Related papers: Line geometry and electromagnetism III: groups of …
A new concept for the geometrisation of electromagnetic interaction is proposed. Instead of the concept "extended field--point sources", interacting Maxwell's and Dirac's fields are considered as a unified closed noneuclidean and…
Field Arithmetic studies the interplay between arithmetical properties of fields and their absolute Galois groups. Here we studies fields satisfying local global principles for rational points of varieties and profinite groups satisfying…
The purpose of this book is to give an exposition of geometry, from a point of view which complements Klein's Erlangen program. The emphasis is on extending the classical Euclidean geometry to the finite case, but it goes beyond that. After…
This note summarizes the physics and mathematics of Lorentz transformations for massless particles, specifically for photons. We provide a complete analytical derivation of Wigner's little group matrix and a closed formula for the…
In this paper we classify in terms of Lie point symmetries the three-dimensional nonrelativistic motion of charged particles in arbitrary time-independent electromagnetic fields. The classification is made on the ground of equivalence…
A simple procedure to obtain complete, closed expressions for Lie algebra invariants is presented. The invariants are ultimately polynomials in the group parameters. The construction of finite group elements require the use of projectors,…
Motivated by Felix Klein's notion that geometry is governed by its group of symmetry transformations, Charles Ehresmann initiated the study of geometric structures on topological spaces locally modeled on a homogeneous space of a Lie group.…
The present work deals with the search of useful physical applications of some generalized groups of metric transformations. We put forward different proposals and focus our attention on the implementation of one of them. Particularly, the…
We present an exposition of our ongoing project in a new area of applicable mathematics: practical computation with finitely generated linear groups over infinite fields. Methodology and algorithms available for practical computation in…
The intimate link between complex geometry and the problem of the pre-metric formulation of electromagnetism is explored. In particular, the relationship between 3+1 decompositions of R4 and the decompositions of the vector space of…
Given a connection on a meromorphic vector bundle over a compact Riemann surface with reductive Galois group, we associate to it a projective variety. Connections such that their associated projective variety are curves can be classified,…
We consider the projective linear group $\mathrm{PSL}(3,\mathbb{H})$. We have investigated the reversibility problem in this group and use the reversibility to offer an algebraic characterization of the dynamical types of…
The group SU(3) is parameterized in terms of generalized ``Euler angles''. The differential operators of SU(3) corresponding to the Lie Algebra elements are obtained, the invariant forms are found, the group invariant volume element is…
A new picture of Quantum Mechanics based on the theory of groupoids is presented. This picture provides the mathematical background for Schwinger's algebra of selective measurements and helps to understand its scope and eventual…
We propose Geometric Clifford Algebra Networks (GCANs) for modeling dynamical systems. GCANs are based on symmetry group transformations using geometric (Clifford) algebras. We first review the quintessence of modern (plane-based) geometric…
Looking to the history of mathematics one could find out two outer approaches to Geometry. First one (algebraic) is due to Descartes and second one (group-theoretic)--to Klein. We will see that they are not rivalling but are tied (by…
The Gale transform, an involution on sets of points in projective space, appears in a multitude of guises, in subjects as diverse as optimization, coding theory, theta-functions, and recently in our proof that certain general sets of points…
In the shape analysis approach to computer vision problems, one treats shapes as points in an infinite-dimensional Riemannian manifold, thereby facilitating algorithms for statistical calculations such as geodesic distance between shapes…
We construct nonlinear representations of the Poincare, Galilei, and conformal algebras on a set of the vector-functions $\Psi =(\vec E, \vec H)$. A nonlinear complex equation of Euler type for the electromagnetic field is proposed. The…
We study linearizability of actions of finite groups on singular cubic threefolds, using cohomological tools, intermediate Jacobians, Burnside invariants, and the equivariant Minimal Model Program.