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Öğe Cayley formula in Minkowski space-time(World Scientific Publ Co Pte Ltd, 2015) Erdogdu, Melek; Ozdemir, MustafaIn this paper, Cayley formula is derived for 4 x 4 semi-skew-symmetric real matrices in E-1(4). For this purpose, we use the decomposition of a semi-skew-symmetric matrix A = theta(1)A(1) + theta(2)A(2) by two unique semi-skew-symmetric matrices A(1) and A(2) satisfying the properties A(1)A(2) = 0, A(1)(3) = A(1) and A(2)(3) = -A(2). Then, we find Lorentzian rotation matrices with semi-skew-symmetric matrices by Cayley formula. Furthermore, we give a way to find the semi-skew-symmetric matrix A for a given Lorentzian rotation matrix R such that R = Cay(A).Öğe Geometry of Hasimoto Surfaces in Minkowski 3-Space(Springer, 2014) Erdogdu, Melek; Ozdemir, MustafaIn this paper, we investigate the Hasimoto surfaces in Minkowski 3-space. We discussed the geometric properties of Hasimoto surfaces in M-3 for three cases. The Gaussian and mean curvature of Hasimoto surface are found for each case. Then, we give the characterization of parameter curves of Hasimoto surfaces in M-3Öğe On Complex Split Quaternion Matrices(Springer Basel Ag, 2013) Erdogdu, Melek; Ozdemir, MustafaIn this paper, we present some important properties of complex split quaternions and their matrices. We also prove that any complex split quaternion has a 4 x 4 complex matrix representation. On the other hand, we give answers to the following two basic questions If AB = I, is it true that BA = I for complex split quaternion matrices? and How can the inverse of a complex split quaternion matrix be found?. Finally, we give an explicit formula for the inverse of a complex split quaternion matrix by using complex matrices.Öğe On Eigenvalues of Split Quaternion Matrices(Springer Basel Ag, 2013) Erdogdu, Melek; Ozdemir, MustafaThe main purpose of this paper is to set a method of finding eigenvalues of split quaternion matrices. In particular, we will give an extension of Gershgorin theorem, which is one of the fundamental theorems of complex matrix theory, for split quaternion matrices.Öğe On exponential of split quaternionic matrices(Elsevier Science Inc, 2017) Erdogdu, Melek; Ozdemir, MustafaThe exponential of a matrix plays an important role in the theory of Lie groups. The main purpose of this paper is to examine matrix groups over the split quaternions and the exponential map from their Lie algebras into the groups. Since the set of split quaternions is a noncommutative algebra, the way of computing the exponential of a matrix over the split quaternions is more difficult than calculating the exponential of a matrix over the real or complex numbers. Therefore, we give a method of finding exponential of a split quaterriion matrix by its complex adjoint matrix. (C) 2017 Elsevier Inc. All rights reserved.Öğe ON REFLECTIONS AND ROTATIONS IN MINKOWSKI 3-SPACE(Inst Biophysics & Biomedical Engineering, Bulgarian Acad Sciences, 2015) Erdogdu, Melek; Ozdemir, MustafaIn this paper, we investigate the reflections in Minkowski three-space by three different approaches. Firstly, we define Lorentzian reflections with Lorentzian inner product. Then, we examine Lorentzian reflections in view of Lorentzian Householder matrices. Finally, we use pure split quaternions to derive Lorentzian reflections. For each case, we find the matrix representation of Lorentzian reflections and characterize the plane of reflection by using this matrix representation. Moreover, we prove that any Lorentzian orthogonal transformation can be represented by the composition of at most six reflections.Öğe On the Eigenvalues and Eigenvectors of a Lorentzian Rotation Matrix by Using Split Quaternions(Springer Basel Ag, 2014) Ozdemir, Mustafa; Erdogdu, Melek; Simsek, HakanIn this paper, we examine eigenvalue problem of a rotation matrix in Minkowski 3 space by using split quaternions. We express the eigenvalues and the eigenvectors of a rotation matrix in term of the coefficients of the corresponding unit timelike split quaternion. We give the characterizations of eigenvalues (complex or real) of a rotation matrix in Minkowski 3 space according to only first component of the corresponding quaternion. Moreover, we find that the casual characters of rotation axis depend only on first component of the corresponding quaternion. Finally, we give the way to generate an orthogonal basis for by using eigenvectors of a rotation matrix.Öğe ON THE ROTATION MATRIX IN MINKOWSKI SPACE-TIME(Pergamon-Elsevier Science Ltd, 2014) Ozdemir, Mustafa; Erdogdu, MelekIn this paper, a Rodrigues-like formula is derived for 4 x 4 semi skew-symmetric real matrices in E-1(4). For this purpose, we use the decomposition of a semi skew-symmetric matrix A = theta(1)A(1) + theta(2)A(2) by two unique semi skew-symmetric matrices A(1) and A(2) satisfying the properties A(1)A(2) = theta, A(1)(3) = A1 and A(2)(3) = A(2). Then, we find Lorentzian rotation matrices with semi skew-symmetric matrices by Rodrigues-like formula. Furthermore, we give a way to find the semi skew-symmetric matrix A for a given Lorentzian rotation matrix R such that R = e(A).Öğe Real matrix representations of complex split quaternions with applications(Wiley, 2020) Erdogdu, Melek; Ozdemir, MustafaIn this paper, we introduce 8x8 real matrix representations of complex split quaternions. Then, the relations between real matrix representations of split and complex split quaternions are stated. Moreover, we investigate some linear split and complex split quaternionic equations with split Fibonacci and complex split Fibonacci quaternion coefficients. Finally, we also give some numerical examples as applications of real matrix representation of complex split quaternions.Öğe Serum Neuroinflammatory Markers' Levels and Evaluation of Peripheral Large Nerve Fibre Function in Patients with Psoriasis(Galenos Yayincilik, 2016) Balevi, Ali; Ozdemir, Mustafa; Guney, Figen; Tol, Huseyin; Okudan, Nilsel; Belviranli, Muaz; Sahin, Kemal TahirObjective: Currently, tumor necrosis factor-alpha (TNF-alpha) inhibitors are widely used all over the world, especially in the treatment of severe psoriasis. However, multiple sclerosis (MS) and peripheral neuropathy occurrence associated with TNF-a inhibitors have been reported in the treatment of patients with psoriasis. It is not known whether TNF-a inhibitors reveal underlying neurological disorders in patients with psoriasis. The aim of this study was to investigate whether there is an association between psoriasis and MS or peripheral neuropathy. Methods: In this study, 50 psoriasis patients and 25 healthy controls were enrolled. Neurological examination and needle electromyography test were performed. Serum levels of matrix metalloproteinase-2 (MMP-2), matrix metalloproteinase-9 (MMP-9), S-100 calcium-binding protein (S100B) and neuron specific enolase (NSE), which are involved in neuroinflammatory process, were measured. Results: Serum MMP-9 levels were higher in psoriasis patients compared to healthy controls. Serum MMP-2 and NSE levels were decreased in psoriasis patients compared to healthy controls. There was no significant difference in S100B levels between two groups. All neurophysiological parameters in each individual patient and control were found to be within normal limits and there was no difference between groups. Conclusion: NSE and MMP-9 might play role in the occurrence of MS in psoriasis patients. Especially, we are in the opinion that it might be useful to measure the serum levels of these markers before the administration of TNF-a inhibitors in psoriasis patients with family history of MS.Öğe Split Quaternion Matrix Representation of Dual Split Quaternions and Their Matrices(Springer Basel Ag, 2015) Erdogdu, Melek; Ozdemir, MustafaIn this paper, we present some important properties of matrices over dual split quaternions. We examine dual split quaternion matrices by their split quaternion matrix representation. Then, we give some interesting results for 2 x 2 split quaternion matrix representation of a dual split quaternion. Finally, we prove that the group and are isomorphic.Öğe Two-sided linear split quaternionic equations with n unknowns(Taylor & Francis Ltd, 2015) Erdogdu, Melek; Ozdemir, MustafaIn this paper, we investigate linear split quaternionic equations with the terms of the form axb. We give a new method of solving general linear split quaternionic equations with one, two and n unknowns. Moreover, we present some examples to show how this procedure works.
