Timelike Bertrand Curves in Semi-Euclidean Space

In this paper, it is proved that, no special timelike Frenet curve is a Bertrand curve in $\mathbb{E}_2^4$ and also, in $\mathbb{E}_\nu^{n+1}$ $({n \ge 3})$, such that the notion of Bertrand curve is definite only in $\mathbb{E}_1^2$ and $\mathbb{E}_1^3$. Therefore, a generalization of timelike Bertrand curve is defined and called as timelike (1,3)-Bertrand curve in $\mathbb{E}_2^4$. Moreover, the characterization of timelike (1,3)-Bertrand curve is given in $\mathbb{E}_2^4$

Lamarle Formula in 3-Dimensional Lorentz Space

The Lamarle Formula, given by Kruppa in \cite{Kr}, is known as a relationship between the Gaussian curvature and the distribution parameter of a ruled surface in the surface theory. The ruled surfaces were investigated in 3 different classes with respect to the character of base curves and rulings, \cite{Tu1},\cite{Tu2}. In this paper on account of these studies, the relationships between the Gaussian curvatures and distribution parameters of spacelike ruled surface, timelike ruled surface with spacelike ruling and timelike ruled surface with timelike ruling are obtained, respectively. These relationships are called as Lorentzian Lamarle formulas. Finally some examples concerning with these relations are given

Commutative Quaternion Matrices

In this study, we introduce the concept of commutative quaternions and commutative quaternion matrices. Firstly, we give some properties of commutative quaternions and their Hamilton matrices. After that we investigate commutative quaternion matrices using properties of complex matrices. Then we define the complex adjoint matrix of commutative quaternion matrices and give some of their properties