Structure of spherically symmetric objects: a study based on structure scalars

The aim of this paper is to explore the consequences of extra curvature terms mediated from f(R, T, Q) (where Q ≡ R μ ν T μ ν ) theory on the formation of scalar functions and their importance in the study of populations who are crowded with regular relativistic objects. For this purpose, we model our system comprising of non-rotating spherical geometry formed due to gravitation of locally anisotropic and radiating sources. After considering a particular f(R, T, Q) model, we form a peculiar relation among Misner-Sharp mass, tidal forces, and matter variables. Through structure scalars, we have modeled shear, Weyl, and expansion evolutions equations. The investigation for the causes of the irregular distribution of energy density is also performed with and without constant curvature conditions. It is deduced that our computed one of the f(R, T, Q) structure scalars (Y T ) has a vital role to play in understanding celestial mechanisms in which gravitational interactions cause singularities to emerge.

Comprehensive quasi-Einstein spacetime with application to general relativity

The aim of this paper is to extend the notion of all known quasi-Einstein (QE) manifolds like generalized QE, mixed generalized QE manifold, pseudo generalized QE manifold and many more and name it comprehensive QE manifold [Formula: see text]. We investigate some geometric and physical properties of the comprehensive QE manifolds [Formula: see text] under certain conditions. We study the conformal and conharmonic mappings between [Formula: see text] manifolds. Then we examine the [Formula: see text] with harmonic Weyl tensor. We define the manifold of comprehensive quasi-constant curvature and prove that conformally flat [Formula: see text] is manifold of comprehensive quasi-constant curvature and vice versa. We study the general two viscous fluid spacetime [Formula: see text] and find out some important consequences about [Formula: see text]. We study [Formula: see text] with vanishing space matter tensor. Finally, we prove the existence of such manifolds by constructing nontrivial example.

On Backlund transformation and motion of null Cartan curves

The main scope of this paper is to examine null Cartan curves especially the ones with constant torsion. In accordance with this scope, the position vector of a null Cartan curve is stated by a linear combination of the vector fields of its pseudo-orthogonal frame with differentiable functions. However, the most important difference that distinguishes this study from the other studies is that the Bertrand curve couples (timelike, spacelike or null) of null Cartan curves are also examined. Consequently, it is seen that all kinds of Bertrand couples of a given null Cartan curve with constant curvature functions have also constant curvature functions. This result is the most valuable result of the study, but allows us to introduce a transformation on null Cartan curves. Then, it is proved that aforesaid transformation is a Backlund transformation which is well recognized in modern physics. Moreover, motion of an inextensible null Cartan curve is investigated. By considering time evolution of null Cartan curve, the angular momentum vector is examined. And three different situations are given depending on the character of the angular momentum vector [Formula: see text] In the case of [Formula: see text] we discuss the solution of the system which is obtained by compatibility conditions. Finally, we provide the relation between torsion of the curve and the velocity vector components of the moving curve [Formula: see text]

A New Approach for Solving the Inverse Kinematics of Continuum Robot Based on Piecewise Constant Curvature Model

The inverse kinematics of continuum robot is an important factor to guarantee the motion accuracy. How to construct a concise inverse kinematics model is very essential for the motion control of continuum robot. In this paper, a new method for solving the inverse kinematics of continuum robot is proposed based on the geometric and numerical method. Assumed that the deformation of the continuum robot is Piecewise Constant Curvature model (PCC), the envelope surface of the continuum robot based on single-segment is modeled and calculated. The clustering method is used to calculate the intersection of the curves. Then, a distinct sequence is designed for solving the inverse kinematics of continuum robot, and it is also suitable for the multi-segment continuum robots in space. Finally, the accuracy of the inverse kinematics algorithm is verified by the simulation and numerical experiment. The experiment results illustrate that this algorithm is with higher accuracy compared with the Jacobian iterative algorithm.

Support theorems for Funk-type isodistant Radon transforms on constant curvature spaces

A connected maximal submanifold in a constant curvature space is called isodistant if its points are in equal distances from a totally geodesic of codimension 1. The isodistant Radon transform of a suitable real function f on a constant curvature space is the function on the set of the isodistants that gives the integrals of f over the isodistants using the canonical measure. Inverting the isodistant Radon transform is severely overdetermined because the totally geodesic Radon transform, which is a restriction of the isodistant Radon transform, is invertible on some large classes of functions. This raises the admissibility problem that is about finding reasonably small subsets of the set of the isodistants such that the associated restrictions of the isodistant Radon transform are injective on a reasonably large set of functions. One of the main results of this paper is that the Funk-type sets of isodistants are admissible, because the associated restrictions of the isodistant Radon transform, we call them Funk-type isodistant Radon transforms, satisfy appropriate support theorems on a large set of functions. This unifies and sharpens several earlier results for the sphere, and brings to light new results for every constant curvature space.

Applying Differential Forms and the Generalized Sundman Transformations in Linearizing the Equation of Motion of a Free Particle in a Space of Constant Curvature

Equation of motion of a free particle in a space of constant curvature applies to many fields, such as the fixed reduction of the second member of the Burgers classes, the study of fusion of pellets, equations of Yang-Baxter, the concept of univalent functions as well as spheres of gaseous stability to mention but a few. In this study, the authors want to examine the linearization of the said equation using both point and non-point transformation methods. As captured in the title, the methods under examination here are the differential forms (DF) and the generalized Sundman transformations (GST), which are point and non-point transformation methods respectively. The comparative analysis of the solutions obtained via the two linearizability methods is also taken into account.