Special Curves According to Bishop Frame in Minkowski 3-Space

Muhammad Abubakar Isah 1  and Mihriban Alyamaç Külahçı 1
  • 1 Department of Mathematics, Elazig, Turkey
Muhammad Abubakar Isah and Mihriban Alyamaç Külahçı

Abstract

Pseudo null curves were studied by some geometers in both Euclidean and Minkowski spaces, but some special characters of the curve are not considered. In this paper, we study weak AW (k) – type and AW (k) – type pseudo null curve in Minkowski 3-space E13 . We define helix and slant helix according to Bishop frame in E13 . Furthermore, the necessary and sufficient conditions for the slant helix and helix in Minkowski 3-space are obtained.

1 Introduction

The ability to “ride” along a three-dimensional space curve and illustrate the properties of the curve, such as curvature and torsion, would be a great asset to Mathematicians. The classic Serret-Frenet frame provides such ability. The tangent normal, and binormal vector fields are called the Frenet–Serret frame or T NB frame. But, the curve might not be continuous at some points, which is undefined when the second derivative of the curve vanishes [9]. In 1975, Richard Lawrence Bishop first introduced the parallel frame as a new frame which is well defined even if the curve has vanishing second derivative, then the parallel frame came to be called the Bishop frame [8, 9, 10]. Bishop frame contains the tangential vector field T and two normal vector fields N1 and N2. The Bishop frame may have applications in the area of Biology and Computer Graphics. For example, it may be possible to compute information about the shape of sequences of DNA using a curve defined by the Bishop frame. It also provides a new way to control virtual cameras in computer animation [12]. Some applications of the Bishop frames in Minkowski spaces can be found in [3, 4].

In differential geometry, a general helix or a curve of constant slope in Euclidean 3-space E3 is defined in such a way that the tangent makes a constant angle with a fixed direction. A classical result stated by M. A. Lancret in 1802 and first proved by B. de Saint Venant in 1845, [5, 7, 16, 18]. For helical structures in nature, helices arise in nano-springs, carbon nano-tubes, DNA double and collagen triple helix, lipid bilayers, bacterial flagella in salmonella and escherichia coli, aerial hyphae in actinomycetes, bacterial shape in spirochetes, horns, tendrils, vines, screws, springs, helical staircases and seashells [1, 2, 11]. Helical structures are used in fractal geometry, for instance, hyper-helices.

In [7], a slant helix in Euclidean 3-space was defined by the property that the principal normal makes a constant angle with a fixed direction. Moreover, Izumiya and Takeuchi showed that α is a slant helix in E3 if and only if the geodesic curvature of the principal normal of a space curve α is a constant function [15, 17, 19].

2 Preliminaries

Definition 1

The Minkowski 3-spaceE13is the real vector space E3which is endowed with the standard indefinite flat metric 〈.,.〉 defined by

u,v=u1v1+u2v2+u3v3,
for any two vectors u = (u1,u2,u3) and v = (v1,v2,v3) in E13 . Since〈.,.〉 is an indefinite metric, an arbitrary vector u ∈E13 {0} can have one of three causal characters:
  1. i)it can be space-like, ifu,u1> 0,
  2. ii)time-like, ifu,u1< 0 or
  3. iii)light-like or isotropic or null vector, ifu,u1 = 0 but u ≠0.

In particular, the norm (length) of a non-lightlike vector u ∈E13is given by

u=|u,u|˙

Given a regular curveα:IE13can locally be spacelike, timelike or null (lightlike), if all of its velocity vectors α(t) satisfyα(t),α(t)〉1> 0, 〈α(t),α(t)〉1< 0 orα(t),α(t)〉1= 0 respectively, at any tɛI, whereα(t)=dαdt .

Definition 2

A spacelike curveα:IE13is called a pseudo null curve, if its principal normal vector field N and binormal vector filed B are null vector fields satisfying the conditionN,B〉 = 1. The Frenet formulae of a non-geodesic pseudo null curve α has the form

[TNB]=[0k00τ0k0τ][TNB],
where the curvature k(s) = 1 and the torsion τ(s) is an arbitrary function in arc-length parameter s of α. The Frenet’s frame vectors satisfy the equations
N,B=1,T,N=T,B=0,T,T=1,N,N=B,B=0
and
T×N=N,N×B=T,B×T=B.

The frame {T,N,B} is positively oriented, if det (T,N,B) = [T,N,B] = 1 [10].

2.1 The Bishop Frame

The Bishop frame or relatively parallel adapted frame {T,N1,N2} of a regular curve in Euclidean 3-space contains a tangential vector field T and two normal vector fields N1 and N2, which can be obtained by rotating the Frenet vectors N and B in the normal plane T of the curve, in such a way that they become relatively parallel. This means that their derivatives N1 and N2 with respect to the arc-length parameter s of the curve are collinear with the tangential vector field T [10].

Remark 1

We can also define N1and N2to be relatively parallel if the normal component T1 = span{N1,N2} of their derivatives N1and N2is zero, which implies that the mentioned derivatives are collinear with T1.

2.2 The Bishop frame of a pseudo null curve in E13

The Bishop frame {T1,N1,N2} of a pseudo null curve in E13 is positively oriented pseudo orthonormal frame consisting of the tangential vector field T1 and two relatively parallel lightlike normal vector fields N1 and N2. Bishop vector N1 (of the first Bishop frame) can be obtained by applying the hyperbolic rotation to the principal normal vector N, while the normal Bishop vector N2 (of the first Bishop frame) can be obtained by applying the composition of three rotations about two lightlike and one spacelike axis to the binormal vector B [10].

Theorem 1

Let α be a pseudo null curve inE13parameterized by arc-length parameter s with the curvature k(s) = 1 and the torsion τ(s). Then the Bishop frame {T1,N1,N2} and the Frenet frame {T,N,B} of α are related by

[T1N1N2]=[10001k2000k2][TNB],
and the Frenet equations of α according to the Bishop frame read
[T1N1N2]=[0k2k1k100k200][T1N1N2],
where k1(s) = 0 and k2(s) = c0eτ(s)ds, c0R0+ .

Which satisfies the conditions

T1,T1=1,N1,N1=N2,N2=0,N1,N2=1,T1,N2=T1,N1=0[10]˙

3 Curves of AW(k)- type

Proposition 1

Let α be a Frenet curve of osculating order 3 inE13 , by using the Bishop frame of pseudo null curve(2.6), then we have

α(s)=T1(s),α(s)=k2N1+k1N2,α(s)=2k1k2T1+k2N1+k1N2,α(s)=(3k1k23k1k2)T1+(k22k1k22)N1+(k12k12k2)N2˙

Notation 1

Let us write

M1(s)=k2N1+k1N2,
M2(s)=k2N1+k1N2,
M3(s)=(k22k1k22)N1+(k12k12k2)N2.

Corollary 1

α (s), α (s), α″′ (s) and α″″ (s) are linearly dependent if and only if M1 (s), M2 (s) and M3 (s) are linearly dependent.

Definition 3

Frenet curves of osculating order 3 are :

  1. i)of type weak AW (2) if they satisfy
    M3(s)=M3(s),M2(s)M2(s),
  2. ii)of type weak AW (3) if they satisfy
    M3(s)=M3(s),M1(s)M1(s),
    where
    M1(s)=M1(s)M1(s),
    M2(s)=M2(s)M2(s),M1(s)M1(s)M2(s)M2(s),M1(s)M1(s),
  3. iii)of type AW (1) if they satisfy
    M3(s)=0,
  4. iv)of type AW (2) if they satisfy
    M2(s)2M3(s)=M3(s),M2(s)M2(s),
  5. v)of type AW (3) if they satisfy
    M1(s)2M3(s)=M3(s),M1(s)M1(s).

Proposition 2

Suppose that α is a Frenet curve of osculating order 3 inE13, then α is AW (1)-type if and only if

k1=2k12k2,
k2=2k1k22.

Proof

Since α is a curve of AW (1)-type, then α must satisfy (3.8)

M3(s)=(k22k1k22)N1+(k12k12k2)N2,0=(k22k1k22)N1+(k12k12k2)N2˙

Since N1 and N2 are linearly independent, then we have

k22k1k22=0k2=2k1k22,
and
k12k12k2=0k1=2k12k2,
which completes the proof of the proposition.

Proposition 3

Let α be a Frenet curve of osculating order 3 inE13 , then α is AW (2)-type if and only if

k1k22k12k2k2=k1k22k1k1k22.

Proof

Suppose that α is a Frenet curve of osculating order 3. From (2.14) and (2.15) we can write

M2(s)=β(s)N1+γ(s)N2,M3(s)=δ(s)N1+η(s)N2,
where β (s), γ (s), δ (s) and η (s) are differential functions. Since M2 (s) and M3 (s) are linearly dependent, then the determinant of the coefficients of N1 and N2 is equal to zero and hence one can write
|β(s)γ(s)δ(s)η(s)|=0,
where
β(s)=k2,γ(s)=k1,δ(s)=k22k1k22,η(s)=k12k12k2.

By substituting (3.15) in (3.14), we can obtain (3.13), which completes the proof of the proposition.

Proposition 4

Let α be a Frenet curve of osculating order 3 inE13, then α is AW (3)-type if and only if

k1k2=k1k2.

Proof

Suppose that α is a Frenet curve of osculating order 3, then

M1(s)=β(s)N1+γ(s)N2,M3(s)=δ(s)N1+η(s)N2,
where β (s), γ (s), δ (s) and η (s) are differential functions. Since M2 (s) and M3 (s) are linearly dependent, then the determinant of the coefficients of N1 and N2 is equal to zero and hence we can write
β(s)η(s)γ(s)δ(s)=0
where
β(s)=k2,γ(s)=k1,δ(s)=k22k1k22,η(s)=k12k12k2˙

Considering these equations in (3.17), we get (3.16), which completes the proof of the proposition.

Proposition 5

Let α be a Frenet curve of osculating order 3 inE13, then α is of weak AW (2)−type if and only if

(k22k1k22)=pp2+q2[(k22k1k22)q+(k12k12k2)p],
(k12k12k2)=qp2+q2[(k22k1k22)q+(k12k12k2)p],
where
p=(k22+k12)k2(k1k2)k2andq=(k22+k12)k1(k1k2)k1˙

Proposition 6

Let α be a Frenet curve of osculating order 3 inE13, then α is of weak AW (3)−type if and only if

k22k1k22=k2k22+k12[k1k2+k1k24k12k22],
k12k12k2=k1k22+k12[k1k2+k1k24k12k22].

4 The Slant helices according to Bishop frame of the pseudo null curve in Minkowski 3-space

Definition 4

Helix is a curve whose tangent lines make a constant angle with a fixed direction. Helices are characterized by the fact that the ratiok1k2is constant along the curve.

Definition 5

A unit speed curve α is called a slant helix if there exists a non-zero constant vector field U ɛE13such that the functionN (s),Uis constant.

It is important to point out, in contrast to what happens in E3, we cannot define the angle between two vectors (except that both vectors are of time-like). For this reason, we avoid to say about the angle between the normal vector field N (s) and U [14].

Theorem 2

Let α be a pseudo null curve inE13, then α is a general helix if and only ifk1k2is constant.

Proof

Let α be a general helix. The slope axis of the curve α is shown as sp{U} . note that

T,U=c(cisconstant).

If we differentiate both sides of the equation (4.1), then we have

T,U=0˙

By using (2.10) and (4.44)

k2N1+k1N2,U=0,k2cosθ+k1sinθ=0,k1k2=cotθ(constant),
as desired.

Theorem 3

Let α be a pseudo null curve inE13, then α is a slant helix if and only ifk1k2is constant.

Proof

Let α be a slant helix in E13 and 〈N (s),U〉 is constant. Then α is a slant helix, from the definition we have

N(s),U=c(cisaconstant),
where U is a constant vector in E13 . By differentiating (4.4) and using (2.6)
N(s),U=0,k1T1,U=0,k10.

Hence

T1,U=0,
uɛsp{N1,N2}, therefore u=cosθ N1 + sinθ N2. U is a linear combination of N1 and N2. By differentiating (4.5) and using (2.6)
T,U=0,k2cosθ+k1sinθ=0,k1k2=cotθ,
as desired.

Theorem 4

Let α be a pseudo null curve inE13, then α is a slant helix if and only if

det(N1,N1,N1)=0.

Proof

(⇒) Suppose that k1k2 be constant. We have equalities as

N1=k1T,N1=k1Tk1k2N1k12N2,N1=(2k12k2k1)T+(2k1k2k1k2)N13k1k1N2˙

So we get

det(N1,N1,N1)=|k100k1k1k2k12(2k12k2k1)(2k1k2+k1k2)3k1k1|,det(N1,N1,N1)=k13k22(k1k2)˙
Since α is a slant helix and k1k2 is constant. Hence, we have
det(N1,N1,N1)=0,butk20.
(⇐) Suppose that det(N1,N1,N″′1) = 0, then it is clear that the k1k2 is constant, since (k1k2) is zero. Hence the theorem is proved.

Theorem 5

Let α be a pseudo null curve inE13, then α is a slant helix if and only if

det(N2,N2,N2)=0.

Proof

(⇒) Suppose that k1k2 be constant. From (2.6) we have

N2=k2T,
therefore
N2=k2Tk22N1k1k2N2,N2=(2k1k22k2)T+(k222k2k2)N1+(k1k2k1k2k1k2)N2˙

So we get

det(N2,N2,N2)=|k200k2k22k1k2(k1k22k2)(22+2k2k2)(k21+k1k2+k1k2)|,det(N2,N2,N2)=k25(k1k2).

Since α is a slant helix and k1k2 is constant. Hence, we have

det(N2,N2,N2)=0,butk20.
(⇐) Suppose that det(N2,N2,N″′2) = 0, then it is clear that the k1k2 is constant, since (k1k2) is zero. Hence the theorem is proved.

From (2.6)

α(s)=T,DTT=k2N1+k1N2,DTN1=k1T1,DTN2=k2T1,

Theorem 6

Letα:IE13be a unit speed pseudo null curve on M1then α is a general slant helix if and only if

DT(DTDTN1)+3k1DTT=(k12k12k2)1k1DTN1.

Proof

(⇒) Suppose that α is a general slant helix. Then, from (4.8), we have

DTN1=k1T1,DT(DTN1)=k1T1k1k2N1k12N2,DT(DTDTN1)=(k12k2k1)T1k1DTT(k1k2+k1k2)N1
k1k2DTN12k1k1N2.

Since α is a general helix

k1k2=ccisconstant,
by differentiating (4.12), we get
(k1k2)=2k1k2,
but
DTN1=N1=k1T,T=1k1DTN1.

By substituting (4.13) and (4.14) in (4.11) we get

DT(DTDTN1)=(k12k12k2)(1k1DTN1)3k1DTT,DT(DTDTN1)+3k1DTT=(k12k12k2)(1k1DTN1).
(⇐) We will now show that pseudo null curve α is a slant helix. By differentianting (4.14) covariantly
T=1k1DTN1
DTT=k1k12DTN11k1DTDTN1,
DTDTT=(k1k12)DTN1+2k1k12DTDTN11k1DTDTDTN1.

By substituting (4.15) in (4.17) we get

DTDTT=[(k1k12)1k12(k12k12k2)]DTN1+2k1k12DTDTN1+3k1k1DTT,
by substituting (4.8) and (4.10) in (4.18) we get
DTDTT=[(k1k12)1k12(k12k12k2)]DTN12(k1k1)2T+k1k2k1N1+k1N2.

From (4.8)

DTT=k2N1+k1N2,DTDTT=k2N1+k2DTN1+k1N2k1k2T1.

By comparing (4.19) and (4.20)

k1k2k1=k2,k1k1=k2k2,
by integrating (4.21) we get
k1k2=ec(constant)˙

Hence α is a general slant helix.

Theorem 7

Letα:IE13be a unit speed pseudo null curve on M1, then α is a general slant helix if and only if

DT(DTDTN2)+3k2DTT=(k2k22k1k2)DTN2.

Proof

(⇒) Suppose that α is a general slant helix. Then, from (4.8), we have

DTN2=k2T1,DTDTN2=k2T1k22N1k1k2N2,DT(DTDTN2)=(k22k1k2)T1k2DTT2k2k2N1
(k1k2+k1k2)N2k1k2DTN2.

Since α is a general helix

k1k2=ccisconstant,
by differentiating the above equation we get
(k1k2)=2k1k2,
but
T=1k2DTN2.

By substituting (4.25) and (4.26) in (4.24) we get

DT(DTDTN2)=(k2k22k1k2)DTN23k2DTT,DT(DTDTN2)+3k2DTT=(k2k22k1k2)DTN2.
(⇐) We will show that pseudo null curve α is a slant helix. So by differentianting (4.26) covariantly we get
DTT=k2k22DTN21k2DTDTN2,DTDTT=(k2k22)DTN2+2k2k22DTDTN2
1k2DTDTDTN2.

By substituting (4.27) in (4.29) we get

DTDTT=[(k2k22)(k2k22k1k2)1k2]DTN2+2k2k22DTDTN2+3k2k2DTT,
by substituting (4.8) and (4.23) in (4.30) we get
DTDTT=[(k2k22)(k2k22k1k2)1k2]DTN22(k2k2)2T1+k2N1+k1k2k2N2.

From (4.8)

DTT=k2N1+k1N2,DTDTT=k2N1+k2DTN1+k1N2k1k2T1.

By comparing (4.31) and (4.32)

k1k2k2=k1k1k1=k2k2,
by integrating (4.33) we get
k1k2=ecconstant˙

Hence α is a general slant helix, the theorem is now proved.

References

  • [1]

    A.A. Lucas, P. Lambin, Diffraction by DNA, carbon nanotubes and other helical nanostructures, Rep. Prog. Phys., 1181–1249, 68 (2005).

  • [2]

    Ahmad T. Ali, Position vectors of general helices in Euclidean 3-space, Bulletin of Mathematical Analysis and Applications, Volume 3, 2, Pages 198–205(2011).

  • [3]

    B. Bükcu, M.K. Karacan, Bishop motion and Bishop Darboux rotation axis of the timelike curve in Minkowski 3-space, Kochi J. Math. 4,109–117, (2009).

  • [4]

    B. Bukcu, M.K. Karacan, On the slant helices according to Bishop frame of the timelike curve in Lorentzian space, Tamkang J. Math. 39 (3), 255–262, (2008).

  • [5]

    C. Özgür and F. Gezgin, “On some curves of AW(k)-type,” Differential Geometry—Dynamical Systems, vol. 7, pp. 74–80,2005.

  • [6]

    K.L. Duggal, D.H. Jin, ”Null Curves and Hypersurfaces of Semi-Riemannian Manifolds”, World Scientific, Singapore, 2007.

  • [7]

    L. Kula, N. Ekmekci, Y. Yaylı and K. Ilarslan, Characterizations of slant helices in Euclidean 3-space. Turk J Math 34, 261 – 273 (2010).

  • [8]

    L. R. Bishop, There is more than one way to frame a curve, Amer. Math. Monthly 82 (1975) 246–251.

  • [9]

    M. Erdogdu, Parallel frame of non-lightlike curves in Minkowski space–time, Int. J. Geom. Methods Mod. Phys,16 pages, 12 (2015).

  • [10]

    M. Grbović and E. Nesovic, ”On the Bishop frames of pseudo null and null Cartan curves in Minkowski 3-space,” Journal of Mathematical Analysis and Applications, Vol. 461, pp. 219–233, 2018.

  • [11]

    N. Chouaieb, A. Goriely, J.H. Maddocks, Helices, PANS, 103, 9398–9403, (2006).

  • [12]

    S. Buyukkutuk and G. Ozturk, Constant Ratio Curves According to Bishop Frame in Euclidean 3-Space. Vol. 28, No. 1, pp.81–91, May 2015.

  • [13]

    Ahmad T. Ali and Rafael Lopez, Slant Helices in Minkowski Space E 1 3 [E_1^3 , J. Korean Math. Soc. Vol. 48, No. 1, pp. 159–167, (2011)

  • [14]

    M. Külahcı, M. Bektaş and M. Ergüt, On Harmonic Curvatures of Null Curves of the AW(k)-Type in Lorentzian Space Z. Naturforsch. 63a, 248 – 252 (2008).

  • [15]

    Yildirim Yilmaz M., Külahcı M. and Öğrenmiş A. O., A Slant Helix Characterization in Riemann- Otsuki Space, Mathematica moravica vol. 16–2, 99–106, (2012).

  • [16]

    Yildirim Yilmaz M., Biharmonic General Helices in Three Dimensional Finsler Manifold F 3, Karaelmas Fen ve Müh. Derg. 7(1):1–4, 2017.

  • [17]

    Yildirim Yilmaz M. and Bektaş M., Slant helices of (k, m)-type in E 4 Acta Univ. Sapientiae, Mathematica, 10, pp 395–401, 2 (2018).

  • [18]

    Yildirim Yilmaz M. and Bektaş M., Helices of the 3- dimensional Finsler Manifold, Journal of Adv. Math. Studies 2(1), 107–113, 2009.

  • [19]

    S. Izumiya, N. Takeuchi, New Special Curves and Developable Surfaces, Turk J Math, 153–163, 28 (2004).

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    A.A. Lucas, P. Lambin, Diffraction by DNA, carbon nanotubes and other helical nanostructures, Rep. Prog. Phys., 1181–1249, 68 (2005).

  • [2]

    Ahmad T. Ali, Position vectors of general helices in Euclidean 3-space, Bulletin of Mathematical Analysis and Applications, Volume 3, 2, Pages 198–205(2011).

  • [3]

    B. Bükcu, M.K. Karacan, Bishop motion and Bishop Darboux rotation axis of the timelike curve in Minkowski 3-space, Kochi J. Math. 4,109–117, (2009).

  • [4]

    B. Bukcu, M.K. Karacan, On the slant helices according to Bishop frame of the timelike curve in Lorentzian space, Tamkang J. Math. 39 (3), 255–262, (2008).

  • [5]

    C. Özgür and F. Gezgin, “On some curves of AW(k)-type,” Differential Geometry—Dynamical Systems, vol. 7, pp. 74–80,2005.

  • [6]

    K.L. Duggal, D.H. Jin, ”Null Curves and Hypersurfaces of Semi-Riemannian Manifolds”, World Scientific, Singapore, 2007.

  • [7]

    L. Kula, N. Ekmekci, Y. Yaylı and K. Ilarslan, Characterizations of slant helices in Euclidean 3-space. Turk J Math 34, 261 – 273 (2010).

  • [8]

    L. R. Bishop, There is more than one way to frame a curve, Amer. Math. Monthly 82 (1975) 246–251.

  • [9]

    M. Erdogdu, Parallel frame of non-lightlike curves in Minkowski space–time, Int. J. Geom. Methods Mod. Phys,16 pages, 12 (2015).

  • [10]

    M. Grbović and E. Nesovic, ”On the Bishop frames of pseudo null and null Cartan curves in Minkowski 3-space,” Journal of Mathematical Analysis and Applications, Vol. 461, pp. 219–233, 2018.

  • [11]

    N. Chouaieb, A. Goriely, J.H. Maddocks, Helices, PANS, 103, 9398–9403, (2006).

  • [12]

    S. Buyukkutuk and G. Ozturk, Constant Ratio Curves According to Bishop Frame in Euclidean 3-Space. Vol. 28, No. 1, pp.81–91, May 2015.

  • [13]

    Ahmad T. Ali and Rafael Lopez, Slant Helices in Minkowski Space E 1 3 [E_1^3 , J. Korean Math. Soc. Vol. 48, No. 1, pp. 159–167, (2011)

  • [14]

    M. Külahcı, M. Bektaş and M. Ergüt, On Harmonic Curvatures of Null Curves of the AW(k)-Type in Lorentzian Space Z. Naturforsch. 63a, 248 – 252 (2008).

  • [15]

    Yildirim Yilmaz M., Külahcı M. and Öğrenmiş A. O., A Slant Helix Characterization in Riemann- Otsuki Space, Mathematica moravica vol. 16–2, 99–106, (2012).

  • [16]

    Yildirim Yilmaz M., Biharmonic General Helices in Three Dimensional Finsler Manifold F 3, Karaelmas Fen ve Müh. Derg. 7(1):1–4, 2017.

  • [17]

    Yildirim Yilmaz M. and Bektaş M., Slant helices of (k, m)-type in E 4 Acta Univ. Sapientiae, Mathematica, 10, pp 395–401, 2 (2018).

  • [18]

    Yildirim Yilmaz M. and Bektaş M., Helices of the 3- dimensional Finsler Manifold, Journal of Adv. Math. Studies 2(1), 107–113, 2009.

  • [19]

    S. Izumiya, N. Takeuchi, New Special Curves and Developable Surfaces, Turk J Math, 153–163, 28 (2004).

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