UID:
almahu_9947367760802882
Format:
1 online resource (391 p.)
ISBN:
1-281-79805-3
,
9786611798055
,
0-08-087252-2
Series Statement:
North-Holland mathematics studies ; 141
Content:
In this volume, a tower of surreal number fields is defined, each being a real-closed field having a canonical formal power series structure and many other higher order properties. Formal versions of such theorems as the Implicit Function Theorem hold over such fields. The Main Theorem states that every formal power series in a finite number of variables over a surreal field has a positive radius of hyper-convergence within which it may be evaluated. Analytic functions of several surreal and surcomplex variables can then be defined and studied. Some first results in the one variable case are d
Note:
Description based upon print version of record.
,
Front Cover; Foundation of Analysis Over Surreal Number Fields; Copyright Page; Preface; Table of Contents; CHAPTER 0. INTRODUCTION; 0.00 The real numbers; 0.01 nξ - fields; 0.02 The ξ -topology on an nξ -set; 0.03 Conway's field No of surreal numbers; 0.04 Valuation theory and surreal number fields; 0.05 Neumann's theorem and hyper-convergence; 0.06 The main theorem; 0.07 Applications of the main theorem; 0.10 Exposition versus research; 0.11 References and indexing; 0.20 Prerequisites; 0.30 Acknowledgements; CHAPTER 1. PRELIMINARIES; 1.00 Class theory and set theory
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1.01 Ordered sets and order types1.02 Well-ordered sets: Cantor's and von Neumann's ordinal numbers; 1.03 Equipotent sets, choice, and cardinal numbers; 1.10 The interval topology; 1.11 The relative topology; 1.20 Cuts and gaps; 1.30 Cofinal and coinitial sets, characters and saturation; 1.40 nξ-classes; 1.50 Canpact ordered spaces; 1.60 Ordered Abelian groups; 1.61 Hahn valuations on ordered groups; 1.62 Pseudo-convergent sequences in Abelian groups with valuation; 1.63 Skeletons, Hahn groups, and extensions of ordered groups; 1.64 Hahn's embedding theorem; 1.65 Ordered direct sums in ξH
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1.66 Camplete and incomplete ordered groups1.70 Ordered rings and fields; 1 .71 The Artin-Schreier theory of real-closed fields; 1.72 Polynomials in one variable over real-closed fields; 1.73 Rational functions in one variable over real-closed fields; 1.74 Rolle's theorem and applications; 1.75 Embedding an ordered field in a real-closed nξ-field; CHAPTER 2. THE ξ-TOPOLOGY; 2.00 The interval topology on an nξ -class; 2.01 The ξ -topology; 2.02 A comparison of ξ-topologies and wξ -additive spaces 5; 2.10 The ξ-topology on ordered sets and classes; 2.11 ξ-closed subclasses of X
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2.12 The relative ξ-topology2.13 On the possible non-existence of ξ-closures and ξ-interiors; 2.20 The main theorem on ξ-connected subspaces of nξ -classes; 2.21 That open subclasses of nξ-classes are ξ-locally connected; 2.30 The main theorem on ξ-compact subspaces of nξ -classes; 2.31 ξ-compact subspaces that are not ξ-closed; 2.40 ξ-continuous maps of ordered classes; 2.41 An additional theorem on ξ-continuous maps; CHAPTER 3. THE ξ-TOPOLOGY ON AFFINE n-SPACE; 3.00 The strong topology and semi-algebraic sets; 3.10 The affine line; 3.20 The ξ-topology on Rn
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3.21 ξ-continuous maps between affine spaces3.30 ξ-connected subspaces of ξRn; 3.40 R as a topological field in the ξ-topology; 3.41 Rn as a topological vector space over R , in the ξ-topology; 3.42 The field C = R( i ), as a topological field; 3.43 Other examples of ξ-continuous maps; CHAPTER 4. INTRODUCTION TO THE SURREAL FIELD No; 4.00 Surreal numbers; 4.01 Conway's construction; 4.02 The Cuesta Dutari construction of No; 4.03 An abstract characterization of a full class of surreal numbers; 4.04 Subtraction in No; 4.05 Addition in No; 4.06 Multiplication in No
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4.07 Order and multiplication in No
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English
Additional Edition:
ISBN 0-444-70226-1
Language:
English
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