In this article, firstly we study about geometrical applications of split quaternions. Then, we obtain Hamitonian mechanical systems with Split quaternions. Quaternionic and Coquaternionic (split analoque of quaternions) extensions of Hamiltonian mechanics are introduced and are shown as offer a unifying framework for quantum mechanics. This study leads to the possibility of employing algebraic techniques of quaternions and coquaternions to absorbing in quantum mechanics. The founded equations are compared with the Hamiltonian energy equations generally are known and the Hamilton energy equations are obtained in Minkowski space.

In this paper, we studied about a detailed analysis of fuzzy ellipse. In the previously studies, some methods for fuzzy parabola are discussed (Ghosh and Chakraborty,2019). To define the fuzzy ellipse, it is necessary to modify the method applied for the fuzzy parabola. First, need to get five same points with the same membership grade to create crisp ellipse and the union of crisp ellipses passing through these points will form the fuzzy ellipse. Although it is difficult to determine the points with this property, it is important for constructing the fuzzy ellipse equation. In this study, we determine the points that satisfy this condition and prove the properties required to obtain the fuzzy ellipse to be formed by using these points. We have drawn a graph of a fuzzy ellipse and depicted the geometric location of fuzzy points with different membership grades on graph. We have also shown some geometric application on examples. In the third part of this study, it has been shown that the determinants defined in the calculation of the coefficients of the fuzzy ellipse can be calculated using the maple program for different points and angles with the examples given, thus different fuzzy ellipses can be obtained.

We build the concept of fuzzy split quaternion numbers of a natural extension of fuzzy real numbers in this study. Then, we give some differential geometric properties of this fuzzy quaternion. Moreover, we construct the Frenet frame for fuzzy split quaternions. We investigate Serret-Frenet derivation formulas by using fuzzy quaternion numbers.

The aim of this paper is to apply the necessary and sufficient conditions of well-known Lagrangian equations with time dependent case for Minkowski 4-space. Many fundamental geometrical properties for time dependent Minkowski 4-space have been obtained in this paper. The energy equations have been applied to the numerical example in order to test its performance. In the numerical examples, we have studied with two time parameters (earth and space time) for accordance to Minkowski 4-space coordinates. This idea is an interesting approach to energy function with Earth-time and Space-time in physical comment. Moreover, velocity and two time dimensions for energy movement equations have been presented a new concept. This study show some physical application of those equations and interpretations are made in Minkowski space too. Results showed that Lagrangian functions for any surface are same type and depend on time coordinates.

Abstract:In this study, the higher order vertical and complete lifts of the tensor fields, which are defined on the extended vector bundles has been obtained. Keywords:The extended vector bundles, The vertical and complete lifts of the tensor fields.