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Murat Tosun

Publications and source records attributed to Murat Tosun.

15 recordsLinked to original sources
arXiv2024

Padovan and Perrin Hyperbolic Spinors

In this study, we intend to bring together Padovan and Perrin number sequences, which are one of the most popular third-order recurrence sequences, and hyperbolic spinors, which are used in several disciplines from physics to mathematics, with the help of the split quaternions. This paper especially improves the relationship between hyperbolic spinors both a physical and mathematical concept, and number theory. For this aim, we combine the hyperbolic spinors and Padovan and Perrin numbers concerning the split Padovan and Perrin quaternions, and we determine two new special recurrence sequences named Padovan and Perrin hyperbolic spinors. Then, we give Binet formulas, generating functions, exponential generating functions, Poisson generating functions, and summation formulas. Additionally, we present some matrix and determinant equations with respect to them. Then, we construct some numerical algorithms for these special number systems, as well. Further, we give an introduction for $(s,t)$-Padovan and $(s,t)$-Perrin hyperbolic spinors.

math.GM↗
arXiv2024

Generalized Tribonacci Hyperbolic Spinors

In this study, we introduce the generalized Tribonacci hyperbolic spinors and properties of this new special numbers system by the generalized Tribonacci numbers, which are one of the most general form of the third-order recurrence sequences, generalized Tribonacci quaternions, and hyperbolic spinors, which have quite an importance and framework from mathematics to physics. This study especially improves the relations between the hyperbolic spinors and generalized Tribonacci numbers with the help of the generalized Tribonacci split quaternions. Furthermore, we examine some special cases of them and construct both new equalities and fundamental properties such as recurrence relation, Binet formula, generating function, exponential generating function, Poisson generating function, summation formulas, special determinant properties, matrix formula, and special determinant equations. Also, we give some numerical algorithms with respect to the obtained materials. In addition to these, we give a brief introduction for further research: generalized Tribonacci polynomial hyperbolic spinor sequence.

math.GM↗
arXiv2014

Consimilarity of Split Quaternion Matrices and a Solution of the Split Quaternion Matrix Equation X-AX_B=C

In this paper, the consimilarity of complex matrices is generalized for the split quaternions. In this regard, coneigenvalue and coneigenvector are defined for split quaternion matrices. Also, the existence of solution to the split quaternion matrix equation X-AXB = C is characterized and the solution of the equation in the explicit form are derived via its real representation.

math.AC↗
arXiv2013

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.

math.AG↗
arXiv2012

Spinor Bishop Equations of Curves in Euclidean 3-Space

In this paper, we study spinor Bishop equations of curves in E^3. We research the spinor formulations of curves according to Bishop frames in E^3. Also, the relation between spinor formulations of Bishop frames and Frenet frame are expressed.

math.DG↗
arXiv2012

A note on inextensible flows of curves in Een

In this paper, we investigate the general formulation for inextensible flows of curves in En. The necessary and sufficient conditions for inextensible curve flow are expressed as a partial differential equation involving the curvatures.

math.DG↗
arXiv2011

On Mannheim Partner Curves of $AW(k)-$type

In this study, firstly, Mannheim curves with $κ_1 (s) \ne 0$, $κ_2 (s) \ne 0$ are considered and the conditions are obtained for Mannheim curve to be slant helix. Moreover, the necessary and sufficient conditions are investigated for Mannheim curve to be AW(2), AW(3) and weak AW(2)-types, respectively. Lastly, it is shown that there is no such a Mannheim curve of AW(1)-type.

math.DG↗
arXiv2010

Generalized Mannheim Curves in Minkowski space-time $E_1^4$

In this paper, the definition of generalized spacelike Mannheim curve in Minkowski space-time $E_1^4$ is given. The necessary and sufficient conditions for the generalized spacelike Mannheim curve are obtained. Also, some characterizations of Mannheim curve are given.

math.DG↗
arXiv2010

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 $\mathbb{E}_ν^{n+1}$ $ ({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 $\mathbb{E}_2^4$.

math.DG↗
arXiv2010

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.

math.DG↗
arXiv2009

The Euler-Savary Formula for One-Parameter Planar Hyperbolic Motion

One-parameter hyperbolic planar motion was first studied by S. Y$\ddot{\texttt{u}}$ce and N. Kuruo$\tilde{\texttt{g}}$lu. Moreover, they analyzed the relationships between the absolute, relative and sliding velocities of one-parameter hyperbolic planar motion as well as the related pole curves, \cite{Yuc}. One-parameter planar motions in the Euclidean plane $\mathbb{E}^2$ and the Euler-Savary formula in one-parameter planar motions were given by M$\ddot{\texttt{u}}$ller, \cite{Mul}. In the present article, one hyperbolic plane moving relative to two other hyperbolic planes, one moving and the other fixed, was taken into consideration and the relation between the absolute, relative and sliding velocities of this movement was obtained. In addition, a canonical relative system for one-parameter hyperbolic planar motion was defined. Euler-Savary formula, which gives the relationship between the curvature of trajectory curves, was obtained with the help of this relative system.

math.DG↗