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Ultraproducts in Analysis
 IN ANALYSIS AND LOGIC, VOLUME 262 OF LONDON MATHEMATICAL SOCIETY LECTURE NOTES
, 2002
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HILBERT’S FIFTH PROBLEM FOR LOCAL GROUPS
, 708
"... Abstract. We solve Hilbert’s fifth problem for local groups: every locally euclidean local group is locally isomorphic to a Lie group. Jacoby claimed a proof of this in 1957, but this proof is seriously flawed. We use methods from nonstandard analysis and model our solution after a treatment of Hilb ..."
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Abstract. We solve Hilbert’s fifth problem for local groups: every locally euclidean local group is locally isomorphic to a Lie group. Jacoby claimed a proof of this in 1957, but this proof is seriously flawed. We use methods from nonstandard analysis and model our solution after a treatment of Hilbert’s fifth problem for global groups by Hirschfeld. 1.
Nonstandard Analysis and Applications UC Davis Mathematics StudentRun Seminar Presentation notes
, 2006
"... Leibniz and Newton, both independently credited as inventors of calculus, relied on the concept of an infinitesimal (nonzero “numbers ” that were “infinitely small”) in their development. Our standard rigorous treatment of calculus involves an “arbitrary epsilon ” limit definiton. There’s an alterna ..."
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Leibniz and Newton, both independently credited as inventors of calculus, relied on the concept of an infinitesimal (nonzero “numbers ” that were “infinitely small”) in their development. Our standard rigorous treatment of calculus involves an “arbitrary epsilon ” limit definiton. There’s an alternative rigorous study of calculus beyond the limits of real analysis. In 1961, Robinson constructed the “hyperreal line ” as a direct consequence of the compactness theorem of first order logic. We will examine some typical proofs of known statements in advanced calculus and extend the nonstandard framework to other mathematical fields. 1 Introduction to FirstOrder Logic 1.1 FirstOrder Languages Define a firstorder language to be a set of symbols, as a base containing a symbol for the logical NAND, quantifiers ∃ and ∀, equality (=), grouping parenthesies/brackets, and variables (as many as are needed). Though NAND is all
Nonstandard Analysis and Ultraproducts in Banach Spaces and Functional Analysis
"... this article is also available on the author's Internet home page (address above). The approach to NSA described here is simpler than what one finds in most expositions of the subject. In particular, there are no logical formulas at the beginning and no set theoretic structures. However, the formali ..."
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this article is also available on the author's Internet home page (address above). The approach to NSA described here is simpler than what one finds in most expositions of the subject. In particular, there are no logical formulas at the beginning and no set theoretic structures. However, the formalism of NSA presented here is adequate for all applications; for more details about how other formalisms for NSA can be interpreted in the one presented here, see [8].
NONSTANDARD ANALYSIS IN POINTSET TOPOLOGY
"... Abstract We present Nonstandard Analysis by three axioms: the Extension, Transfer and Saturation Principles in the framework of the superstructure of a given infinite set. We also present several applications of this axiomatic approach to pointset topology. Some of the topological topics such as th ..."
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Abstract We present Nonstandard Analysis by three axioms: the Extension, Transfer and Saturation Principles in the framework of the superstructure of a given infinite set. We also present several applications of this axiomatic approach to pointset topology. Some of the topological topics such as the Hewitt realcompactification and the nonstandard characterization of the sober spaces seem to be new in the literature on nonstandard analysis. Others have already close counterparts but they are presented here with essential simplifications.
NONSTANDARD HULLS OF LOCALLY EXPONENTIAL LIE ALGEBRAS
, 804
"... Abstract. We show how to construct the nonstandard hull of certain infinitedimensional Lie algebras in order to generalize a theorem of Pestov on the enlargeability of BanachLie algebras. In the process, we consider a nonstandard smoothness condition on functions between locally convex spaces to e ..."
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Abstract. We show how to construct the nonstandard hull of certain infinitedimensional Lie algebras in order to generalize a theorem of Pestov on the enlargeability of BanachLie algebras. In the process, we consider a nonstandard smoothness condition on functions between locally convex spaces to ensure that the induced function between the nonstandard hulls is smooth. We also discuss some conditions on a function between locally convex spaces which guarantee that its nonstandard extension maps finite points to finite points. 1.
ARITHMETIC NULLSTELLENSATZ AND NONSTANDARD METHODS
"... Abstract. In this study we find height bounds for polynomial rings over integral domains. We apply nonstandard methods and hence our constants will be ineffective. Furthermore we consider unique factorization domains and possible bounds for valuation rings and arithmetical functions. hal00738713, v ..."
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Abstract. In this study we find height bounds for polynomial rings over integral domains. We apply nonstandard methods and hence our constants will be ineffective. Furthermore we consider unique factorization domains and possible bounds for valuation rings and arithmetical functions. hal00738713, version 2 17 Oct 2012 1.