Local Functions : Algebras, Ideals, and Reduced Power Algebras

📝 Abstract

A further significant extension is presented of the infinitely large class of differential algebras of generalized functions which are the basic structures in the nonlinear algebraic theory listed under 46F30 in the AMS Mathematical Subject Classification. These algebras are constructed as {\it reduced powers}, when seen in terms of Model Theory. The major advantage of these differential algebras of generalized functions is that they allow their elements to have singularities on {\it dense} subsets of their domain of definition, and {\it without} any restrictions on the respective generalized functions in the neighbourhood of their singularities. Their applications have so far been in 1) solving large classes of systems of nonlinear PDEs, 2) highly singular problems in Differential Geometry, with respective applications in modern Physics, including General Relativity and Quantum Gravity. These infinite classes of algebras contain as a particular case the Colombeau algebras, since in the latter algebras rather strongly limiting growth conditions, namely, of polynomial type, are required on the generalized functions in the neighbourhood of their singularities.

💡 Analysis

A further significant extension is presented of the infinitely large class of differential algebras of generalized functions which are the basic structures in the nonlinear algebraic theory listed under 46F30 in the AMS Mathematical Subject Classification. These algebras are constructed as {\it reduced powers}, when seen in terms of Model Theory. The major advantage of these differential algebras of generalized functions is that they allow their elements to have singularities on {\it dense} subsets of their domain of definition, and {\it without} any restrictions on the respective generalized functions in the neighbourhood of their singularities. Their applications have so far been in 1) solving large classes of systems of nonlinear PDEs, 2) highly singular problems in Differential Geometry, with respective applications in modern Physics, including General Relativity and Quantum Gravity. These infinite classes of algebras contain as a particular case the Colombeau algebras, since in the latter algebras rather strongly limiting growth conditions, namely, of polynomial type, are required on the generalized functions in the neighbourhood of their singularities.

📄 Content

arXiv:0912.4049v3 [math.GM] 28 Jun 2010 Local Functions : Algebras, Ideals, and Reduced Power Algebras Elem´er E Rosinger Department of Mathematics and Applied Mathematics University of Pretoria Pretoria 0002 South Africa eerosinger@hotmail.com Dedicated to Marie-Louise Nykamp Abstract A further significant extension is presented of the infinitely large class of differential algebras of generalized functions which are the basic structures in the nonlinear algebraic theory listed under 46F30 in the AMS Mathematical Subject Classification. These algebras are con- structed as reduced powers, when seen in terms of Model Theory. The major advantage of these differential algebras of generalized functions is that they allow their elements to have singularities on dense subsets of their domain of definition, and without any restrictions on the re- spective generalized functions in the neighbourhood of their singular- ities. Their applications have so far been in 1) solving large classes of systems of nonlinear PDEs, 2) highly singular problems in Differential Geometry, with respective applications in modern Physics, including General Relativity and Quantum Gravity. These infinite classes of algebras contain as a particular case the Colombeau algebras, since in the latter algebras rather strongly limiting growth conditions, namely, of polynomial type, are required on the generalized functions in the neighbourhood of their singularities. 1 ”We do not possess any method at all to derive systematically solutions that are free of singularities…” Albert Einstein : The Meaning of Relativity. Princeton Univ. Press, 1956, p. 165 0. Preliminaries The nonlinear theory of generalized functions, see 46F30 in the AMS Mathematical Subject Classification of the AMS, has known a wide range of applications in solving large classes of nonlinear PDEs, see [6-8,10-12,15-17,20,22,33,34,36,40,44,54] and the literature cited there, as well as in Abstract Differential Geometry with applications to mod- ern Physics, including General Relativity and Quantum Gravity, [9,13- 19,21,56-61], as well as in the study of manifolds, [31]. These algebras are constructed as reduced powers, when seen in terms of Model Theory, [80], and as such, they belong to the same kind of general and rather simple construction as their earlier and more par- ticular cases in [1-22,25-32,36-39,41-45,47-49,53,54]. A major interest in the large classes of algebras in the mentioned liter- ature as well as in this paper comes from the following three properties which are typical and exclusive to these algebras, namely that, their generalized functions • are constructed in a rather simple way, requiring only Algebra 101, and specifically, basic results and methods in ring theory, plus a few facts about filters on arbitrary infinite sets, • can have singularities on dense subsets of their domain of defi- nition of generalized functions, thus can have sets of singularity points with larger cardinal than that of their sets of regular, that is, non-singular points, since the only restriction on singularity sets is that their complement, that is, the sets of nonsingular points be dense in the domain of generalized functions, 2 • are not subjected to any conditions, in particular, not to growth conditions, in the neighbourhood of their singularities. This nonlinear theory of generalized functions has the further advan- tage of being given by an infinite variety of differential algebras of gen- eralized functions, algebras which contain as rather small subspaces the linear space of Schwartz distributions. Consequently, there is a wide liberty in choosing one or another such algebra when dealing with specific problems involving singularities. And not seldom, such a liberty of choice is welcome since it facilitates the appropriate ap- proach to the specific singularities at hand, even if traditionally, one may tend to think that one should rather be given one single algebra, an algebra which would be universally and equally useful in dealing with all possible kind of singularities. As it happens, however, the variety of possible singularities turns out to be so wide as to require more than one single algebra for its proper treatment. Indeed, the essence of this phenomenon is related to the following simple yet fundamental fact • the operation of addition does not appear to branch into alter- natives, when dealing with singularities, on the other hand, however • the operation of multiplication does naturally and inevitably branch into infinitely many different alternatives, when deal- ing with singularities, as shown by most simple algebraic, more precisely, ring theoretic arguments, [7,8,10]. The respective algebras do in fact extend, or in other words general- ize, large classes of functions f : X −→E, where X is a domain in an Euclidean space or a finite dimensional manifold, while E is any real or complex, commutative or non-commutative unital Banach al- gebra, in particular, the usual filed R of

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