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Local Functions : Algebras, Ideals, and Reduced Power Algebras

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.