Hausdorff continuous solutions of arbitrary continuous nonlinear PDEs through the order completion method

In 1994 we showed that very large classes of systems of nonlinear PDEs have solutions which can be assimilated with usual measurable functions on the Euclidean domains of definition of the respective equations. Recently, the regularity of such solutions has significantly been improved by showing that they can in fact be assimilated with Hausdorff continuous functions. The method of solution of PDEs is based on the Dedekind order completion of spaces of smooth functions which are defined on the domains of the given equations. In this way, the method does not use functional analytic approaches, or any of the customary distributions, hyperfunctions, or other generalized functions

New Symmetry Groups for Generalized Solutions of ODEs

It is shown for a simple ODE that it has many symmetry groups beyond its usual Lie group symmetries, when its generalized solutions are considered within the nowhere dense differential algebra of generalized functions

Scattering in highly singular potentials

Recently, in Quantum Field theory, there has been an interest in scattering in highly singular potentials. Here, solutions to the stationary Schroedinger equation are presented when the potential is a multiple of an arbitrary positive power of the Dirac delta distribution. The one dimensional, and the spherically symmetric three dimensional cases are dealt with

Difficulties with Learning and Teaching Calculus

Several thoughts are presented on the long ongoing difficulties both students and academics face related to Calculus 101. Some of these thoughts may have a more general interest

Extending Mappings between Posets

A variety of possible extensions of mappings between posets to their Dedekind order completion is presented. One of such extensions has recently been used for solving large classes of nonlinear systems of partial differential equations with possibly associated initial and/or boundary value problems

Brief Notes on Sheaves Theory

Essentials of sheaves are briefly presented, followed by related comments on presheaves, bundles, manifolds and singularities, aiming to point to their differences not only in their different formal mathematical structures, but also in the very purposes for which they were introduced in the first place

On the Safe Use of Inconsistent Mathematics

A method is presented for using the consistent part of inconsistent axiomatic systems

An extension of the Eilenberg - Mac Lane concept of category in the case of nonrigid structures

Nonrigid mathematical structures may no longer form usual Eilenberg - Mac Lane categories, but more general ones, as illustrated by pseudo-topologies. A rather general concept of pseudo-topology was used in constructing differential algebras of generalized functions containing the Schwartz distributions, [4-6,8-11]. These algebras proved to be convenient in solving large classes of nonlinear partial differential equations, see [12,13] and the literature cited there, as well as section 46F30 in the Subject Classification 2000 of the American Mathematical Society, at www.ams.org/index/msc/46Fxx.html The totality of such pseudo-topologies no longer constitutes a usual Eilenberg - Mac Lane category, but an extended one which is presented here. Other nonrigid mathematical structures are mentioned and treated briefly. This is a revised and augmented version of the earlier published paper [7]

What is wrong with von Neumann's theorem on "no hidden variables"

It is shown that, although correct mathematically, the celebrated 1932 theorem of von Neumann which is often interpreted as proving the impossibility of the existence of "hidden variables" in Quantum Mechanics, is in fact based on an assumption which is physically not reasonable. Apart from that, the alleged conclusion of von Neumann proving the impossibility of the existence of "hidden variables" was already set aside in 1952 by the counterexample of the possibility of a physical theory, such as given by what is usually called the "Bohmian Mechanics". Similar arguments apply to other two well known mathematical theorems, namely, of Gleason, and of Kochen and Specker, which have often been seen as equally proving the impossibility of the existence of "hidden variables" in Quantum Mechanics

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