The possibility to apply the Frenet trihedron and formulas for the complex movement of a point on a plane with the predefined plane displacement

Material particles interact with the working moving surfaces of machines in various technological processes. Mechanics considers a technique to describe the movement of a point and decompose the speed and acceleration into single unit vectors of the accompanying trajectory trihedron for simple movement. The shape of the spatial curve uniquely sets the movement of the accompanying Frenet trihedral as a solid body. This paper has considered the relative movement of a material particle in the static plane of the accompanying Frenet trihedron, which moves along a flat curve with variable curvature. Frenet formulas were used to build a system of differential equations of relative particle movement. In contrast to the conventional approach, the chosen independent variable was not the time but the length of the arc of the guide curve along which the trihedron moves. The system of equations has been built in the projections onto the unit vectors of the moving trihedron; it has been solved by numerical methods. The use of the accompanying curve trihedron as a moving coordinate system makes it possible to solve the problems of the complex movement of a point. The shape of the curve guide assigned by parametric equations in its length function determines the portable movement of the trihedron and makes it possible to use Frenet formulas to describe the relative movement of a point in the trihedron system. This approach enables setting the portable movement of the trihedron osculating plane along a curve with variable curvature, thereby revealing additional possibilities for solving problems on a complex movement of a point at which rotational motion around a fixed axis is a partial case. The proposed approach has been considered using an example of the relative movement of cargo in the body of a truck moving along the road with a curvilinear axis of variable curvature. The charts of the relative trajectory of cargo slip and the relative speed for the predefined speed of the truck have been constructed

Classification of non-conformally flat static plane symmetric perfect fluid solutions via proper conformal vector fields in f(T) gravity

The aim of this paper is to classify non-conformally flat static plane symmetric (SPS) perfect fluid solutions via proper conformal vector fields (CVFs) in [Formula: see text] gravity. For this purpose, first we explore some SPS perfect fluid solutions of the Einstein field equations (EFEs) in [Formula: see text] gravity. Second, we utilize these solutions to find proper CVFs. In this study, we found 16 cases. A detailed study of each case reveals that in three of these cases, the space-times admit proper CVFs whereas in the rest of the cases, either the space-times become conformally flat or they admit proper homothetic vector fields (HVFs) or Killing vector fields (KVFs). The dimension of CVFs for non-conformally flat space-times in [Formula: see text] gravity is four, five or six.

A note on classification of static plane symmetric perfect fluid space-times via proper conformal vector fields in f(G) theory of gravity

In the [Formula: see text] theory of gravity, we classify static plane symmetric perfect fluid space-times via proper conformal vector fields (CVFs) using algebraic and direct integration approaches. During this classification, we found eight cases. Studying each case in detail, we found that the dimensions of CVFs are 4, 5, 6 or 15. In the cases when the space-time admits 15 independent CVFs it becomes conformally flat.

Conformal vector fields in proper non-static plane symmetric spacetimes in f(R) gravity

Nonstatic plane symmetric spacetimes are considered to study conformal vector fields (VFs) in the [Formula: see text] theory of gravity. Firstly, we investigate some proper nonstatic plane symmetric spacetimes by solving the Einstein field equations (EFEs) in the [Formula: see text] theory of gravity using algebraic techniques. Secondly, we find CVFs of the obtained spacetimes by means of the direct integration approach. There exist seven cases. Studying each case in detail, we find that the CVFs are of dimension three, five, six and fifteen.

Nuclear Plasticity Increases Susceptibility to Damage During Confined Migration

Large nuclear deformations during migration through confined spaces have been associated with nuclear membrane rupture and DNA damage. However, the stresses associated with nuclear damage remain unclear. Here, using a quasi-static plane strain finite element model, we map evolution of nuclear shape and stresses during confined migration of a cell through a deformable matrix. Plastic deformation of the nucleus observed for a cell with stiff nucleus transiting through a stiffer matrix lowered nuclear stresses, but also led to kinking of the nuclear membrane. In line with model predictions, transwell migration experiments with fibrosarcoma cells showed that while nuclear softening increased invasiveness, nuclear stiffening led to plastic deformation and higher levels of DNA damage. In addition to highlighting the advantage of nuclear softening during confined migration, our results suggest that plastic deformations of the nucleus during transit through stiff tissues may lead to bending-induced nuclear membrane disruption and subsequent DNA damage.

A study of energy conditions in static plane symmetric space–times via homothetic symmetries

The aim of this study is twofold. First, we use a new approach to study the homothetic vector fields (HVFs) of static plane symmetric space–times by an algorithm which we have developed using the Maple platform. The interesting feature of this algorithm is that it provides the most general form of metrics admitting HVFs as compared to those obtained in an earlier study where direct integration techniques were used. Second, the obtained metrics are used in Einstein’s field equations to compute the energy–momentum tensor and it is shown how the parameters involved in the obtained space–time metrics are associated with certain important energy conditions.

Homothetic matter collineations of static plane symmetric spacetimes

In this paper, we present a complete classification of static plane symmetric spacetimes via their homothetic symmetries of the energy–momentum tensor, known as homothetic matter collineations (HMCs). The HMC equations for these spacetimes are derived and then solved by considering the degeneracy and non-degeneracy of the energy–momentum tensor. In the former case, we have obtained 6, 11 and infinite number of HMCs, while in the latter case, the solution of HMC equations yields 6-, 7-, 8-, 10- and 11-dimensional algebra of HMCs. The obtained HMCs generate some differential constraints involving the components of the energy–momentum tensor. Some examples of static plane symmetric spacetime metrics satisfying these constraints are provided and the physical interpretations of these metrics are discussed.