This study proposes to improve the Lagrangian energy equations for instructed complex jet bundles on Kähler manifolds. The coordinates on the bundle structure of Kähler manifolds have been given for real and imaginary dimensions. For given bundle structures, all fundamental geometrical properties have been investigated and applications to complex bundle structures are carried out. The energy equations have been applied to the numerical example in order to test its performance. Moreover, velocity and time dimensions for energy movement equations have been presented as a new concept. This study shows some physical applications of those equations and interpretations are made. Results show that the imaginary coordinates are same as real coordinates. One of the interesting conclusion of this study is that the Lagrangian energy of a movement particle is static when the particle moves in a large velocity.

In this paper, we studied about a detailed analysis of fuzzy hyperbola. In the previous studies, some methods for fuzzy parabola are discussed (Ghosh and Chakraborty, 2019). To define the fuzzy hyperbola, it is necessary to modify the method applied for the fuzzy parabola. To obtain a conic, it is necessary to know at least five points on this curve. First of all, in this study, we examined how to detect these five fuzzy points. We have discussed in detail the impact of points in this examination on finding fuzzy membership degrees and determining the curve. We show the use of the algorithm for calculating the coefficients in the conic equation on the examples. We make detailed drawings of all the fuzzy hyperbolas found and depicted the geometric location of fuzzy points with different membership degrees on the graph. As can be seen from the figures in our study, the importance of membership degrees in fuzzy space is that it causes us to find different numbers of hyperbola curves for the five points we study with. In addition, finding the membership of a given point to the fuzzy hyperbola is possible by solving nonlinear equations under different angular approaches. This examination is shown in detail in this study, and the results in the examples are evaluated by geometric comments. The systems formed by the fuzzy hyperbola curves are found to have different areas of use, as presented in the Conclusion. Some of usage areas of fuzzy hyperbola are radar systems, scanning devices, photosynthesis, heat, and CO2 distribution of plants.

The aim of this article is firstly to improve time-dependent Lagrangian energy equations using the super jet bundles on supermanifolds. Later, we adapted this study to the graph bundle. Thus, we created a graph bundle by examining the graph manifold structure in superspace. The geometric structures obtained for the mechanical energy system with superbundle coordinates were reexamined with the graph bundle coordinates. Thus, we were able to calculate the energy that occurs during the motion of a particle when we examine this motion with graph points. The supercoordinates on the superbundle structure of supermanifolds have been given for body and soul and also even and odd dimensions. We have given the geometric interpretation of this property in coordinates for the movement on graph points. Lagrangian energy equations have been applied to the presented example, and the advantage of examining the movement with graph points was presented. In this article, we will use the graph theory to determine the optimal motion, velocity, and energy of the particle, due to graph points. This study showed a physical application and interpretation of supervelocity and supertime dimensions in super-Lagrangian energy equations utilizing graph theory.

The aim of this paper is to improve Hamiltonian energy equations for the complex jet bundles using Kähler manifolds. The coordinates on the bundle structure of Kähler manifolds have been given for real and imaginary dimensions. For given bundle structures, all fundamental geometrical properties have been investigated in Hamiltonian energy equations and applications to complex bundle structures. The improved Hamiltonian energy equations have been applied to the presented example in order to test its performance. Moreover, we have presented a new concept of velocity and time dimensions for energy movement equations. Results showed that imaginary Hamiltonian values are opposed to real Hamiltonian values in 2n dimensions. Finally, this study showed a physical application and interpretation of velocity and time dimensions in Hamiltonian energy equations for given examples.

No abstract