Geometrical transformations and development of geometry

A way of systematizing different geometries by geometrical transformations is presented by using Klein’s perspective. Symmetry is considered the basic precedent, and different groups of transformations are introduced by using Erlangen program ideas. So, to focus on the importance of geometric transformations, two purposes are shown: first, the classification of geometries is based on the invariants of geometric transformations; and second, the construction of geometry unlike the theory of deduction can be done by means of geometric transformations.

The new electromagnetic tetrads, infinite tetrad nesting and the non-trivial emergence of complex numbers in real theories of gravitation

How complex numbers get into play in a non-trivial way in real theories of gravitation is relevant since in a unified structure they should be able to relate in a natural way with quantum theories. For a long time this issue has been lingering on both relativistic formulations and quantum theories. We will analyze this fundamental subject under the light of new group isomorphism theorems linking local internal groups of transformations and local groups of spacetime transformations. The bridge between these two kinds of transformations is represented by new tetrads introduced previously. It is precisely through these local tetrad structures that we will provide a non-trivial answer to this old issue. These new tetrads have two fundamental building components, the skeletons and the gauge vectors. It is these constructive elements that provide the mathematical support that allows to prove group isomorphism theorems. In addition to this, we will prove a unique new property, the infinite tetrad nesting, alternating the nesting with non-Abelian tetrads in the construction of the tetrad gauge vectors. As an application we will demonstrate an alternative proof of a new group isomorphism theorem.

Modeling the Cluster Organization of Al2O3 for Obtaining more Perfect Materials

This paper considers the issues of building a model of the cluster organization of Al2O3 on principles immediately associated with the involvement of a non-Euclidian technique of describing. As a model space which most adequately describes the internal structure of a real crystal, a finite closed space with elliptic metric and constant positive Gaussian curvature (K=1) was chosen, which assumes the realization of lattice systems in compliance with Fedorov groups of transformations. An algorithm for generating the cluster structures is presented, which determines the sequence of filling of the model space with cations and anions, taking into account the symmetry of a microstructure (a Fedorov group) and the electrostatic parameters of ions. Calculations are given of the geometrical sizes of nanoand microcrystalline complexes of Al2O3, taking into account the structural features of their formation at the cluster level. Practical applications connected with improving the structural characteristics of crystalline materials are discussed.

Some Exact Solutions for a Klein Gordon Equation

In solving practical problems in science and engineering arises as a direct consequence differential equations that explains the dynamics of the phenomena. Finding exact solutions to this equations provides importan information about the behavior of physical systems. The Lie symmetry method allows tofind invariant solutions under certain groups of transformations for differential equations.This method not very well known and used is of great importance in the scientific community. By this approach it was possible to find several exactinvariant solutions for the Klein Gordon Equation uxx − utt = k(u). A particularcase, The Kolmogorov equation uxx − utt = k1u + k2un was considered.These equations appear in the study of relativistic and quantum physics. The general solutions found, could be used for future explorations on the study for other specific K(u) functions.