?œ€ = −1, and đ?œ…đ?‘” (đ?‘ ) is the geodesic curvature of đ?›ź on đ?‘†12 , which is given by đ?œ…đ?‘” (đ?‘ ) = det(đ?›ź(đ?‘ ), đ?‘Ą(đ?‘ ), đ?‘Ąó¸€ (đ?‘ )), where đ?‘ is the arc length parameter of đ?›ź. This relation is also given by [10, 11] for đ?œ€ = 1. In particular, by using equation (ii), the following relations hold: đ?œ€đ?›ź = đ?‘Ą ∧ đ?œ‚,

Let {đ?›ź, đ?‘Ą, đ?œ‚} and {đ?›˝, đ?‘Ąđ?›˝ , đ?œ‚đ?›˝ } be the moving Sabban frames of đ?›ź and đ?›˝, respectively. Then we have the following definitions and theorems of Smarandache curves đ?›˝ = đ?›˝(đ?‘ (đ?‘ )) given in Section 3. In Section 3, we deal with Smarandache curves on de Sitter and hyperbolic spaces for timelike curves. Similar results are given for spacelike curves in the Appendix.

đ?›˝ (đ?‘ (đ?‘ )) =

ó¸€

0 1 0 đ?›ź đ?›ź [ đ?‘Ąó¸€ ] = [ đ?œ€ 0 đ?œ€đ?œ…đ?‘” ] [ đ?‘Ą ] , ó¸€ [ đ?œ‚ ] [0 đ?œ€đ?œ…đ?‘” 0 ] [ đ?œ‚ ]

(a) the Smarandache curve đ?›˝(đ?‘ (đ?‘ )) may be a timelike curve on đ?‘†12 ,

(c) the Smarandache curve đ?›˝(đ?‘ (đ?‘ )) is in đ??ť02 for all đ?‘ ∈ đ??ź.

where đ?‘Ľâƒ— = (đ?‘Ľ0 , đ?‘Ľ1 , đ?‘Ľ2 ), đ?‘Śâƒ— = (đ?‘Ś0 , đ?‘Ś1 , đ?‘Ś2 ), đ?‘§âƒ— = (đ?‘§0 , đ?‘§1 , đ?‘§2 ) ∈ đ??¸13 . We now define de Sitter 2-space by

đ??ť02 = {đ?‘Ľâƒ— ∈ R31 : −đ?‘Ľ02 + đ?‘Ľ12 + đ?‘Ľ22 = −1, đ?‘Ľ0 > 0} .

Based on this definition, if a regular unit speed curve đ?›ź : đ??ź ⊂ R → đ?‘†12 is lying fully on đ?‘†12 for all đ?‘ ∈ đ??ź and its position vector đ?›ź is a unit spacelike, then the Smarandache curve đ?›˝ = đ?›˝(đ?‘ (đ?‘ )) of curve đ?›ź is a regular unit speed curve lying fully on đ?‘†12 or đ??ť02 . In this case we have the following:

(b) the Smarandache curve đ?›˝(đ?‘ (đ?‘ )) may be a spacelike curve on đ?‘†12 , or

and also the following relations hold: 󾄨󾄨 đ?‘Ľ0 đ?‘Ľ1 đ?‘Ľ2 󾄨󾄨 (i) â&#x;¨đ?‘Ľâƒ— ∧ đ?‘Ś,⃗ đ?‘§â&#x;Šâƒ— đ??ż = 󾄨󾄨󾄨 đ?‘Śđ?‘§0 đ?‘Śđ?‘§1 đ?‘Śđ?‘§2 󾄨󾄨󾄨, 󾄨 0 1 2󾄨 (ii) đ?‘Ľâƒ— ∧ (đ?‘Śâƒ— ∧ đ?‘§)⃗ = â&#x;¨đ?‘Ľ,⃗ đ?‘Śâ&#x;Šâƒ— đ??ż đ?‘§âƒ— − â&#x;¨đ?‘Ľ,⃗ đ?‘§â&#x;Šâƒ— đ??ż đ?‘Ś,⃗

đ?‘†12 = {đ?‘Ľâƒ— ∈ R31 : −đ?‘Ľ02 + đ?‘Ľ12 + đ?‘Ľ22 = 1}

Definition 1. A unit speed regular curve đ?›˝(đ?‘ (đ?‘ )) lying fully in Minkowski 3-space, whose position vector is associated with Sabban frame vectors on another regular curve đ?›ź(đ?‘ ), is called a Smarandache curve [1].

(5)

is called the đ?›źđ?œ‚-Smarandache curve of đ?›ź and fully lies on đ?‘†12 , where đ?‘?1 , đ?‘?2 ∈ R \ {0} and đ?œ€(đ?‘?12 + đ?‘?22 ) = 2. If đ?œ€ = −1; then the hyperbolic đ?›źđ?œ‚-Smarandache curve is undefined since the equation (đ?‘?12 + đ?‘?22 ) = −2 has no solution in R. Theorem 3. Let đ?›ź : đ??ź ⊂ R → đ?‘†12 be a regular unit speed timelike curve lying fully on đ?‘†12 with the Sabban frame {đ?›ź, đ?‘Ą, đ?œ‚} and geodesic curvature đ?œ…đ?‘” . If đ?›˝ : đ??ź ⊂ R → đ?‘†12 is the đ?›źđ?œ‚-timelike Smarandache curve of đ?›ź, then the relationships

Journal of Applied Mathematics

3

between the Sabban frames of đ?›ź and its đ?›źđ?œ‚-Smarandache curve are given by đ?‘? đ?‘?1 0 2 √ √ [ 2 2] đ?›ź đ?›˝ ] [ ] [đ?‘Ą] 0 đ?œ€ 0 [ đ?‘Ąđ?›˝ ] = [ , ] [ ] [ [đ?œ‚đ?›˝ ] [ đ?‘?2 đ?œ€ đ?‘?1 đ?œ€ ] [ đ?œ‚ ] 0 √2 ] [ √2

It is easily seen that đ?œ‚đ?›˝ is a unit spacelike vector. On the other hand, differentiating (12) with respect to đ?‘ , we find đ?‘‘đ?‘Ąđ?›˝ đ?‘‘đ?‘ đ?‘‘đ?‘ đ?‘‘đ?‘

(7)

√2đ?œ€ đ?‘Ąđ?›˝ó¸€ = 󾄨󾄨 󾄨 (đ?›ź + đ?œ…đ?‘” đ?œ‚) . 󾄨󾄨đ?‘?1 + đ?‘?2 đ?œ…đ?‘” 󾄨󾄨󾄨 󾄨 󾄨 (8)

Proof. By taking the derivative of (6) with respect to đ?‘ and by using (4), we get đ?›˝ó¸€ (đ?‘ (đ?‘ )) =

đ?‘‘đ?›˝ đ?‘‘đ?‘ 1 (đ?‘? + đ?‘? đ?œ… ) đ?‘Ą = đ?‘‘đ?‘ đ?‘‘đ?‘ √2 1 2 đ?‘”

đ?‘‘đ?‘ 1 (đ?‘? + đ?‘? đ?œ… ) đ?‘Ą, = đ?‘‘đ?‘ √2 1 2 đ?‘”

Consequently, the geodesic curvature đ?œ…đ?‘”đ?›˝ of the curve đ?›˝ = đ?›˝(đ?‘ ) is given by

(9)

(10)

Corollary 4. Let đ?›ź : đ??ź ⊂ R → đ?‘†12 be a regular unit speed timelike curve lying fully on đ?‘†12 . Then the đ?›źđ?œ‚-spacelike Smarandache curve đ?›ź : đ??ź ⊂ R → đ?‘†12 of đ?›ź does not exist.

where 󾄨󾄨 󾄨󾄨 đ?‘‘đ?‘ 󾄨󾄨󾄨đ?‘?1 + đ?‘?2 đ?œ…đ?‘” 󾄨󾄨󾄨 . = √2 đ?‘‘đ?‘

(11)

(12)

where đ?œ€ = 1 if đ?‘?1 + đ?‘?2 đ?œ…đ?‘” > 0 for all đ?‘ and đ?œ€ = −1 if đ?‘?1 + đ?‘?2 đ?œ…đ?‘” < 0 for all đ?‘ . From (6) and (12) we get đ?œ‚đ?›˝ = đ?›˝ ∧ đ?‘Ąđ?›˝ =

đ?œ€ (đ?‘? đ?›ź + đ?‘?1 đ?œ‚) . √2 2

(13)

(16)

Definition 5. Let đ?›ź = đ?›ź(đ?‘ ) be a regular unit speed timelike curve lying fully on đ?‘†12 . Then the đ?›źđ?‘Ą-Smarandache curve đ?›˝ : đ??ź ⊂ R → đ?‘†12 (đ?›˝ : đ??ź ⊂ R → đ??ť02 ) of đ?›ź is defined by đ?›˝ (đ?‘ (đ?‘ )) =

Hence, the unit timelike tangent vector of the curve đ?›˝ is given by đ?‘Ąđ?›˝ = đ?œ€đ?‘Ą,

(15)

đ?‘?1 đ?œ…đ?‘” − đ?‘?2 đ?œ…đ?‘”đ?›˝ = det (đ?›˝, đ?‘Ąđ?›˝ , đ?‘Ąđ?›˝ó¸€ ) = 󾄨󾄨 󾄨. 󾄨󾄨đ?‘?1 + đ?‘?2 đ?œ…đ?‘” 󾄨󾄨󾄨 󾄨 󾄨

or equivalently đ?‘Ąđ?›˝

(14)

and by combining (11) and (14) we have

where đ?œ€ = Âą1 and its geodesic curvature đ?œ…đ?‘”đ?›˝ is given by đ?‘?1 đ?œ…đ?‘” − đ?‘?2 đ?œ…đ?‘”đ?›˝ = 󾄨󾄨 󾄨. 󾄨󾄨đ?‘?1 + đ?‘?2 đ?œ…đ?‘” 󾄨󾄨󾄨 󾄨 󾄨

= đ?œ€ (đ?›ź + đ?œ…đ?‘” đ?œ‚)

1 (đ?‘? đ?›ź (đ?‘ ) + đ?‘?2 đ?‘Ą (đ?‘ )) , √2 1

(17)

where đ?‘?1 , đ?‘?2 ∈ R \ {0} and đ?œ€(đ?‘?12 − đ?‘?22 ) = 2. Theorem 6. Let đ?›ź : đ??ź ⊂ R → đ?‘†12 be a regular unit speed timelike curve lying fully on đ?‘†12 with the Sabban frame {đ?›ź, đ?‘Ą, đ?œ‚} and geodesic curvature đ?œ…đ?‘” . If đ?›˝ : đ??ź ⊂ R → đ?‘†12 (đ?›˝ : đ??ź ⊂ R → đ??ť02 ) is the đ?›źđ?‘Ą-timelike (hyperbolic) Smarandache curve of đ?›ź, then its frame {đ?›˝, đ?‘Ąđ?›˝ , đ?œ‚đ?›˝ } is given by đ?‘?2 √2 đ?‘?1

đ?‘?1 √2 đ?‘?2

0 [ ] [ ] đ?œ… đ?‘? [ ] 2 đ?‘” đ?›˝ [ ] đ?›ź ] [ đ?‘Ąđ?›˝ ] = [ 2 đ?œ…2 ) 2 đ?œ…2 ) ] [ đ?‘Ą ] . [ √2 − đ?œ€ (đ?‘?22 đ?œ…đ?‘”2 ) √2 √2 − đ?œ€ (đ?‘? − đ?œ€ (đ?‘? 2 đ?‘” 2 đ?‘” [ ] đ?œ‚ ][ ] [đ?œ‚đ?›˝ ] [ −đ?‘?22 đ?œ…đ?‘” −đ?‘?1 đ?‘?2 đ?œ…đ?‘” [ ] 2 [ ] √2 (2 − đ?œ€ (đ?‘?22 đ?œ…đ?‘”2 )) √2 (2 − đ?œ€ (đ?‘?22 đ?œ…đ?‘”2 )) √2 (2 − đ?œ€ (đ?‘?22 đ?œ…đ?‘”2 )) [ ] The geodesic curvature đ?œ…đ?‘”đ?›˝ of curve đ?›˝ is given by

(18)

where đ?œ† 1 = đ?œ€đ?‘?23 đ?œ…đ?‘” đ?œ…đ?‘”ó¸€ + đ?‘?1 (2 − đ?œ€đ?‘?22 đ?œ…đ?‘”2 ) ,

đ?œ…đ?‘”đ?›˝

=

1 (2 − đ?œ€ (đ?‘?22 đ?œ…đ?‘”2 ))

(đ?‘?22 đ?œ…đ?‘” đ?œ† 1 5/2

− đ?‘?1 đ?‘?2 đ?œ…đ?‘” đ?œ† 2 + 2đ?œ† 3 ) ,

đ?œ† 2 = đ?œ€đ?‘?1 đ?‘?22 đ?œ…đ?‘” đ?œ…đ?‘”ó¸€ + (đ?‘?2 + đ?‘?2 đ?œ…đ?‘”2 ) (2 − đ?œ€đ?‘?22 đ?œ…đ?‘”2 ) , (19)

đ?œ† 3 = đ?œ€đ?‘?23 đ?œ…đ?‘”2 đ?œ…đ?‘”ó¸€ + (đ?‘?1 đ?œ…đ?‘” + đ?‘?2 đ?œ…đ?‘”ó¸€ ) (2 − đ?œ€đ?‘?22 đ?œ…đ?‘”2 ) .

(20)

4

Journal of Applied Mathematics

Proof. We take đ?œ€ = 1. By taking the derivative of (17) with respect to đ?‘ and by using (4), we get đ?›˝ó¸€ (đ?‘ (đ?‘ )) =

đ?‘‘đ?›˝ đ?‘‘đ?‘ 1 (đ?‘? đ?›ź + đ?‘?1 đ?‘Ą + đ?‘?2 đ?œ…đ?‘” đ?œ‚) = đ?‘‘đ?‘ đ?‘‘đ?‘ √2 2

Consequently, we have đ?œ…đ?‘”đ?›˝ = det (đ?›˝, đ?‘Ąđ?›˝ , đ?‘Ąđ?›˝ó¸€ )

(21)

=

or equivalently đ?‘‘đ?‘ 1 (đ?‘? đ?›ź + đ?‘?1 đ?‘Ą + đ?‘?2 đ?œ…đ?‘” đ?œ‚) . đ?‘Ąđ?›˝ = đ?‘‘đ?‘ √2 2

2 2

(23)

(24)

Therefore, the unit timelike tangent vector of the curve đ?›˝ is given by đ?‘Ąđ?›˝ =

1

(đ?‘?2 đ?›ź + đ?‘?1 đ?‘Ą + đ?‘?2 đ?œ…đ?‘” đ?œ‚) .

√2 − đ?‘?22 đ?œ…đ?‘”2

(25)

On the other hand, from (17) and (25) it can be easily seen that đ?œ‚đ?›˝ = đ?›˝ ∧ đ?‘Ąđ?›˝ =

1 √4 − 2đ?‘?22 đ?œ…đ?‘”2

(−đ?‘?22 đ?œ…đ?‘” đ?›ź − đ?‘?1 đ?‘?2 đ?œ…đ?‘” đ?‘Ą + 2đ?œ‚)

(26)

is a unit spacelike vector. Differentiating (25) with respect to đ?‘ , we obtain đ?‘‘đ?‘Ąđ?›˝ đ?‘‘đ?‘

=

đ?‘‘đ?‘ đ?‘‘đ?‘

1 (2 − đ?‘?22 đ?œ…đ?‘”2 )

3/2

(đ?œ† 1 đ?›ź + đ?œ† 2 đ?‘Ą + đ?œ† 3 đ?œ‚) ,

(27)

đ?œ† 2 = đ?‘?1 đ?‘?22 đ?œ…đ?‘” đ?œ…đ?‘”ó¸€ + (đ?‘?2 + đ?‘?2 đ?œ…đ?‘”2 ) (2 − đ?‘?22 đ?œ…đ?‘”2 ) , + (đ?‘?1 đ?œ…đ?‘” +

đ?‘Ąđ?›˝ó¸€ =

2

(2 − đ?‘?22 đ?œ…đ?‘”2 )

đ?‘?2 √2 đ?‘?1

đ?‘?1 √2 đ?‘?2

0 [ ] [ ] đ?‘?2 đ?œ…đ?‘” [ ] [ ] [ ] = [ √đ?‘?22 đ?œ…đ?‘”2 − 2 √đ?‘?22 đ?œ…đ?‘”2 − 2 √đ?‘?22 đ?œ…đ?‘”2 − 2 ] [ ] [ ] −đ?‘?22 đ?œ…đ?‘” −đ?‘?1 đ?‘?2 đ?œ…đ?‘” [ ] 2 [ ] 2 đ?œ…2 − 2) √2 (đ?‘?2 đ?œ…2 − 2) √2 (đ?‘?2 đ?œ…2 − 2) √2 (đ?‘? 2 đ?‘” 2 đ?‘” 2 đ?‘” [ ] đ?›ź Ă— [đ?‘Ą] . [đ?œ‚]

(31)

The geodesic curvature đ?œ…đ?‘”đ?›˝ of curve đ?›˝ is given by 1 (đ?‘?22 đ?œ…đ?‘”2

− 2)

5/2

(đ?‘?22 đ?œ…đ?‘” đ?œ† 1 − đ?‘?1 đ?‘?2 đ?œ…đ?‘” đ?œ† 2 + 2đ?œ† 3 ) ,

(32)

Definition 8. Let đ?›ź : đ??ź ⊂ R → đ?‘†12 be a regular unit speed timelike curve lying fully on đ?‘†12 . Then the đ?‘Ąđ?œ‚-Smarandache curve đ?›˝ : đ??ź ⊂ R → đ?‘†12 (đ?›˝ : đ??ź ⊂ R → đ??ť02 ) of đ?›ź is defined by

đ?‘?2 đ?œ…đ?‘”ó¸€ ) (2

−

(28) đ?›˝ (đ?‘ (đ?‘ )) =

đ?‘?22 đ?œ…đ?‘”2 ) ,

(đ?œ† 1 đ?›ź + đ?œ† 2 đ?‘Ą + đ?œ† 3 đ?œ‚) .

1 (đ?‘? đ?‘Ą (đ?‘ ) + đ?‘?2 đ?œ‚ (đ?‘ )) , √2 1

(33)

where đ?‘?1 , đ?‘?2 ∈ R \ {0} and đ?œ€(−đ?‘?12 + đ?‘?22 ) = 2.

and by combining (24) and (26) we get √2

đ?›˝ [ đ?‘Ąđ?›˝ ] [đ?œ‚đ?›˝ ]

where đ?œ† 1 , đ?œ† 2 , and đ?œ† 3 can be calculated as in Theorem 6.

đ?œ† 1 = đ?‘?23 đ?œ…đ?‘” đ?œ…đ?‘”ó¸€ + đ?‘?1 (2 − đ?‘?22 đ?œ…đ?‘”2 ) ,

đ?œ†3 =

Corollary 7. Let đ?›ź : đ??ź ⊂ R → đ?‘†12 be a regular unit speed timelike curve lying fully on đ?‘†12 with the Sabban frame {đ?›ź, đ?‘Ą, đ?œ‚} and geodesic curvature đ?œ…đ?‘” . If đ?›˝ : đ??ź ⊂ R → đ?‘†12 is the đ?›źđ?‘Ąspacelike Smarandache curve of đ?›ź, then its frame {đ?›˝, đ?‘Ąđ?›˝ , đ?œ‚đ?›˝ } is given by

đ?œ…đ?‘”đ?›˝ =

where

đ?‘?23 đ?œ…đ?‘”2 đ?œ…đ?‘”ó¸€

(30)

The following corollary is proved by the same methods as the above theorem.

If đ?‘?22 đ?œ…đ?‘”2 < 2, then đ?‘Ąđ?›˝ is a timelike vector. So đ?‘‘đ?‘ √ 2 − đ?‘?2 đ?œ…đ?‘” = . đ?‘‘đ?‘ 2

(đ?‘?22 đ?œ…đ?‘” đ?œ† 1 − đ?‘?1 đ?‘?2 đ?œ…đ?‘” đ?œ† 2 + 2đ?œ† 3 ) .

(22)

2

đ?‘‘đ?‘ 1 ) = (đ?‘?22 đ?œ…đ?‘”2 − 2) . đ?‘‘đ?‘ 2

(2 − đ?‘?22 đ?œ…đ?‘”2 )

5/2

The proof of đ?œ€ = −1 case is similar.

Taking the Lorentzian inner product in (22) we have â&#x;¨đ?‘Ąđ?›˝ , đ?‘Ąđ?›˝ â&#x;Šđ??ż (

1

(29)

Theorem 9. Let đ?›ź : đ??ź ⊂ R → đ?‘†12 be a regular unit speed timelike curve lying fully on đ?‘†12 with the Sabban frame {đ?›ź, đ?‘Ą, đ?œ‚} and geodesic curvature đ?œ…đ?‘” . If đ?›˝ : đ??ź ⊂ R → đ?‘†12

Journal of Applied Mathematics

5

(đ?›˝ : đ??ź ⊂ R → đ??ť02 ) is the đ?‘Ąđ?œ‚-timelike (hyperbolic) Smarandache curve of đ?›ź, then its frame {đ?›˝, đ?‘Ąđ?›˝ , đ?œ‚đ?›˝ } is given by đ?›˝ [ đ?‘Ąđ?›˝ ] [đ?œ‚đ?›˝ ]

1

đ?œ‚đ?›˝ = đ?›˝ ∧ đ?‘Ąđ?›˝ = đ?‘?1 √2

0

đ?‘?2 √2

[ ] [ ] [ ] đ?‘?2 đ?œ…đ?‘” đ?‘?1 đ?œ…đ?‘” [ ] đ?‘?1 [ ] ] =[ 2 2 2 [ √2đ?œ…đ?‘”2 − đ?œ€đ?‘?1 √2đ?œ…đ?‘”2 − đ?œ€đ?‘?1 √2đ?œ…đ?‘”2 − đ?œ€đ?‘?1 ] [ ] [ ] đ?œ€2đ?œ…đ?‘” −đ?‘?12 [ ] đ?‘?1 đ?‘?2 [ ] 2 2 2 2 2 2 √2 (2đ?œ…đ?‘” − đ?œ€đ?‘?1 ) √2 (2đ?œ…đ?‘” − đ?œ€đ?‘?1 ) √2 (2đ?œ…đ?‘” − đ?œ€đ?‘?1 ) [ ] đ?›ź Ă— [đ?‘Ą] . [đ?œ‚]

√4đ?œ…đ?‘”2 − 2đ?‘?12

(2đ?œ…đ?‘” đ?›ź + đ?‘?1 đ?‘?2 đ?‘Ą − đ?‘?12 đ?œ‚)

(34)

1 5/2

(2đ?œ…đ?‘”2 − đ?œ€đ?‘?12 )

(−đ?œ€2đ?œ…đ?‘” đ?œ† 1 + đ?‘?1 đ?‘?2 đ?œ† 2 − đ?‘?12 đ?œ† 3 ) ,

(42)

is a unit spacelike vector. Differentiating (41) with respect to đ?‘ , we find đ?‘‘đ?‘Ąđ?›˝ đ?‘‘đ?‘ đ?‘‘đ?‘ đ?‘‘đ?‘

=

1 3/2

(2đ?œ…đ?‘”2 − đ?‘?12 )

(đ?œ† 1 đ?›ź + đ?œ† 2 đ?‘Ą + đ?œ† 3 đ?œ‚) ,

(43)

where đ?œ† 1 = − 2đ?‘?1 đ?œ…đ?‘” đ?œ…đ?‘”ó¸€ + đ?‘?2 đ?œ…đ?‘” (2đ?œ…đ?‘”2 − đ?‘?12 ) , đ?œ† 2 = − 2đ?‘?2 đ?œ…đ?‘”2 đ?œ…đ?‘”ó¸€ + (đ?‘?1 + đ?‘?2 đ?œ…đ?‘”ó¸€ + đ?‘?1 đ?œ…đ?‘”2 ) (2đ?œ…đ?‘”2 − đ?‘?12 ) ,

The geodesic curvature đ?œ…đ?‘”đ?›˝ of curve đ?›˝ is given by đ?œ…đ?‘”đ?›˝ =

On the other hand, from (33) and (41) it can be easily seen that

(44)

đ?œ† 3 = − 2đ?‘?1 đ?œ…đ?‘”2 đ?œ…đ?‘”ó¸€ + (đ?‘?2 đ?œ…đ?‘”2 + đ?‘?1 đ?œ…đ?‘”ó¸€ ) (2đ?œ…đ?‘”2 − đ?‘?12 ) , and by combining (40) and (43) we get đ?‘Ąđ?›˝ó¸€ =

(35)

where

√2 2

(2 đ?œ…đ?‘”2 − đ?‘?12 )

(đ?œ† 1 đ?›ź + đ?œ† 2 đ?‘Ą + đ?œ† 3 đ?œ‚) .

(45)

As a result, we have

đ?œ†1 = −

2đ?‘?1 đ?œ…đ?‘” đ?œ…đ?‘”ó¸€

+

đ?‘?2 đ?œ…đ?‘” (2đ?œ…đ?‘”2

−

đ?œ€đ?‘?12 ) ,

đ?œ† 2 = − 2đ?‘?2 đ?œ…đ?‘”2 đ?œ…đ?‘”ó¸€ + (đ?‘?1 + đ?‘?2 đ?œ…đ?‘”ó¸€ + đ?‘?1 đ?œ…đ?‘”2 ) (2đ?œ…đ?‘”2 − đ?œ€đ?‘?12 ) ,

đ?œ…đ?‘”đ?›˝ = det (đ?›˝, đ?‘Ąđ?›˝ , đ?‘Ąđ?›˝ó¸€ ) (36)

đ?œ† 3 = − 2đ?‘?1 đ?œ…đ?‘”2 đ?œ…đ?‘”ó¸€ + (đ?‘?2 đ?œ…đ?‘”2 + đ?‘?1 đ?œ…đ?‘”ó¸€ ) (2đ?œ…đ?‘”2 − đ?œ€đ?‘?12 ) . Proof. Let đ?œ€ = 1. By taking the derivative of (33) with respect to đ?‘ and by using (4), we get đ?‘‘đ?›˝ đ?‘‘đ?‘ 1 (đ?‘?1 đ?›ź + đ?‘?2 đ?œ…đ?‘” đ?‘Ą + đ?‘?1 đ?œ…đ?‘” đ?œ‚) = đ?›˝ (đ?‘ (đ?‘ )) = √ đ?‘‘đ?‘ đ?‘‘đ?‘ 2 ó¸€

(37)

or equivalently đ?‘Ąđ?›˝

đ?‘‘đ?‘ 1 (đ?‘? đ?›ź + đ?‘?2 đ?œ…đ?‘” đ?‘Ą + đ?‘?1 đ?œ…đ?‘” đ?œ‚) . = đ?‘‘đ?‘ √2 1

(38)

Taking the Lorentzian inner product in (38) we have â&#x;¨đ?‘Ąđ?›˝ , đ?‘Ąđ?›˝ â&#x;Šđ??ż (

đ?‘‘đ?‘ 2 1 2 ) = (đ?‘?1 − 2đ?œ…đ?‘”2 ) đ?‘‘đ?‘ 2

(39)

and đ?‘Ąđ?›˝ is a unit timelike vector for 2đ?œ…đ?‘”2 > đ?‘?12 . It follows that 2 2 đ?‘‘đ?‘ √ 2đ?œ…đ?‘” − đ?‘?1 = . đ?‘‘đ?‘ 2

(40)

Therefore, the unit timelike tangent vector of the curve đ?›˝ is given by đ?‘Ąđ?›˝ =

1 √2đ?œ…đ?‘”2

− đ?‘?12

(đ?‘?1 đ?›ź + đ?‘?2 đ?œ…đ?‘” đ?‘Ą + đ?‘?1 đ?œ…đ?‘” đ?œ‚) .

(41)

=

1 5/2

(2đ?œ…đ?‘”2 − đ?‘?12 )

(−2đ?œ…đ?‘” đ?œ† 1 + đ?‘?1 đ?‘?2 đ?œ† 2 − đ?‘?12 đ?œ† 3 ) .

(46)

The proof of đ?œ€ = −1 case is similar. Corollary 10. Let đ?›ź : đ??ź ⊂ R → đ?‘†12 be a regular unit speed timelike curve lying fully on đ?‘†12 with the Sabban frame {đ?›ź, đ?‘Ą, đ?œ‚} and geodesic curvature đ?œ…đ?‘” . If đ?›˝ : đ??ź ⊂ R → đ?‘†12 is the đ?‘Ąđ?œ‚spacelike Smarandache curve of đ?›ź, then its frame {đ?›˝, đ?‘Ąđ?›˝ , đ?œ‚đ?›˝ } is given by đ?›˝ [ đ?‘Ąđ?›˝ ] [đ?œ‚đ?›˝ ] đ?‘?2 đ?‘?1 0 [ ] √2 √2 [ ] [ ] đ?‘?2 đ?œ…đ?‘” đ?‘?1 đ?œ…đ?‘” [ ] đ?‘?1 [ ] [ ] = [ √đ?‘?2 − 2đ?œ…2 √đ?‘?12 − 2đ?œ…đ?‘”2 √đ?‘?12 − 2đ?œ…đ?‘”2 ] đ?‘” 1 [ ] [ ] đ?œ€2đ?œ…đ?‘” −đ?‘?12 [ ] đ?‘?1 đ?‘?2 [ ] √2 (đ?‘?12 − 2đ?œ…đ?‘”2 ) √2 (đ?‘?12 − 2đ?œ…đ?‘”2 ) √2 (đ?‘?12 − 2đ?œ…đ?‘”2 ) [ ] đ?›ź Ă— [đ?‘Ą] . [đ?œ‚]

(47)

6

Journal of Applied Mathematics

The geodesic curvature đ?œ…đ?‘”đ?›˝ of curve đ?›˝ is given by

đ?œ…đ?‘”đ?›˝

=

1 (đ?‘?12 − 2đ?œ…đ?‘”2 )

5/2

(−2đ?œ…đ?‘” đ?œ† 1 + đ?‘?1 đ?‘?2 đ?œ† 2 −

đ?‘?12 đ?œ† 3 ) ,

curve đ?›˝ : đ??ź ⊂ R → đ?‘†12 (đ?›˝ : đ??ź ⊂ R → đ??ť02 ) of đ?›ź is defined by đ?›˝ (đ?‘ (đ?‘ )) = (48)

where đ?œ† 1 , đ?œ† 2 , and đ?œ† 3 can be calculated as in Theorem 9. Definition 11. Let đ?›ź : đ??ź ⊂ R → đ?‘†12 be a regular unit speed timelike curve lying fully on đ?‘†12 . Then the đ?›źđ?‘Ąđ?œ‚-Smarandache

1 (đ?‘? đ?›ź (đ?‘ ) + đ?‘?2 đ?‘Ą (đ?‘ ) + đ?‘?3 đ?œ‚ (đ?‘ )) , √3 1

where đ?‘?1 , đ?‘?2 , đ?‘?3 ∈ R \ {0} and đ?œ€(đ?‘?12 − đ?‘?22 + đ?‘?32 ) = 3. Theorem 12. Let đ?›ź : đ??ź ⊂ R → đ?‘†12 be a regular unit speed timelike curve lying fully on đ?‘†12 with the Sabban frame {đ?›ź, đ?‘Ą, đ?œ‚} and geodesic curvature đ?œ…đ?‘” . If đ?›˝ : đ??ź ⊂ R → đ?‘†12 (đ?›˝ : đ??ź ⊂ R → đ??ť02 ) is the đ?›źđ?‘Ąđ?œ‚-timelike (hyperbolic) Smarandache curve of đ?›ź, then its frame {đ?›˝, đ?‘Ąđ?›˝ , đ?œ‚đ?›˝ } is given by

đ?‘?3 đ?‘?2 đ?‘?1 √ √ √ 3 3 3 [ ] [ ] đ?›˝ đ?‘?1 + đ?‘?3 đ?œ…đ?‘” đ?‘?2 đ?œ…đ?‘” [ ] đ?›ź đ?‘?2 [ ] [đ?‘Ą] [ đ?‘Ąđ?›˝ ] = [ , ] √đ?œ€đ??´ √đ?œ€đ??´ √đ?œ€đ??´ [ ] đ?œ‚ đ?œ‚ [ ] [ đ?›˝ ] [ −đ?‘?2 đ?œ… + đ?‘? (đ?‘? + đ?‘? đ?œ… ) −đ?‘? đ?‘? đ?œ… + đ?‘? đ?‘? đ?‘? (đ?‘? + đ?‘? đ?œ… ) − đ?‘?2 ] [ ] 3 1 3 đ?‘” 1 1 3 đ?‘” 1 2 đ?‘” 2 3 2 đ?‘” 2 [

√3đ?œ€đ??´

√3đ?œ€đ??´

where đ??´ = (đ?‘?1 + đ?‘?3 đ?œ…đ?‘” )2 −đ?‘?22 −đ?‘?22 đ?œ…đ?‘”2 . If we take đ?œ€ = 1 đ?‘œđ?‘&#x; −1, then the Smarandache curve đ?›˝ is timelike or hyperbolic, respectively. Furthermore, the geodesic curvature đ?œ…đ?‘”đ?›˝ of curve đ?›˝ is given by

√3đ?œ€đ??´

đ?›˝ó¸€ (đ?‘ (đ?‘ )) =

+ (đ?‘?12 + đ?‘?1 đ?‘?3 đ?œ…đ?‘” − đ?‘?22 ) đ?œ† 3 )

đ?‘Ąđ?›˝ 5/2 −1

2

where đ?œ†1 =

(đ?‘?1 + đ?‘?3 đ?œ…đ?‘” ) +

]

đ?‘‘đ?›˝ đ?‘‘đ?‘ 1 (đ?‘? đ?›ź + (đ?‘?1 + đ?‘?3 đ?œ…đ?‘” ) đ?‘Ą + đ?‘?2 đ?œ…đ?‘” đ?œ‚) (53) = đ?‘‘đ?‘ đ?‘‘đ?‘ √3 2

đ?‘‘đ?‘ 1 (đ?‘? đ?›ź + (đ?‘?1 + đ?‘?3 đ?œ…đ?‘” ) đ?‘Ą + đ?‘?2 đ?œ…đ?‘” đ?œ‚) . = đ?‘‘đ?‘ √3 2

(54)

) , (51)

đ?œ€ (đ?‘?2 (−đ?‘?3 đ?œ…đ?‘”ó¸€

(50)

or equivalently

đ?œ…đ?‘”đ?›˝ = ((đ?‘?22 đ?œ…đ?‘” − đ?‘?32 đ?œ…đ?‘” − đ?‘?1 đ?‘?3 ) đ?œ† 1 + (−đ?‘?1 đ?‘?2 đ?œ…đ?‘” + đ?‘?2 đ?‘?3 ) đ?œ† 2

Ă— ((đ?œ€ ((đ?‘?1 + đ?‘?3 đ?œ…đ?‘” ) − đ?‘?22 − đ?‘?22 đ?œ…đ?‘”2 ))

(49)

Taking the Lorentzian inner product in (54) we have

â&#x;¨đ?‘Ąđ?›˝ , đ?‘Ąđ?›˝ â&#x;Šđ??ż (

đ?‘?22 đ?œ…đ?‘” đ?œ…đ?‘”ó¸€ ) 2

2 đ?‘‘đ?‘ 2 1 2 2 2 ) = (đ?‘?2 + đ?‘?2 đ?œ…đ?‘” − (đ?‘?1 + đ?‘?3 đ?œ…đ?‘” ) ) . đ?‘‘đ?‘ 3

(55)

For (đ?‘?1 + đ?‘?3 đ?œ…đ?‘” )2 > đ?‘?22 + đ?‘?22 đ?œ…đ?‘”2 , đ?‘Ąđ?›˝ is a unit timelike vector. It follows that

+ (đ?‘?1 + đ?‘?3 đ?œ…đ?‘” ) ((đ?‘?1 + đ?‘?3 đ?œ…đ?‘” ) − đ?‘?22 − đ?‘?22 đ?œ…đ?‘”2 )) , đ?œ† 2 = đ?œ€ ( (đ?‘?1 + đ?‘?3 đ?œ…đ?‘” ) (−đ?‘?3 đ?œ…đ?‘”ó¸€ (đ?‘?1 + đ?‘?3 đ?œ…đ?‘” ) + đ?‘?22 đ?œ…đ?‘” đ?œ…đ?‘”ó¸€ )

2

2 2 2 đ?‘‘đ?‘ √ (đ?‘?1 + đ?‘?3 đ?œ…đ?‘” ) − đ?‘?2 − đ?‘?2 đ?œ…đ?‘” = . đ?‘‘đ?‘ 3

2

+ (đ?‘?2 + đ?‘?3 đ?œ…đ?‘”ó¸€ + đ?‘?2 đ?œ…đ?‘”2 ) ((đ?‘?1 + đ?‘?3 đ?œ…đ?‘” ) − đ?‘?22 − đ?‘?22 đ?œ…đ?‘”2 )) , đ?œ† 3 = đ?œ€ (đ?‘?2 đ?œ…đ?‘” (−đ?‘?3 đ?œ…đ?‘”ó¸€ (đ?‘?1 + đ?‘?3 đ?œ…đ?‘” ) + đ?‘?22 đ?œ…đ?‘” đ?œ…đ?‘”ó¸€ )

(56)

Therefore, the unit timelike tangent vector of the curve đ?›˝ is given by

+ (đ?œ…đ?‘” (đ?‘?1 + đ?‘?3 đ?œ…đ?‘” ) + đ?‘?2 đ?œ…đ?‘”ó¸€ ) 2

Ă— ((đ?‘?1 + đ?‘?3 đ?œ…đ?‘” ) − đ?‘?22 − đ?‘?22 đ?œ…đ?‘”2 )) . (52) Proof. We take đ?œ€ = 1. By taking the derivative of (49) with respect to đ?‘ and using (4), we get

đ?‘Ąđ?›˝ =

1 2

√ (đ?‘?1 + đ?‘?3 đ?œ…đ?‘” ) − đ?‘?22 − đ?‘?22 đ?œ…đ?‘”2

(đ?‘?2 đ?›ź + (đ?‘?1 + đ?‘?3 đ?œ…đ?‘” ) đ?‘Ą + đ?‘?2 đ?œ…đ?‘” đ?œ‚) . (57)

Journal of Applied Mathematics

7 đ?œ† 3 = đ?‘?2 đ?œ…đ?‘” (−đ?‘?3 đ?œ…đ?‘”ó¸€ (đ?‘?1 + đ?‘?3 đ?œ…đ?‘” ) + đ?‘?22 đ?œ…đ?‘” đ?œ…đ?‘”ó¸€ )

On the other hand, taking the cross-product of (49) with (57) it can be easily seen that

+ (đ?œ…đ?‘” (đ?‘?1 + đ?‘?3 đ?œ…đ?‘” ) + đ?‘?2 đ?œ…đ?‘”ó¸€ )

đ?œ‚đ?›˝ = đ?›˝ ∧ đ?‘Ąđ?›˝

2

Ă— ((đ?‘?1 + đ?‘?3 đ?œ…đ?‘” ) − đ?‘?22 − đ?‘?22 đ?œ…đ?‘”2 ) ,

= ((−đ?‘?22 đ?œ…đ?‘” + đ?‘?3 (đ?‘?1 + đ?‘?3 đ?œ…đ?‘” )) đ?›ź + (đ?‘?2 đ?‘?3 − đ?‘?1 đ?‘?2 đ?œ…đ?‘” ) đ?‘Ą + (đ?‘?1 (đ?‘?1 + đ?‘?3 đ?œ…đ?‘” ) − đ?‘?22 ) đ?œ‚)

(60) (58)

−1

2

đ?‘Ąđ?›˝ó¸€ =

Ă— (√ 3 (đ?‘?1 + đ?‘?3 đ?œ…đ?‘” ) − 3đ?‘?22 − 3đ?‘?22 đ?œ…đ?‘”2 ) . This means that the đ?œ‚đ?›˝ is a unit spacelike vector. In order to obtain the tangent vector of đ?›˝ let us differentiate (57) with respect to đ?‘ . We find đ?‘‘đ?‘Ąđ?›˝ đ?‘‘đ?‘ đ?‘‘đ?‘ đ?‘‘đ?‘

=

1 2

((đ?‘?1 + đ?‘?3 đ?œ…đ?‘” ) −

đ?‘?22

−

3/2 đ?‘?22 đ?œ…đ?‘”2 )

and by combining (56) and (59) we get √3 2

((đ?‘?1 + đ?‘?3 đ?œ…đ?‘” ) − đ?‘?22 − đ?‘?22 đ?œ…đ?‘”2 )

2

(đ?œ† 1 đ?›ź + đ?œ† 2 đ?‘Ą + đ?œ† 3 đ?œ‚) . (61)

Finally, the geodesic curvature đ?œ…đ?‘”đ?›˝ of the curve đ?›˝ = đ?›˝(đ?‘ (đ?‘ )) is given by đ?œ…đ?‘”đ?›˝ = det (đ?›˝, đ?‘Ąđ?›˝ , đ?‘Ąđ?›˝ó¸€ ) = ((đ?‘?22 đ?œ…đ?‘” − đ?‘?32 đ?œ…đ?‘” − đ?‘?1 đ?‘?3 ) đ?œ† 1 + (−đ?‘?1 đ?‘?2 đ?œ…đ?‘” + đ?‘?2 đ?‘?3 ) đ?œ† 2

(đ?œ† 1 đ?›ź + đ?œ† 2 đ?‘Ą + đ?œ† 3 đ?œ‚) ,

2

Ă— (((đ?‘?1 + đ?‘?3 đ?œ…đ?‘” ) − đ?‘?22 − đ?‘?22 đ?œ…đ?‘”2 )

where

+ (đ?‘?1 + đ?‘?3 đ?œ…đ?‘” ) ((đ?‘?1 + đ?‘?3 đ?œ…đ?‘” ) −

5/2 −1

) .

The proof of đ?œ€ = −1 case is similar.

đ?œ† 1 = đ?‘?2 (−đ?‘?3 đ?œ…đ?‘”ó¸€ (đ?‘?1 + đ?‘?3 đ?œ…đ?‘” ) + đ?‘?22 đ?œ…đ?‘” đ?œ…đ?‘”ó¸€ ) 2

(62)

+ (đ?‘?12 + đ?‘?1 đ?‘?3 đ?œ…đ?‘” − đ?‘?22 ) đ?œ† 3 )

(59)

đ?‘?22

−

đ?‘?22 đ?œ…đ?‘”2 ) ,

đ?œ† 2 = (đ?‘?1 + đ?‘?3 đ?œ…đ?‘” ) (−đ?‘?3 đ?œ…đ?‘”ó¸€ (đ?‘?1 + đ?‘?3 đ?œ…đ?‘” ) + đ?‘?22 đ?œ…đ?‘” đ?œ…đ?‘”ó¸€ ) 2

+ (đ?‘?2 + đ?‘?3 đ?œ…đ?‘”ó¸€ + đ?‘?2 đ?œ…đ?‘”2 ) ((đ?‘?1 + đ?‘?3 đ?œ…đ?‘” ) − đ?‘?22 − đ?‘?22 đ?œ…đ?‘”2 ) ,

Corollary 13. Let đ?›ź : đ??ź ⊂ R → đ?‘†12 be a regular unit speed timelike curve lying fully on đ?‘†12 with the Sabban frame {đ?›ź, đ?‘Ą, đ?œ‚} and geodesic curvature đ?œ…đ?‘” . If đ?›˝ : đ??ź ⊂ R → đ?‘†12 is the đ?›źđ?‘Ąđ?œ‚spacelike Smarandache curve of đ?›ź, then, for đ??´ = đ?‘?22 + đ?‘?22 đ?œ…đ?‘”2 − (đ?‘?1 + đ?‘?3 đ?œ…đ?‘” )2 , its frame {đ?›˝, đ?‘Ąđ?›˝ , đ?œ‚đ?›˝ } is given by

đ?‘?3 đ?‘?2 đ?‘?1 √3 √3 √3 [ ] [ ] [ ] đ?›˝ đ?‘?1 + đ?‘?3 đ?œ…đ?‘” đ?‘?2 đ?œ…đ?‘” [ ] đ?›ź đ?‘?2 [ ] [đ?‘Ą] [ đ?‘Ąđ?›˝ ] = [ . ] √ √ √ đ??´ đ??´ đ??´ [ ] đ?œ‚ đ?œ‚ ][ ] [ đ?›˝] [ [ 2 ] [ −đ?‘?2 đ?œ…đ?‘” + đ?‘?3 (đ?‘?1 + đ?‘?3 đ?œ…đ?‘” ) −đ?‘?1 đ?‘?2 đ?œ…đ?‘” + đ?‘?2 đ?‘?3 đ?‘?1 (đ?‘?1 + đ?‘?3 đ?œ…đ?‘” ) − đ?‘?22 ] √3đ??´

[

√3đ??´

Furthermore, the geodesic curvature đ?œ…đ?‘”đ?›˝ of curve đ?›˝ is given by

(64) −1 2 5/2

Ă— ((đ?‘?22 + đ?‘?22 đ?œ…đ?‘”2 − (đ?‘?1 + đ?‘?3 đ?œ…đ?‘” ) )

) ,

]

Example 14. Let us consider a unit speed timelike curve đ?›ź on đ?‘†12 defined by đ?›ź (đ?‘ ) = (√2 sinh đ?‘ , cosh đ?‘ , sinh đ?‘ ) .

đ?œ…đ?‘”đ?›˝ = ((đ?‘?22 đ?œ…đ?‘” − đ?‘?32 đ?œ…đ?‘” − đ?‘?1 đ?‘?3 ) đ?œ† 1 + (−đ?‘?1 đ?‘?2 đ?œ…đ?‘” + đ?‘?2 đ?‘?3 ) đ?œ† 2 + (đ?‘?12 + đ?‘?1 đ?‘?3 đ?œ…đ?‘” − đ?‘?22 ) đ?œ† 3 )

√3đ??´

(65)

Then the orthonormal Sabban frame {đ?›ź(đ?‘ ), đ?‘Ą(đ?‘ ), đ?œ‚(đ?‘ )} of đ?›ź can be calculated as follows: đ?›ź (đ?‘ ) = (√2 sinh đ?‘ , cosh đ?‘ , sinh đ?‘ ) , đ?‘Ą (đ?‘ ) = (√2 cosh đ?‘ , sinh đ?‘ , cosh đ?‘ ) ,

where đ?œ† 1 , đ?œ† 2 , and đ?œ† 3 can be calculated as in Theorem 12.

(63)

đ?œ‚ (đ?‘ ) = (−1, 0, −√2) .

(66)

8

Journal of Applied Mathematics

The geodesic curvature of đ?›ź is 0. In terms of the definitions, we obtain Smarandache curves according to Sabban frame on đ?‘†12 . Firstly, when we take đ?‘?1 = 1 and đ?‘?2 = 1, then the timelike đ?›źđ?œ‚-Smarandache curve is given by đ?›˝ (đ?‘ (đ?‘ )) =

1 √ ( 2 sinh đ?‘ − 1, cosh đ?‘ , sinh đ?‘ − √2) √2

(68)

1 (3√2 sinh đ?‘ + √14 cosh đ?‘ , √2 3 cosh đ?‘ + √7 sinh đ?‘ , 3 sinh đ?‘ + √7 cosh đ?‘ ) , (69)

and if we take đ?‘?1 = √7 and đ?‘?2 = 3, then the hyperbolic đ?›źđ?‘ĄSmarandache curve is given by đ?›˝ (đ?‘ (đ?‘ )) =

1 √ ( 14 sinh đ?‘ + 3√2 cosh đ?‘ , √2 √7 cosh đ?‘ + 3 sinh đ?‘ , √7 sinh đ?‘ + 3 cosh đ?‘ ) . (70)

Thirdly, when we take đ?‘?1 = √2 and đ?‘?2 = 2, then the spacelike đ?‘Ąđ?œ‚-Smarandache curve is given by đ?›˝ (đ?‘ (đ?‘ )) = (√2 (cosh đ?‘ − 1) , sinh đ?‘ , cosh đ?‘ − 2)

(71)

and if we take đ?‘?1 = 2 and đ?‘?2 = √2, then the hyperbolic đ?‘Ąđ?œ‚Smarandache curve is given by đ?›˝ (đ?‘ (đ?‘ )) = (2 cosh đ?‘ − 1, √2 sinh đ?‘ , √2 (cosh đ?‘ − 1)) . (72) Finally, when we take đ?‘?1 = 2, đ?‘?2 = √2, and đ?‘?3 = 1, then the đ?›źđ?‘Ąđ?œ‚-Smarandache curve is a timelike curve and given by đ?›˝ (đ?‘ (đ?‘ )) =

1 (2√2 sinh đ?‘ + 2 cosh đ?‘ − 1, √3 2 cosh đ?‘ + √2 sinh đ?‘ , 2 sinh đ?‘ + √2 cosh đ?‘ − √2) ,

1 (2 sinh đ?‘ + 4 cosh đ?‘ − √3, √3 (74)

√2 cosh đ?‘ + 2√2 sinh đ?‘ , √2 sinh đ?‘ + 2√2 cosh đ?‘ − √6) .

On the other hand, in the last case, if we take đ?‘?1 = 1, đ?‘?2 = √2, and đ?‘?3 = 2 (i.e., đ?‘?1 < đ?‘?2 < đ?‘?3 ), then the đ?›źđ?‘Ąđ?œ‚-Smarandache curve is spacelike and given by đ?›˝ (đ?‘ (đ?‘ )) =

1 √ ( 2 sinh đ?‘ + 2 cosh đ?‘ − 2, √3 (75)

cosh đ?‘ + √2 sinh đ?‘ ,

and its geodesic curvature đ?œ…đ?‘”đ?›˝ is −1. Here the hyperbolic đ?›źđ?œ‚Smarandache curve is undefined. Secondly, when we take đ?‘?1 = 3 and đ?‘?2 = √7, then the timelike đ?›źđ?‘Ą-Smarandache curve is given by đ?›˝ (đ?‘ (đ?‘ )) =

đ?›˝ (đ?‘ (đ?‘ )) =

(67)

and the Sabban frame of the đ?›źđ?œ‚-Smarandache curve is given by 1 1 [ √2 0 √2 ] [ ] đ?›ź đ?›˝ [ ] [ đ?‘Ąđ?›˝ ] = [ 0 1 0 ] [ đ?‘Ą ] , [ ] [ ] [đ?œ‚đ?›˝ ] [ 1 0 1 ] [ đ?œ‚ ] √2 √2 [ ]

and if we take đ?‘?1 = √2, đ?‘?2 = 2√2, and đ?‘?3 = √3, then the đ?›źđ?‘Ąđ?œ‚-Smarandache curve is a hyperbolic curve and given by

(73)

sinh đ?‘ + √2 cosh đ?‘ − 2√2) . The Sabban frames and geodesic curvatures of đ?›źđ?‘Ą, đ?‘Ąđ?œ‚, and đ?›źđ?‘Ąđ?œ‚-Smarandache curves can be easily obtained by using a similar way to the above. Also we give the curve đ?›ź and its Smarandache partners in Figure 1.

Appendix In this section, we give as a table different Smarandache curves on de Sitter space or on hyperbolic space for spacelike curves in Minkowski-space. We give the theorem about undefined Smarandache curve below. The other đ?›źđ?œ‚-, đ?›źđ?‘Ą-, đ?‘Ąđ?œ‚-, and đ?›źđ?‘Ąđ?œ‚-Smarandache curves on đ?‘†12 and đ?›źđ?œ‚-, đ?‘Ąđ?œ‚-, đ?›źđ?‘Ąđ?œ‚Smarandache curves on đ??ť02 and their corresponding Sabban frames and geodesic curvatures are similar to those in the previous section. Theorem A.1. Let đ?›ź : đ??ź ⊂ R → đ?‘†12 be a regular unit speed spacelike curve lying fully on đ?‘†12 . Then the đ?›źđ?‘Ą-Smarandache curve đ?›˝ : đ??ź ⊂ R → đ??ť02 of đ?›ź does not exist. Proof. Let đ?›ź : đ??ź ⊂ R → đ?‘†12 be a regular unit speed spacelike curve lying fully on đ?‘†12 . Then the đ?›źđ?‘Ą-Smarandache curve đ?›˝ : đ??ź ⊂ R → đ??ť02 of đ?›ź can be written as follows: đ?›˝ (đ?‘ (đ?‘ )) =

1 (đ?‘? đ?›ź (đ?‘ ) + đ?‘?2 đ?‘Ą (đ?‘ )) , √2 1

(A.1)

đ?‘?1 , đ?‘?2 ∈ R \ {0}, and đ?‘?12 + đ?‘?22 = −2 which is contradiction. The Smarandache curves of a regular unit spacelike curve đ?›ź are given in Table 1. Example A.2. Let us consider a unit speed spacelike curve đ?›ź on đ?‘†12 defined by đ?›ź (đ?‘ ) = (

(đ?‘ − 1)2 (đ?‘ − 1)2 , − 1, đ?‘ − 1) . 2 2

(A.2)

Journal of Applied Mathematics

9

� is a timelike curve on S12 � is a timelike or spacelike curve

đ?›˝ is a hyperbolic curve

đ?›źđ?œ‚-Smarandache curve

đ?›źđ?œ‚-Smarandache curve is undefined

�t-Smarandache curve

�t-Smarandache curve

tđ?œ‚-Smarandache curve (spacelike)

tđ?œ‚-Smarandache curve

đ?›źđ?œ‚t-Smarandache curve (timelike)

đ?›źđ?œ‚t-Smarandache curve

Figure 1: Smarandache curves of a timelike curve �.

10

Journal of Applied Mathematics Table 1: Classification of the Smarandache curves for spacelike curve đ?›ź. đ?›ź is a spacelike curve on đ?‘†12 đ?›˝ is a spacelike or timelike curve 1 (đ?‘? đ?›ź + đ?‘?2 đ?œ‚), đ?›˝ (đ?‘ (đ?‘ )) = √2 1 đ?‘?12 − đ?‘?22 = 2 1 (đ?‘? đ?›ź + đ?‘?2 đ?‘Ą), đ?›˝ (đ?‘ (đ?‘ )) = √2 1 đ?‘?12 + đ?‘?22 = 2 1 (đ?‘? đ?‘Ą + đ?‘?2 đ?œ‚), đ?›˝ (đ?‘ (đ?‘ )) = √2 1 đ?‘?12 − đ?‘?22 = 2 1 đ?›˝ (đ?‘ (đ?‘ )) = (đ?‘? đ?›ź + đ?‘?2 đ?‘Ą + đ?‘?3 đ?œ‚), √3 1 đ?‘?12 + đ?‘?22 − đ?‘?32 = 3

đ?›źđ?œ‚

��

đ?‘Ąđ?œ‚

đ?›źđ?‘Ąđ?œ‚

Then the orthonormal Sabban frame {đ?›ź(đ?‘ ), đ?‘Ą(đ?‘ ), đ?œ‚(đ?‘ )} of đ?›ź can be calculated as follows: đ?›ź (đ?‘ ) = (

(đ?‘ − 1)2 (đ?‘ − 1)2 , − 1, đ?‘ − 1) , 2 2

đ?‘Ą (đ?‘ ) = (đ?‘ − 1, đ?‘ − 1, 1) , đ?œ‚ (đ?‘ ) = (

đ?›˝ (đ?‘ (đ?‘ )) = (A.4)

√2 + 1 ) (đ?‘ − 1)2 + 1, 2

√2 + 1 ( ) (đ?‘ − 1)2 − √2, (đ?‘ − 1) (√2 + 1) ) , 2 (A.5) and also when we take đ?‘?1 = √2 and đ?‘?2 = 2, then the hyperbolic đ?›źđ?œ‚-Smarandache curve is given by

(

The {đ?›˝, đ?‘Ąđ?›˝ , đ?œ‚đ?›˝ } Sabban frames and geodesic curvatures đ?œ…đ?‘”đ?›˝ are similar to the above section. Secondly, we take đ?‘?1 = 1 and đ?‘?2 = 1; then the spacelike Smarandache-đ?›źđ?‘Ą curve is given by 1 (đ?‘ − 1)2 (đ?‘ − 1)2 ( + đ?‘ − 1, + đ?‘ − 2, đ?‘ ) . √2 2 2 (A.7)

Here the hyperbolic đ?›źđ?‘Ą-Smarandache curve is undefined. Thirdly, we take đ?‘?1 = 2 and đ?‘?2 = √2; then the spacelike đ?‘Ąđ?œ‚-Smarandache curve is given by

In terms of the definitions, we obtain Smarandache curves according to Sabban frame on đ?‘†12 . Firstly, we take đ?‘?1 = 2 and đ?‘?2 = √2; then the spacelike đ?›źđ?œ‚-Smarandache curve is given by

đ?›˝ (đ?‘ (đ?‘ )) = ((

1 (đ?‘? đ?‘Ą + đ?‘?2 đ?œ‚), √2 1 đ?‘?12 − đ?‘?22 = −2 1 đ?›˝ (đ?‘ (đ?‘ )) = (đ?‘? đ?›ź + đ?‘?2 đ?‘Ą + đ?‘?3 đ?œ‚), √3 1 đ?‘?12 + đ?‘?22 − đ?‘?32 = −3 đ?›˝ (đ?‘ (đ?‘ )) =

(A.3)

The geodesic curvature of � is expressed as

đ?›˝ (đ?‘ (đ?‘ )) = ((

undefined

đ?›˝ (đ?‘ (đ?‘ )) =

(đ?‘ − 1)2 (đ?‘ − 1)2 + 1, , đ?‘ − 1) . 2 2

đ?œ…đ?‘” (đ?‘ ) = −1.

đ?›˝ is a hyperbolic curve 1 (đ?‘? đ?›ź + đ?‘?2 đ?œ‚), đ?›˝ (đ?‘ (đ?‘ )) = √2 1 đ?‘?12 − đ?‘?22 = −2

√2 + 1 ) (đ?‘ − 1)2 + √2, 2 √2 + 1 ) (đ?‘ − 1)2 − 1, (đ?‘ − 1) (√2 + 1) ) . 2 (A.6)

1 √2 ( (đ?‘ − 1)2 + 2đ?‘ − 2 + √2, √2 2 √2 (đ?‘ − 1)2 + 2đ?‘ − 2, √2 (đ?‘ − 1) + 2) , 2 (A.8)

and also when we take đ?‘?1 = √2 and đ?‘?2 = 2, then the hyperbolic đ?‘Ąđ?œ‚-Smarandache curve is given by đ?›˝ (đ?‘ (đ?‘ )) =

1 ((đ?‘ − 1)2 + √2 (đ?‘ − 1) + 2, √2 (đ?‘ − 1)2 + √2 (đ?‘ − 1) , 2 (đ?‘ − 1) + √2) . (A.9)

Finally, when đ?‘?1 = 2, đ?‘?2 = √2, and đ?‘?3 = √3, then the đ?›źđ?‘Ąđ?œ‚Smarandache curve is a spacelike curve and is given by đ?›˝ (đ?‘ (đ?‘ )) =

√3 1 ((đ?‘ − 1)2 ( + 1) + √2 (đ?‘ − 1) + √3, √3 2 (đ?‘ − 1)2 (

√3 + 1) + √2 (đ?‘ − 1) − 2, 2

(đ?‘ − 1) (√3 + 2) + √2) (A.10)

Journal of Applied Mathematics

11

� is a spacelike curve on S12 � is a timelike or spacelike curve

đ?›˝ is a hyperbolic curve

đ?›źđ?œ‚-Smarandache curve

đ?›źđ?œ‚-Smarandache curve

�t-Smarandache curve

�t-Smarandache curve is undefined

tđ?œ‚-Smarandache curve

tđ?œ‚-Smarandache curve

đ?›źđ?œ‚t-Smarandache curve

đ?›źđ?œ‚t-Smarandache curve

Figure 2: Smarandache curves of a spacelike curve �.

12

Journal of Applied Mathematics

and also when we take đ?‘?1 = 2, đ?‘?2 = √2, and đ?‘?3 = 3, then the hyperbolic đ?›źđ?‘Ąđ?œ‚-Smarandache curve is given by đ?›˝ (đ?‘ (đ?‘ )) =

1 5 ( (đ?‘ − 1)2 + √2 (đ?‘ − 1) + 3, √3 2 5 (đ?‘ − 1)2 + √2 (đ?‘ − 1) − 3, 2

(A.11)

5 (đ?‘ − 1) + √2) . The Sabban frames and geodesic curvatures of the đ?›źđ?œ‚, đ?›źđ?‘Ą, đ?‘Ąđ?œ‚ and đ?›źđ?‘Ąđ?œ‚-Smarandache curves can be easily obtained by using methods similar to those in the previous section. Furthermore, we give curve the đ?›ź and its Smarandache partners in Figure 2.

Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.

References [1] M. Turgut and S. Yilmaz, “Smarandache curves in Minkowski space-time,â€? International Journal of Mathematical Combinatorics, vol. 3, pp. 51–55, 2008. [2] A. T. Ali, “Special smarandache curves in the euclidean space,â€? International Journal of Mathematical Combina Torics, vol. 2, pp. 30–36, 2010. [3] K. Tas¸k¨opr¨u and M. Tosun, “Smarandache curves on S2 ,â€? Boletim da Sociedade Paranaense de Matem´atica, vol. 32, no. 1, pp. 51–59, 2014. [4] E. B. Koc Ozturk, U. Ozturk, K. Ilarslan, and E. Nesovic, “On pseudohyperbolical Smarandache curves in Minkowski 3space,â€? International Journal of Mathematics and Mathematical Sciences, vol. 2013, Article ID 658670, 7 pages, 2013. [5] O. Bektas and S. Yuce, “Special smarandache curves according to darboux frame in Euclidean 3-space,â€? Romanian Journal of Mathematics and Computer Science, vol. 3, no. 1, pp. 48–59, 2013. [6] M. Cetin, Y. Tuncer, and M. K. Karacan, “Smarandache curves according to bishop frame in euclidean 3-space,â€? General Mathematics Notes, vol. 20, no. 2, pp. 50–66, 2014. [7] T. Kahraman, M. Onder, and H. H. Ugurlu, “Dual smarandache curves and smarandache ruled surfaces,â€? Mathematical Sciences and Applications E, vol. 2, no. 1, pp. 83–98, 2014. [8] B. O’Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, San Diego, Calif, USA, 1983. [9] T. Sato, “Pseudo-spherical evolutes of curves on a spacelike surface in three dimensional Lorentz-Minkowski space,â€? Journal of Geometry, vol. 103, no. 2, pp. 319–331, 2012. [10] S. Izumiya, D. H. Pei, T. Sano, and E. Torii, “Evolutes of hyperbolic plane curves,â€? Acta Mathematica Sinic, vol. 20, no. 3, pp. 543–550, 2004. [11] V. Asil, T. Korpinar, and S. Bas, “Inextensible ows of timelike curves with Sabban frame in S21,â€? Siauliai Mathematical Seminar, vol. 7, no. 15, pp. 5–12, 2012.

Advances in

Operations Research Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Advances in

Decision Sciences Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Journal of

Applied Mathematics

Algebra

Hindawi Publishing Corporation http://www.hindawi.com

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Journal of

Probability and Statistics Volume 2014

The Scientific World Journal Hindawi Publishing Corporation http://www.hindawi.com

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

International Journal of

Differential Equations Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Volume 2014

Submit your manuscripts at http://www.hindawi.com International Journal of

Advances in

Combinatorics Hindawi Publishing Corporation http://www.hindawi.com

Mathematical Physics Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Journal of

Complex Analysis Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

International Journal of Mathematics and Mathematical Sciences

Mathematical Problems in Engineering

Journal of

Mathematics Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Volume 2014

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Discrete Mathematics

Journal of

Volume 2014

Hindawi Publishing Corporation http://www.hindawi.com

Discrete Dynamics in Nature and Society

Journal of

Function Spaces Hindawi Publishing Corporation http://www.hindawi.com

Abstract and Applied Analysis

Volume 2014

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

International Journal of

Journal of

Stochastic Analysis

Optimization

Hindawi Publishing Corporation http://www.hindawi.com

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Volume 2014

Turn static files into dynamic content formats.

Create a flipbook
Issuu converts static files into: digital portfolios, online yearbooks, online catalogs, digital photo albums and more. Sign up and create your flipbook.