•Quaternionic expressions of the magnetic flux tubes.•The motion of the charged particle in the magnetic flux tubes.•Characterizations of the magnetic field lines of magnetic flux tubes.•Diffusion of ...the magnetic flux tubes.
In the present paper, we give a generalization for the magnetic flux tubes. Firstly, we define the flux tube via quaternions. We obtain the magnetic flux canal surfaces and flux tubes by quaternion product of a unit quaternion and the unit normal vector of the magnetic field line. Besides, we generate these surfaces by the homothetic motion. Then, we determine the magnetic field components by using the quaternionic representation of the flux tube. Secondly, the flux tube is dictated by the magnetic vector field defined at each point on a generating curve. A local orthonormal basis is attached to every point of the generating curve then the magnetic field is given in terms of the local coordinate directions, leading to a geometric viewpoint of the flux tubes. These provide direct access to the magnetic field line and kinematics equations related to the flux tube. Moreover, the magnetic energy function and magnetic diffusion of the flux tube that has an arbitrary cross-section curve are computed. The impacts of the stretch factor of the flux tube on the magnetic trajectories are investigated. Finally, the analytical solutions of the kinematics equations and several examples of the theory are provided.
In the present paper, we investigate the geometric properties of the linearly polarized light wave (LPLW) and the homothetic motion of the polarization plane traveling in optical fiber in ...three-dimensional Riemannian manifold. We examine the behavior of the polarized plane for the conditions that the electric field ε makes a constant angle with the Frenet vectors {e1,e2,e3} of the curve related to the optical fiber that can be considered as a space curve in Riemannian 3-space. Moreover, we give the relation between the Fermi–Walker parallel transportation laws and the homothetic motion of the polarization plane in Riemannian 3-space. The key technique here we use for examining this approach is to use quaternion algebra. We give the parametric equations of the Rytov curves that are traced curves of the polarization vector ε via quaternion product and a matrix that is similar to a Hamilton operator. By means of this matrix a new motion is defined and this motion is proven to be homothetic. For this one-parameter homothetic motion, we prove some theorems about the motion of the polarization plane traveling in optical fiber in three-dimensional Riemannian manifold. Then, we obtain the characterization of the electric field and generate the electromagnetic trajectories (εM-trajectories) along the (LPLW) in the optical fiber using the variational approach. Finally, we give various examples with Maple codes to confirms the theoretical results.
Divergence free vector fields are called magnetic vector fields in three-dimensional semi-Riemannian manifolds. When a charged particle enters the magnetic vector field, it traces a new trajectory ...called magnetic curve by the influenced of magnetic field. In the present paper, we investigate the magnetic curves on the lightlike surfaces corresponding to the Killing magnetic fields in 3D semi-Riemannian manifolds. Moreover, we give some characterizations of these curves. As an application, we determine all magnetic curves on the lightlike cone. Finally, we give various examples to confirm the main results.
This paper aims to investigate the hyperbolically the motion of the polarization plane traveling along the linearly polarized light wave in the optical fiber via the hyperbolic split quaternion ...algebra. The motion of the electric field (polarization vector) about an axis on the general hyperboloid is described by the Lorentzian scalar product space Ra1,a2,a32,1. The hyperbolic geometric (Berry) phase models are generated through the pseudo-spheres of Ra1,a2,a32,1. Then, the parametric representations of the Rytov curves are obtained via the hyperbolic split quaternion product and one-parameter homothetic motion. Then, the hyperbolically motion of the electric field is expressed by the Fermi–Walker parallel transportation law. Moreover, the electromagnetic curves (EM-curves) associated with the electric field E are determined via the hyperbolic geometric phase models in the optical fiber. Furthermore, some motivating examples are presented by using the MAPLE program.
Along with other types of calculus, multiplicative calculus brings an entirely new perspective. Geometry now has a new field as a result of this new understanding. In this study, multiplicative ...differential geometry was used to explore peculiar surfaces. Multiplicative quaternions are also used to depict surfaces. Additionally, multiplicative differential geometry was used to generate the accretive surface subject, which is a developing subject. The derived surfaces' perspective silhouette curve equation is provided. The Bishop multiplicative frame was also established and applied when expressing surfaces along with these. Finally, the surfaces and perspective silhouette curves were visualized using Maple, and the equations were obtained.
In the present paper, we define the three cases of the geometric phase equations associated with a monochromatic linearly polarized light wave traveling along an optical fiber in three dimensional ...Walker manifold (M, g^ε_f). Walker manifolds have many applications in mathematics and theoretical physics. We are working in the context of a pseudo-Riemannian manifold (i.e. a manifold equipped with a non-degenerate arbitrary signature metric tensor). That is, we generalize the motion of the light wave in the optical fiber and the associated electromagnetic curves that describe the motion of a charged particle under the influence of an electromagnetic field over a Walker space defined as a pseudo-Riemannian manifold with a light-like distribution, parallel to the Levi-Civita junction. These manifolds (especially Lorentzian) are important in physics because of their applications in general relativity. Then, we obtain the Rytov curves related to the cases of geometric phase models. Moreover, we give some examples and visualize the evolution of the electric field along the optical fiber in (M, g^ε_f ) via MAPLE program.
In this article, light-like hypersurfaces which are derived by null
Cartan curves are examined and discussed. The singularities of lightlike hy-
persurfaces and light-like focal sets are investigated ...by using the Bishop frame
on the Null Cartan curves. We obtain that the types of these singularities
and the order of contact between the null Cartan curves are closely related
to the Bishop curvatures of the null Cartan curves. Moreover, two examples
of light-like hypersurfaces and light-like focal sets are given to illustrate our
theoretical results.
Background
Pilonidal sinus (PS) is a benign disease for which different treatment modalities are used, varying from non‐surgical treatments to surgical treatments with flap use for big defects. High ...recurrence is the main problem in complicated cases. We presented complete natal cleft excision with kite incision and fasciocutaneous flap (CNCEF with kite incision) in extensive sacrococcygeal PS.
Methods
Seventy‐six patients who underwent CNCEF with kite incision because of extensive PS extending through intergluteal sulcus to the anus, with multiple sinus tracts and recurrent PS were retrospectively analysed. A special incision involving natal cleft and all sinus tracts completely and fasciocutaneous flap, lateralizing median line, was used in all cases. Post‐operative complications and recurrence was recorded.
Results
Sixteen (21.1%) patients were operated because of recurrence. The mean age and body mass index were 33.04 ± 6.78 and 29.86 ± 5.46, respectively. The mean hospital stay period and mean operation time was 2.95 ± 0.76 days and 64.33 ± 8.64 min, respectively. The mean drain removal time was 2.78 ± 0.7 days and mean follow‐up was 13.46 ± 4.31 months. There were flap oedema and seroma in one (1.3%) and four (5.3%) of the patients, respectively. Surgical site infection necessitating antibiotic treatment developed in three (3.9%) patients. The overall post‐operative complication rate was 10.5% and there was recurrence in one patient on follow‐up period (1.3%).
Conclusion
CNCEF with kite incision method, which shifts midline, is a safe and reliable method with acceptable post‐operative complication and recurrence rates in extensive sacrococcygeal PS patients.
Complete natal cleft excision with Kite incision in extensive pilonidal sinus disease.
In this paper, we aim a kinematic method to investigate the behavior of polarized light in an optical fiber. Geometric phase equations are obtained using elliptic quaternions. The behavior of ...polarized light is studied for three conditions where the angle between the polarization vector (electric field) and the Darboux frame fields are constant on the ellipsoid. For these conditions, the polarization vector experiences a homothetic motion on the ellipsoid. Moreover, this motion can be defined by the Fermi Walker parallelism rule. Also, the traced curves of the polarization vector (Rytov curves) are calculated by homothetic motions and Elliptic quaternions. Additionally, the magnetic vector field related to the polarization vector is derived. Then the characterizations of the electromagnetic curve are obtained. To reinforce the theory, the homothetic motions of the polarization vector on the ellipsoid are visualized by giving examples for each case.
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