arXiv subjects

E. E. Rosinger

Publications and source records attributed to E. E. Rosinger.

7 recordsLinked to original sources

arXiv2004

Short introduction to Nonstandard Analysis

These lecture notes, to be completed in a later version, offer a short and rigorous introduction to Nostandard Analysis, mainly aimed to reach to a presentation of the basics of Loeb integration, and in particular, Loeb measures. The Abraham Robinson version of Nostandard Analysis is pursued, with a respective incursion into Superstructures. Two formal languages are used, one simpler at first, and then later, one for the full blown theory. A number of detailed comments, not often found in textbooks, accompanies the text.

math.GM↗

arXiv2004

Two problems in Classical Mechanics

A problem about the present structure of dimensional analysis, and another one about the differences between solids and fluids are suggested. Both problems appear to have certain foundational aspects.

physics.gen-ph↗

arXiv2004

A question in the axiomatic approach to Quantum Mechanics

The classical Hilbert space formulation of the axioms of Quantum Mechanics appears to leave open the question whether the Hermitian operators which are associated with the observables of a finite non-relativistic quantum system are uniquely determined.

quant-ph↗

arXiv2004

Basics of Quantum Computation (Part 1)

The aim of this textbook is to bridge in regard of quantum computation what proves to be a considerable threshold even to the usual science trained readership between the level of science popularization, and on the other hand, the presently available more encyclopedic textbooks. In this respect the present textbook is aimed to be a short, simple, rigorous and direct introduction, addressing itself only to quantum computation proper.

quant-ph↗

arXiv2004

Can there be a general nonlinear PDE theory for the existence of solutions ?

Contrary to widespread perception, there is ever since 1994 a unified, general type independent theory for the existence of solutions for very large classes of nonlinear systems of PDEs. This solution method is based on the Dedekind order completion of suitable spaces of piece-wise smooth functions on the Euclidean domains of definition of the respective PDEs. The method can also deal with associated initial and/or boundary value problems. The solutions obtained can be assimilated with usual measurable functions or even with Hausdorff continuous functions on the respective Euclidean domains. It is important to note that the use of the order completion method does not require any monotonicity condition on the nonlinear systems of PDEs involved. One of the major advantages of the order completion method is that it eliminates the algebra based dichotomy "linear versus nonlinear" PDEs, treating both cases with equal ease. Furthermore, the order completion method does not introduce the dichotomy "monotonous versus non-monotonous" PDEs. None of the known functional analytic methods can exhibit such a performance, since in addition to topology, such methods are significantly based on algebra.

math.AP↗

arXiv2004

How to solve smooth nonlinear PDEs in algebras of generalized functions with dense singularities

As a significant strengthening of properties of earlier algebras of generalized functions, here are presented classes of such algebras which can deal with dense singularities. In fact, the cardinal of the set of singular points can be larger than that of the nonsingular points. This large class of singularities allows the solution of large classes of smooth nonlinear PDEs. This in certain ways overcomes the celebrated 1957 H. Lewy impossibility result.

math.AP↗

arXiv2004

Space-time foam dense singularities and de Rham cohomology

In an earlier paper of the authors it was shown that the sheaf theoretically based recently developed abstract differential geometry of the first author can in an easy and natural manner incorporate singularities on arbitrary closed nowhere dense sets in Euclidean spaces, singularities which therefore can have arbitrary large positive Lebesgue measure. As also shown, one can construct in such a singular context a de Rham cohomology, as well as a short exponential sequence, both of which are fundamental in differential geometry. In this paper, these results are significantly strengthened, motivated by the so called space-time foam structures in general relativity, where singularities can be dense. In fact, this time one can deal with singularities on arbitrary sets, provided that their complementaries are dense, as well. In particular, the cardinal of the set of singularities can be larger than that of the nonsingular points.

math.DG↗