Glimpses beyond the Two Millennia Old Bondage of the Archimedean Axiom: Part 3

Group Invariant Entanglements in Generalized Tensor Products (by Elemer E. Rosinger): Abstract: The group invariance of entanglement is obtained within a very general and simple setup of the latter, given by a recently introduced considerably extended concept of tensor products. This general approach to entanglement - unlike the usual one given in the particular setup of tensor products of vector spaces - turns out not to need any specific algebraic structure. The resulting advantage is that, entanglement being in fact defined by a negation, its presence in a general setup increases the chances of its manifestations, thus also its availability as a resource.

Space-Time Foam Differential Algebras of Generalized Functions and a Global Cauchy-Kovalevskaia Theorem (by Elemer E. Rosinger): Abstract: The new global version of the Cauchy-Kovalevskaia theorem presented here is a strengthening and extension of the regularity of similar global solutions obtained earlier by the author. Recently the space-time foam differential algebras of generalized functions with dense singularities were introduced. A main motivation for these algebras comes from the so called space-time foam structures in General Relativity, where the set of singularities can be dense. A variety of applications of these algebras have been presented elsewhere, including in de Rham cohomology, Abstract Differential Geometry, Quantum Gravity, etc. Here a global Cauchy-Kovalevskaia theorem is presented for arbitrary analytic nonlinear systems of PDEs. The respective global generalized solutions are analytic on the whole of the domain of the equations considered, except for singularity sets which are closed and nowhere dense, and which upon convenience can be chosen to have zero Lebesgue measure. In view of the severe limitations due to the polynomial type growth conditions in the definition of Colombeau algebras, the class of singularities such algebras can deal with is considerably limited. Consequently, in such algebras one cannot even formulate, let alone obtain, the global version of the Cauchy-Kovalevskaia theorem presented in this paper.

Brief Lecture Notes on Self-Referential Mathematics and Beyond (by Elemer E. Rosinger): Abstract: Recently delivered lectures on Self-Referential Mathematics, [2], at the Department of Mathematics and Applied Mathematics, University of Pretoria, are briefly presented. Comments follow on the subject, as well as on Inconsistent Mathematics.