Alling N.L. — Foundations of Analysis over Surreal Number Fields :: Электронная библиотека попечительского совета мехмата МГУ

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Alling N.L. — Foundations of Analysis over Surreal Number Fields

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Название: Foundations of Analysis over Surreal Number Fields

Автор: Alling N.L.

Аннотация:

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 derived. A primer on Conway's field of surreal numbers is also given.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1987

Количество страниц: 373

Добавлена в каталог: 02.02.2010

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Предметный указатель

$0_{\alpha}$ (= the set of all "old" numbers, born before day $\alpha$ )      4.01
$<_{t}$       4.51 4.53
$CC(\cdot)$ (= set of all Conway cuts in $\cdot$ )      14.61
$CD(\cdot)$ (= set of all Cuesta Dutari cuts in $\cdot$ )      4.01 4.02
$D_{\lambda}$ (= the simplest Dedekind-completion of $0_{\lambda}$ )      5.50
$EVC_{\nu}$ (= extended value class)      1.62
$H_{\xi}$ (= Hausdorff's normal $\eta_{\xi}$ -type)      5.10
$max(\xi Cx)$       7.11
$max(\xi No)$       6.56 7.11
$M_{CX}$ (= the maximal ideal in a valuation ring $0_{CX}$ of $\xi Cx$ )      7.11
$M_{\alpha}$ (= the set of all "made" numbers, born on or before day $\alpha$ )      4.01
$N_{\alpha}$ (= the set of all "new" numbers, born on day a)      4.01
$O_{cx}$ (= a valuation ring of $\xi Cx$ )      7.11
$p(\cdot)$ (= the rank of $\cdot$ )      5.40
$supp(\cdot)$ (= the support of $\cdot$ )      1.63 6.20
$supp(\cdot)$ (= the support of a vector $\cdot\varepsilon M^{n}$ )      7.41
$s_{\pm}$       5.51
$U^{\wedge}$ (= U extended to Y)      8.00
$x_{\alpha}$ (= the ctth approximation to x)      4.50
$[\cdot]$ (= the convex subgroup generated by $\cdot$ )      7.21
$\alpha$ (= commensurate)      1.61 4.30 4.40
$\alpha$ (= equipotent)      1.03
$\alpha$ -term      6.50
$\alpha$ th approximation,      50
$\cdot \rightarrow {\cdot}^{d}$ (= $\omega_{\xi}$ -additive closure operator)      2.02
$\cdot^{C}$ (= Kuratowski closure of $\cdot$ )      2.02
$\chi^{(\cdot)}$ (= Cuesta Dutari completion of $\cdot$ )      11.02
$\emptyset$ (= empty set)      1.00
$\eta$ -character (= true $\eta$ -character)      1.1(0
$\eta_{\xi}$ -class      1.10
$\eta_{\xi}$ -field, 0      3
$\omega \cdot S$ (= subsemi-group generated by a subset S of $G^{+}$ )      7.20 7.21
$\omega$ (= first infinite ordinal)      1.02
$\omega$ -power series      6.43
$\omega^{X}$ (= Conway's exponential in No)      4.40
$\omega_{\alpha}$ (= ath infinite initial ordinal)      1.03
$\omega_{\alpha}^{+}$ (= next larger initial ordinal number)      1.03
$\omega_{\xi}$ -additive spaces      2.02
$\omega_{\xi}$ -closure operator $(\cdot \rightarrow \cdot^{d})$       2.02
$\partial/\partial x_{i}$       7.60
$\pi_{A}$ and $\pi_{B}$ (= projection homomorphisms)      1.65
$\pm^{\infty} (in D_{\lambda})$       5.51
$\pm^{\infty}$       1.10
$\sigma$ (= a map form No to its sign expansions in $\Sigma$ )      4.50
$\Sigma$ (= class of all sign expansions)      4.50
$\underline{1}$ (= the identity element of a formal power series field)      6.20
$\underline{\alpha}$ (= order equivalent)      1.01
$\varepsilon$ -transitive      1.02
$\xi #$ --analytic      8.00
$\xi B$ (= the $\xi$ -topology generated by a base B)      2.01
$\xi Cx$ - (= a surcomplex number field)      7.10
$\xi$ --closed subclasses of $R^{n}$       3.20
$\xi$ -closed      2.01
$\xi$ -compact      2.01 2.30
$\xi$ -components      2.01
$\xi$ -connected (spaces)      2.01 2.20
$\xi$ -continuous (mapping)      2.01
$\xi$ -disconnected (spaces)      2.20
$\xi$ -homeomorphism      2.01
$\xi$ -interior (of a set)      2.13
$\xi$ -open subclass      2.01 2.10
$\xi$ -open subclasses of $R^{n}$       3.20
$\xi$ -path      3 — 30
$\xi$ -path connected      3.30
$\xi$ -pseudo complete (fields)      6.41
$\xi$ -topology      2.01
$\xi$ -topology generated (by a base B)      2.01
$\xi$ -topology on $R^{n}$       3.20
$\{ F, <, b, \beta \}$       Ч.ОЗ 4.60
$\{ F, <, b, \}$       4.03 4.60
$\{ No, <, b\}$       4.04
$\{ \cdot | \cdot \}$       11.00 11.01 11.02. 11.60
$|\cdot|$ (= absolute value in an ordered group)      1.60
$|\cdot|$ (= analytic norm in a aurcomplex field)      7.10
$|\cdot|$ (= cardinal number or power)      1.03
$||\cdot||$ (= norm in $R^{n}$ )      3.00
(0) (= order axiom)      4.60
(E)      (eta axiom)
(F)      (= fullness axiom) �
(FE) (= full eta axiom)      4.60
(IN) (= axiom: there is a strongly inaccessible cardinal number i)      1.00
(SD) (= simple density axiom)      4.60
(В) (= birth-order axiom)      4.60
<<      1.60 4.40
A*      (= set of non — zero elements of a ring A)
AC (axiom of choice)      1.00
Addition (in No)      4.05
Addition theorem (for binomial coefficients)      7.33
Additive subgroup ( $(\cdot, +)$ of a ring or a field $\cdot$ )      1.60
Affine line      3.10
Affine n space      3.00
Analytic at a point      8.00
Analytic norm $(= |\cdot|)$       7.10
Analytic on U      8.00
Archimedean (ordered group, ring or field))      1.60
Archimedean complete      1.63
Archimedean extension      1.63
Archimedean ordered group      1.60
Arcsine (over certain formal power series fields)      7.50
Arctangent (over certain formal power series fields)      7.50
Artin — Schreier theory      1.71
Associative law for multiplication (in No)      4.08
b (= birthday map)      4.01 4.02. f.60
Ball about a point of radius greater than g      7.62
Binomial coefficients      7.32
Binomial coefficients (generalized)      7.93
Binomial series      7.33
Birth-order axiom      4.60
Birth-order function      11.01 11.02 11.03 4.60
Birthday      11.01 1402
Born on day 0      4.00
Breadth (of a pseudo-convergent sequence)      1.62 6.40
Canonical (nature of power series structure on No)      6.57
Cantor's normal form      4.04 6.43
Cardinal (number, = power)      1.03
Change sign (a polynomial)      1.72
Class of surreal numbers (No)      11.03
Class of surreal numbers of height $\beta$       11.03
Class theory      1.00
Closed class (in the interval topology)      1.10
Closed semi-algebraic set      3.00
Closed, semi-algebraic set      3.00
Closure operator $\cdot \rightarrow \cdot^{C}$       2.02
Cofinal (subclass in an ordered class)      1.30
Coinitial (subclass in an ordered class)      1.30
Commensurate (elements in an ordered group, $\alpha$ )      4.40
Common predecessor      52
Compact ordered spaces      1.50
Complete (= Dedekind-complete)      1.20
Complete ordered groups      1.66
Conjugate (roots)      1.72
Connected      1.20
Continue to change sign (in $D_{\lambda}$ )      5.51
Convex (subgroup)      1.60
Convex (subspace)      3.30
Conway cut represents, ^      2
Conway cuts      1.20
Conway's Normal Form      6.50
Conway's Simplicity Theorem, ^      2 ^.03 ^.60
Cosecant (over certain formal power series fields)      7.50
Cosine (= extended cosine function over $\xi Cx$ )      7.51
Cosine (over certain formal power series fields)      7.50
Critical point      1.74
Cuesta Dutari completion (= $CD(\cdot)$       4.02
Cuesta Dutari cut      1.20
Cuesta Dutari cut representation (of a point)      4.02
Cut point      1.20
Cuts      1.20
D (= dyadic numbers)      4.21
Decomposition (of formal power series fields)      7.81

Dedekind cut      1. 20
Dedekind-complete      1.20
Dedekind-completion of $0_{\lambda}$       5.50
Degree $\infty$ (polynomial)      7.70
Degree (of $v\in Z(\geq0)^{n}$ )      7.0
Dense (subclass of an ordered class)      1.10
Dense in itself      1.10
Derivative (of a formal power aeries)      7.60
Disconnected      1.20
Distance function ( $d(\cdot, \cdot) = |\cdot - \cdot|$ in a surcomplex field)      7.10
Distinguished base      2.01
Distinguished base of open sets      2.01
Distinguished subbase      2.01
Dyadic      4.21
Embedding of $n_{\xi}$ -fields      1.75
End points (of an interval)      1.10
Equipotent      1.03
Equivalent (sets)      4.02
Eta axiom      4.60
EVS (= extended value set)      1.61 1.10
Exp (= extended exponential function over $\xi Cx$ ))      7.51
Exponential function      7.36
Exponential series      7 — 36
Extended value class $(EVC_{\nu})$ of a group {G,v} with valuation      1.62
Extended value set (= EVS)      1.61
Extension (of a Conway cut)      1(.61
Extension (of a field with valuation)      6.30
Extension (of a set)      8.00
Extension (of an ordered set)      1.10
Factor      1.61
Factorial (generalized)      7.93
Field of formal power series      6.30
Fill (a cut)      4.02
Finite intersection property (= f.i.p.)      2.30
Finite ordinals      1.02
First kind (ordinals)      1.02
Formal implicit function theorem      7.70 7.71
Formal implicit mapping theorem      7.72
Formal inverse mapping theorem      7.40
Formal power series (in n variables)      7.1)0
Formal power series over formal power series fields      7.80
Formally real (field)      1.70
Full (condition)      4.03
Full binary tree of height On      4.50
Full eta axiom (= FE), i      60
Full field of formal power series      6.30
Full Hahn group      1.63
Fullness axiom (= F), i      60
Galaxies      5.40
Game      4.01
GCH (= Generalized Continuum Hypothesis)      1.30
Generalized binomial coefficients      7.93
Generalized factorials      7.93
Hahn group      1.63
Hahn valuation      1.61 6.20
Hahn's Embedding Theorem      1.64
Harzheim's Theorem      4.02
Hausdorff (space)      2.10 7.62
Hausdorff's Normal $\eta_{\xi}$ -type      5.10
Height function      4.50
Hessenberg product (- Natural product)      4.05
Hessenberg sum (= Natural sum)      4.05
Hion's Lemma      1.61
Holder's Theorem      1.60
Hyper-convergent      7.22
i, (= a strongly inaccessible cardinal number)      5.40
Ideal of infinitesimal elements      6.00
Ideal theory of a valuation ring      6.70
Identity Theorem (for normal forms)      6.51
Imaginary part      7.10
Immediate extension      1 .63 6.30
Implicit function theorem (for formal power series)      7.70 7.71
Implicit function theorem (for Neumann series)      7.75
Implicit mapping theorem (for formal power series)      7.73
Implicit mapping theorem (for Neumann series)      7.75
Incomplete (= not Dedekind-complete)      1.20
Incomplete ordered groups      1.66
Independence of representation (of Neumann series)      7.83
Infinitely large      1.60
Infinitely small      1.60
interval      1.10
Interval ( $\xi$ — open)      2.12
interval ( $\xi$ -closed)      2.12
Interval topology      1.10
Inverse mapping theorem (for formal power series)      7.75
Inverse mapping theorem (for Neumann series)      7.75
Isolated (= convex subgroup)      1.60
Jacobian (matrix)      7.73
K-valued coefficient      7.40
Kuratowski closure operator $(\cdot \longrightarrow \cdot^{C})$       2.02
Leader (of y)      4.40
Left character      1.30
Left-option      4.00
Length (of a pseudo-convergent sequence)      6.41 6.44
Lexicographically ordered (set)      1.60
Linear term (of a formal power series)      7.40
Linearly ordered set (= ordered set)      1.01
Locally $\xi$ -connected      2.01
Logarithmic derivative      1.73
Logarithmic series      7.36
Lower character      1 .30
Lower-saturated      1.02 1.30
M (= the maximal ideal in a valuation ring 0)      6.00
MacLane's Theorem      6.23
Maclaurin — Neumann series      7.91
Main Theorem      7.82
Maximal (field with valuation)      6.30
Maximal extension (of a Conway cut)      4.61
Maximum      1.74
Minimum      1.74
Modifie $\xi$ -open (set)      7.64
Modified $\xi$ -topology      7.64
Modified valuation topology      7.64
Monomial (of a formal power series)      7.40
Multiplication (in No)      4.06
Mutually cofinal (sets in an ordered class)      4.02
Mutually coinitial (sets in an ordered class)      4.02
Natural product (= Hessenberg product)      4.05
Natural sum (= Hessenberg sum)      4.05
NBG (set of axioms for set theory)      1.00
Nearest common predecessor      4.52
Neumann series (expansion)      7.22
Neumann series in a finite number of variables      7.^1
Neumann's First Lemma      7.21
Neumann's Lemma      7.20
Neumann's Second Lemma      7.21
Neumann's Theorem      7.22
No (= Conway's field of all surreal numbers)      4.01 4.02 4.03 4.60
Non — Archimedean (ordered group, ring or field)      1.60
Non-decreasing (sequence)      7.21
Non-limit ordinal number      1.02
Non-singular (point)      7.73
Non-zero limit ordinal      1.02
nop (= nearest common predecessor)      4.52
Norm $(||\cdot|| in R^{n})$       3.00
Normal form (Cantor's)      4.04 6.43
Normal form (Conway's in $\xi No$ )      6.53
Normal form (in $\xi Cx$ )      7.10
Normal form (in Cx)      7.11
Not too many (= of power $< \omega_{\xi}$ )      2.10
O (= a valuation ring of a field with valuation)      6.00
On (= class of all ordinal numbers)      1.02
ONAG (= On Numbers and Games), 0      3
Open ball (about about $x_{0}$ of radiu $\geq g$ )      7.64
Open ball (about about $x_{0}$ of radius > g)      7.62
Open class (in the interval topology)      1.10
Open semi-algebraic set      3.00
Open, semi-algebraic set      3.00
Option (right or left)      4.00
Order (of $v\varepsilon Z(\geq0)^{n}$ )      7.40

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