Kinematical modeling of circular and elliptical polarization of the polarized light via screw transformation

In this paper, it is shown that the motion of a polarized light wave in an optical fiber has a special meaning that relates the scalar and vector part of the dual quaternion to the kinematics concept of a screw. We describe the circular and elliptical polarization of a light wave according to how the electric field vector varies in a projection onto a plane perpendicular to the propagation direction using the Clifford algebra of the dual quaternion. Also, we determine the circular-Rytov curve (CR ) and elliptical-Rytov curve (ER ), which described the paths traced out by the tip of the electric field vector as it propagates in space, related to the polarization state of the electric field ε. The elliptical and circular polarization states are expressed by using the four Stokes parameters and their matrix form. Furthermore, some motivating examples are given and visualized their images with the help of the MAPLE program.

Miłość – świadectwo osobowej dojrzałości. Perspektywa filozoficzno-pedagogiczna

Love: An Indicator of Personal Maturity. A Philosophical and Pedagogical Perspective The article is a philosophical analysis of the problem of love from the perspective of human maturity. It aims to recall the philosophical theses (including those by Aristotle, Pieper, Plato, and Wojtyla) that are pedagogically oriented. The article consists of six parts, arranged in six questions, which allow us to define love as a pedagogically desirable attitude. The ontic factors of love are primarily considered, but so too is the contemporary cultural situation that constitutes a context of human relationships. The determinants of this context are individualism, consumerism, metaphysical indifference, and more recently also pandemic-induced isolationism. In Part 1, love is defined as the „being together” of people governed by a personalistic norm. In Part 2, the emphasis is on its corporeal-sensual-spiritual complexity and, in particular, on the aptitude of the will and the intellect. Part 3 considers the principle of love: love is governed by an affirmation, not of a personal feature (or set of features), but of the person’s existence as a person. In Part 4, love is analyzed in terms of its inherent aspect of giving: giving to someone and being given to. Attention is drawn to the need to combine both attitudes in mature love. Part 5 is about friendship, the most perfect form of love. It is noted that although it binds a small group of people (it is exclusive), it also has a socially-oriented vector. Part 6 is devoted to the issue of the deficit of love (indifference, hatred) and its sources in the decline of existential thinking among contemporary people.

Calculation of vectorial derivatives for functions of a quaternion variable and their properties

Various attempts have been made in defining the derivative of a quaternionic function due to the noncommutativity of the product over quaternions. We observe that the difference in the left and right operations caused by the noncommutativity of the quaternion product is determined by the vector part of the quaternion. In this paper, we propose a corresponding derivative to replace the derivative of a quaternion-valued function of a quaternionic variable using the component terms of a quaternion. Further, the analogous constant, product, and quotient rules for the proposed calculations are given. Application of the proposed derivatives is provided to compute the derivatives of elementary functions. Several illustrations are also presented.

METHOD FOR CONSTRUCTING THE COMMUTATIVE ALGEBRA OF QUATERNION AND OCTONION

In this paper, we solve the problem of constructing a commutative algebra of quaternions and octonions. A proof of the theorem is given that the commutativity of quaternions can be ensured by specifying a set of sign coefficients of the directions of reference of the angles between the radius vectors in the coordinate planes of the vector part of the coordinate system of the quaternion space. The method proposed in the development of quaternions possessing the commutative properties of multiplication is used further to construct a commutative octonion algebra. The results obtained on improving the algebra of quaternions and octonions can be used in the development of new hypercomplex numbers with division over the field of real numbers, and can also find application for solving a number of scientific and technical problems in the areas of field theory, physical electronics, robotics, and digital processing of multidimensional signals.

Weak-lensing observables in relativistic N-body simulations

ABSTRACT We present a numerical weak-lensing analysis that is fully relativistic and non-perturbative for the scalar part of the gravitational potential and first order in the vector part, frame dragging. Integrating the photon geodesics backwards from the observer to the emitters, we solve the Sachs optical equations and study in detail the weak-lensing convergence, ellipticity and rotation. For the first time, we apply such an analysis to a high-resolution relativistic N-body simulation, which consistently includes the leading-order corrections due to general relativity on both large and small scales. These are related to the question of gauge choice and to post-Newtonian corrections, respectively. We present the angular power spectra and one-point probability distribution functions for the weak-lensing variables, which we find are broadly in agreement with comparable Newtonian simulations. Our geometric approach, however, is more robust and flexible, and can therefore be applied consistently to non-standard cosmologies and modified theories of gravity.

Dirac Spin and Angular Momentum of the Stationary Axisymmetric Space-Time in the Teleparallel Gravity

In the framework of teleparallel equivalent to general relativity, the stationary axisymmetric space-time in the teleparallel gravity for two different sets of tetrad fields have been investigated. For these sets, we have obtained the expressions for the torsion vector, torsion axial-vector and the angular momentum of the solution. We found that the obtained expressions of the torsion axial-vector and the angular momentum are, in general quite different in both two sets of tetrad fields, while the expressions for the torsion vector have the same value. Moreover, the vector part connected with Dirac spin has been evaluated as well. Finally, special cases of the stationary axisymmetric space-time are discussed.

Unified quaternionic description of charge and spin transport and intrinsic nonlinearity of spin currents

We present a unified theory of charge and spin transport using quaternionic formalism. It is shown that both charge and spin currents can be combined together to form a quaternionic current. The scalar and vector part of quaternionic currents correspond to charge and spin currents, respectively. We formulate a unitarity condition on the scattering matrix for quaternionic current conservation. It is shown that in the presence of spin flip interactions, a weaker quaternionic unitarity condition implying charge flux conservation but spin flux nonconservation is valid. Using this unified theory, we find that spin currents are intrinsically nonlinear. Its implication for recent experimental observation of spin generation far away from the boundaries are discussed.