Geometrical particles and dualities related to curves in null de Sitter 3-sphere

In this paper, we confine the trajectory of geometrical particles to the de Sitter three-dimensional space–time, we model geometrical particles as spacelike tachyons. Using the Legendrian duality theory of pseudo-spheres and contact manifolds theory we establish the dual relationships between spacelike moving trajectory of geometrical particles and future nullcone hypersurfaces.

Dynamics of the Dual Relationships

Nature creates its systems in the form of a sequence of hierarchy levels that unite many heterogeneous objects, based on dual relationships. The article discusses the general principles of work and the manifestation of dual relations, as well as a measuring instrument for their functioning.

Hyperbolic worldsheets and worldlines of null Cartan curves in de Sitter 3-space

The hyperbolic worldsheets and the hyperbolic worldline generated by null Cartan curves are defined and their geometric properties are investigated. As applications of singularity theory, the singularities of the hyperbolic worldsheets and the hyperbolic worldline are classified by using the approach of the unfolding theory in singularity theory. It is shown that under appropriate conditions, the hyperbolic worldsheet is diffeomorphic to cuspidal edge or swallowtail type of singularity and the hyperbolic worldline is diffeomorphic to cusp. An important geometric invariant which has a close relation with the singularities of the hyperbolic worldsheets and worldlines is found such that the singularities of the hyperbolic worldsheets and worldlines can be characterized by the invariant. Meanwhile, the contact of the spacelike normal curve of a null Cartan curve with hyperbolic quadric or world hypersheet is discussed in detail. In addition, the dual relationships between the spacelike normal curve of a null Cartan curve and the hyperbolic worldsheet are described. Moreover, it is demonstrated that the spacelike normal curve of a null Cartan curve and the hyperbolic worldsheet are [Formula: see text]-dual each other.

Ethics in Play Therapy Consultation and Supervision

Ethics in play therapy consultation and supervision can be complex because of the “layered” nature of the relationships involved in each of these indirect services. In this chapter, we discuss the differential obligations the play therapy consultant or supervisor owes to the consulate/supervisee and the client. The ethical constructs of informed consent, competence, confidentiality, and dual relationships are discussed in relationship to the consultee/supervisee and the client. While the obligations are similar in these indirect services, there are also different obligations in each of these areas.