Yazar "Bekar, Murat" seçeneğine göre listele

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    Dual Quaternion Involutions and Anti-Involutions
    (Springer Basel Ag, 2013) Bekar, Murat; Yayi, Yusuf
    An involution or anti-involution is a self-inverse linear mapping. In this paper, we present involutions and anti-involutions of dual quaternions. In order to do this, quaternion conjugate, dual conjugate and total conjugate are defined for a dual quaternion and these conjugates are used in some transformations in order to check whether involution or anti-involution axioms are being satisfied or not by these transformations. Finally, geometric interpretations of real quaternion and dual quaternion involutions and anti-involutions are given.
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    Involutions in Dual Split-Quaternions
    (Springer Basel Ag, 2016) Bekar, Murat; Yayli, Yusuf
    Involutions and anti-involutions are self-inverse linear mappings. In three-dimensional Euclidean space , a reflection of a vector in a plane can be represented by an involution or anti-involution mapping obtained by real-quaternions. A reflection of a line about a line in can also be represented by an involution or anti-involution mapping obtained by dual real-quaternions. In this paper, we will represent involution and anti-involution mappings obtaind by dual split-quaternions and a geometric interpretation of each as rigid-body (screw) motion in three-dimensional Lorentzian space .
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    Involutions in Quasi-Quaternions
    (Amer Scientific Publishers, 2017) Bekar, Murat; Yayli, Yusuf
    In this paper, we present an involution and an anti-involution mapping obtained by using the algebra of quasi-quaternions. Moreover, the geometric interpretations of each these mappings are given in three-dimensional Euclidean space R-3.
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    INVOLUTIONS IN SEMI-QUATERNIONS
    (Inst Biophysics & Biomedical Engineering, Bulgarian Acad Sciences, 2016) Bekar, Murat; Yayli, Yusuf
    Involutions are self-inverse and homomorphic linear mappings. Rotations, reflections and rigid-body (screw) motions in three-dimensional Euclidean space R-3 can be represented by involution mappings obtained by quaternions. For example, a reflection of a vector in a plane can be represented by an involution mapping obtained by real-quaternions, while a reflection of a line about a line can be represented by an involution mapping obtained by dual-quaternions. In this paper, we will consider two involution mappings obtained by semi-quternions, and a geometric interpretation of each as a planar-motion in R-3.
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    Involutions of Complexified Quaternions and Split Quaternions
    (Springer Basel Ag, 2013) Bekar, Murat; Yayli, Yusuf
    An involution or anti-involution is a self-inverse linear mapping. Involutions and anti-involutions of real quaternions were studied by Ell and Sangwine [15]. In this paper we present involutions and antiinvolutions of biquaternions (complexified quaternions) and split quaternions. In addition, while only quaternion conjugate can be defined for a real quaternion and split quaternion, also complex conjugate can be defined for a biquaternion. Therefore, complex conjugate of a biquaternion is used in some transformations beside quaternion conjugate in order to check whether involution or anti-involution axioms are being satisfied or not by these transformations. Finally, geometric interpretations of real quaternion, biquaternion and split quaternion involutions and anti-involutions are given.
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    Lie Algebra of Unit Tangent Bundle
    (Springer Basel Ag, 2017) Bekar, Murat; Yayli, Yusuf
    In this paper, semi-quaternions are studied with their basic properties. Unit tangent bundle of is also obtained by using unit semi-quaternions and it is shown that the set of all unit semi-quaternions based on the group operation of semi-quaternion multiplication is a Lie group. Furthermore, the vector space matrix of angular velocity vectors forming the Lie algebra of the group is obtained. Finally, it is shown that the rigid body displacements obtained by using semi-quaternions correspond to planar displacements in .
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    N-Legendre and N-slant curves in the unit tangent bundle of Minkowski surfaces
    (World Scientific Publ Co Pte Ltd, 2018) Bekar, Murat; Hathout, Fouzi; Yayli, Yusuf
    Let (T1M12, g(1)) be a unit tangent bundle of Minkowski surface (M-1(2), g) endowed with the pseudo-Riemannian induced Sasaki metric. In this present paper, we studied the N-Legendre and N-slant curves in which the inner product of its normal vector and Reeb vector is zero and nonzero constant, respectively, in (T1M12, g(1)) and several important characterizations of these curves are given.
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    N-Legendre and N-Slant curves in the unit tangent bundle of surfaces
    (Academic Publication Council, 2017) Hathout, Fouzi; Bekar, Murat; Yayli, Yusuf
    Let (T1M, g(1)) be a unit tangent bundle of some surface (M, g) endowed with the induced Sasaki metric. In this paper, we define two kinds of curves called N-legendre and N-slant curves in which the inner product of its normal vector and Reeb vector is, respectively, equal to zero and to non-zero constant in (T1M, g(1)). Some several important characterizations of these curves are also obtained.
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    Semi-Euclidean quasi-elliptic planar motion
    (World Scientific Publ Co Pte Ltd, 2016) Bekar, Murat; Yayli, Yusuf
    The aim of this paper is to study the algebra of split semi-quaternions with their basic properties. Also, the results of the Euclidean planar motion given by Blaschke and Grunwald is generalized to semi-Euclidean planar motion by using the algebra of split semi-quaternions.
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    Slant Helix Curves and Acceleration Centers in Minkowski 3-Space E13
    (Amer Scientific Publishers, 2017) Bekar, Murat; Yayli, Yusuf
    In this study, some basic concepts (e.g., instant screw axis (ISA), instantaneous pole points, acceleration pole points) will be given and analyzed about an alternative one-parameter motion of a rigid-body in 3-dimensional Minkowski space. E-1(3) obtained by moving coordinate frame {N, C, W} along a non-null unit speed curve alpha = alpha(t), where N, C and W correspond to unit principal normal vector field, derivative vector field of unit principal normal vector field and Darboux vector field (or angular-velocity vector field) of the non-null unit speed curve alpha, respectively.y