Singularity theory: 1.
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| Previous Title: | Dynamical systems ; 6 |
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| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
1998
|
| Edition: | 1. ed., 2. printing |
| Series: | Encyclopaedia of mathematical sciences
6 |
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| Physical Description: | 245 S. graph. Darst. |
| ISBN: | 3540637117 |
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| 245 | 1 | 0 | |a Singularity theory |n 1. |c V. I. Arnol'd ... |
| 250 | |a 1. ed., 2. printing | ||
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 1998 | |
| 300 | |a 245 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Encyclopaedia of mathematical sciences |v 6 | |
| 490 | 0 | |a Encyclopaedia of mathematical sciences |v ... | |
| 700 | 1 | |a Arnolʹd, V. I. |d 1937-2010 |e Sonstige |0 (DE-588)119540878 |4 oth | |
| 773 | 0 | 8 | |w (DE-604)BV017436526 |g 1 |
| 780 | 0 | 0 | |i 1. Aufl. auch als |t Dynamical systems ; 6 |
| 830 | 0 | |a Encyclopaedia of mathematical sciences |v 6 |w (DE-604)BV024126459 |9 6 | |
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CONTENTS
FOREWORD
.
7
CHAPTER
1.
CRITICAL
POINTS
OF
FUNCTIONS
.
10
§
1.
INVARIANTS
OF
CRITICAL
POINTS
.
10
1.1.
DEGENERATE
AND
NONDEGENERATE
CRITICAL
POINTS
.
10
1.2.
EQUIVALENCE
OF
CRITICAL
POINTS
.
11
1.3.
STABLE
EQUIVALENCE
.
12
1.4.
THE
LOCAL
ALGEBRA
AND
THE
MULTIPLICITY
OF
A
SINGULARITY
.
13
1.5.
FINITE
DETERMINACY
OF
AN
ISOLATED
SINGULARITY
.
14
1.6.
LIE
GROUP
ACTIONS
ON
MANIFOLDS
.
15
1.7.
VERSAL
DEFORMATIONS
OF
A
CRITICAL
POINT
.
16
1.8.
INFINITESIMAL
VERSALITY
.
17
1.9.
THE
MODALITY
OF
A
CRITICAL
POINT
.
18
1.10.
THE
LEVEL
BIFURCATION
SET
.
19
1.11.
TRUNCATED
VERSAL
DEFORMATIONS
AND
THE
FUNCTION
BIFURCATION
SET
.
20
§2.
THE
CLASSIFICATION
OF
CRITICAL
POINTS
.
22
2.1.
NORMAL
FORMS
.
22
2.2.
CLASSES
OF
LOW
MODALITY
.
23
2.3.
SINGULARITIES
OF
MODALITY
2
.
24
2.4.
SIMPLE
SINGULARITIES
AND
KLEIN
SINGULARITIES
.
25
2.5.
RESOLUTION
OF
SIMPLE
SINGULARITIES
.
26
2.6.
UNIMODAL
AND
BIMODAL
SINGULARITIES
.
28
2.7.
ADJACENCY
OF
SINGULARITIES
.
30
2.8.
REAL
SINGULARITIES
.
32
2
CONTENTS
§
3.
REDUCTION
TO
NORMAL
FORMS
.
34
3.1.
THE
NEWTON
DIAGRAM
.
35
3.2.
QUASIHOMOGENEOUS
FUNCTIONS
AND
FILTRATIONS
.
36
3.3.
THE
MULTIPLICITY
AND
THE
GENERATORS
OF
THE
LOCAL
ALGEBRA
OF
A
SEMI-QUASIHOMOGENEOUS
FUNCTION
.
38
3.4.
QUASIHOMOGENEOUS
MAPS
.
38
3.5.
QUASIHOMOGENEOUS
DIFFEOMORPHISMS
AND
VECTOR
FIELDS
.
40
3.6.
THE
NORMAL
FORM
OF
A
SEMI-QUASIHOMOGENEOUS
FUNCTION
.
42
3.7.
THE
NORMAL
FORM
OF
A
QUASIHOMOGENEOUS
FUNCTION
.
43
3.8.
THE
NEWTON
FILTRATION
.
45
3.9.
THE
SPECTRAL
SEQUENCE
.
47
3.10.
THEOREMS
ON
NORMAL
FORMS
FOR
THE
SPECTRAL
SEQUENCE
.
49
CHAPTER
2.
MONODROMY
GROUPS
OF
CRITICAL
POINTS
.
50
§
1.
THE
PICARD-LEFSCHETZ
THEORY
.
51
1.1.
TOPOLOGY
OF
THE
NONSINGULAR
LEVEL
MANIFOLD
.
51
1.2.
THE
CLASSICAL
MONODROMY
AND
THE
VARIATION
OPERATOR
.
53
1.3.
THE
MONODROMY
OF
A
MORSE
SINGULARITY
.
54
1.4.
THE
MONODROMY
GROUP
OF
AN
ISOLATED
SINGULARITY
.
56
1.5.
VANISHING
CYCLES
AND
DISTINGUISHED
BASES
.
58
1.6.
THE
INTERSECTION
MATRIX
OF
A
SINGULARITY
.
61
1.7.
STABILIZATION
OF
SINGULARITIES
.
63
1.8.
DYNKIN
DIAGRAMS
.
64
1.9.
TRANSFORMATIONS
OF
A
BASIS
AND
OF
ITS
DYNKIN
DIAGRAM
.
64
1.10.
THE
MILNOR
FIBRATION
OVER
THE
COMPLEMENT
OF
THE
LEVEL
BIFURCATION
SET
.
68
1.11.
THE
TOPOLOGICAL
TYPE
OF
A
SINGULARITY
ALONG
THE
P-CONSTANT
STRATUM
.
70
§
2.
DYNKIN
DIAGRAMS
AND
MONODROMY
GROUPS
.
72
2.1.
INTERSECTION
MATRICES
OF
SINGULARITIES
OF
FUNCTIONS
OF
TWO
VARIABLES
.
72
2.2.
THE
INTERSECTION
MATRIX
OF
A
DIRECT
SUM
OF
SINGULARITIES
.
75
2.3.
PHAM
SINGULARITIES
.
77
2.4.
THE
POLAR
CURVE
AND
THE
INTERSECTION
MATRIX
.
77
2.5.
MODALITY
AND
QUADRATIC
FORMS
OF
SINGULARITIES
.
82
2.6.
THE
MONODROMY
GROUP
AND
THE
INTERSECTION
FORM
.
84
2.7.
THE
MONODROMY
GROUP
IN
THE
SKEW-SYMMETRIC
CASE
.
87
§
3.
COMPLEX
MONODROMY
AND
PERIOD
MAPS
.
88
3.1.
THE
COHOMOLOGY
BUNDLE
AND
THE
GAUSS-MANIN
CONNECTION
.
88
3.2.
SECTIONS
OF
THE
COHOMOLOGY
BUNDLE
.
89
3.3.
THE
VANISHING
COHOMOLOGY
BUNDLE
.
90
3.4.
THE
PERIOD
MAP
.
91
3.5.
THE
RESIDUE
FORM
.
91
3.6.
TRIVIALIZATIONS
OF
THE
COHOMOLOGY
BUNDLE
.
92
CONTENTS
3
3.7.
THE
CLASSICAL
COMPLEX
MONODROMY
.
94
3.8.
DIFFERENTIAL
EQUATIONS
AND
ASYMPTOTICS
OF
INTEGRALS
.
95
3.9.
NONDEGENERATE
PERIOD
MAPS
.
98
3.10.
STABILITY
OF
PERIOD
MAPS
.
100
3.11.
PERIOD
MAPS
AND
INTERSECTION
FORMS
.
101
3.12.
THE
CHARACTERISTIC
POLYNOMIAL
AND
THE
ZETA
FUNCTION
OF
THE
MONODROMY
OPERATOR
.
102
§4.
THE
MIXED
HODGE
STRUCTURE
IN
THE
VANISHING
COHOMOLOGY
.
105
4.1.
THE
PURE
HODGE
STRUCTURE
.
105
4.2.
THE
MIXED
HODGE
STRUCTURE
.
106
4.3.
THE
ASYMPTOTIC
HODGE
FILTRATION
IN
THE
FIBRES
OF
THE
COHOMOLOGY
BUNDLE
.
108
4.4.
THE
WEIGHT
FILTRATION
.
108
4.5.
THE
ASYMPTOTIC
MIXED
HODGE
STRUCTURE
.
110
4.6.
THE
HODGE
NUMBERS
AND
THE
SPECTRUM
OF
A
SINGULARITY
.
ILL
4.7.
COMPUTING
THE
SPECTRUM
.
112
4.8.
SEMICONTINUITY
OF
THE
SPECTRUM
.
114
4.9.
THE
SPECTRUM
AND
THE
GEOMETRIC
GENUS
.
115
4.10.
THE
MIXED
HODGE
STRUCTURE
AND
THE
INTERSECTION
FORM
.
116
4.11.
THE
NUMBER
OF
SINGULAR
POINTS
OF
A
COMPLEX
PROJECTIVE
HYPERSURFACE
.
116
4.12.
THE
GENERALIZED
PETROVSKII-OLEINIK
INEQUALITIES
.
118
§5.
SIMPLE
SINGULARITIES
.
119
5.1.
REFLECTION
GROUPS
.
119
5.2.
THE
SWALLOWTAIL
OF
A
REFLECTION
GROUP
.
122
5.3.
THE
ARTIN-BRIESKORN
BRAID
GROUP
.
124
5.4.
CONVOLUTION
OF
INVARIANTS
OF
A
COXETER
GROUP
.
125
5.5.
ROOT
SYSTEMS
AND
WEYL
GROUPS
.
127
5.6.
SIMPLE
SINGULARITIES
AND
WEYL
GROUPS
.
129
5.7.
VECTOR
FIELDS
TANGENT
TO
THE
LEVEL
BIFURCATION
SET
.
130
5.8.
THE
COMPLEMENT
OF
THE
FUNCTION
BIFURCATION
SET
.
132
5.9.
ADJACENCY
AND
DECOMPOSITION
OF
SIMPLE
SINGULARITIES
.
132
5.10.
FINITE
SUBGROUPS
OF
SU
2
,
SIMPLE
SINGULARITIES,
AND
WEYL
GROUPS
.
133
5.11.
PARABOLIC
SINGULARITIES
.
134
§6.
TOPOLOGY
OF
COMPLEMENTS
OF
DISCRIMINANTS
OF
SINGULARITIES
.
138
6.1.
COMPLEMENTS
OF
DISCRIMINANTS
AND
BRAID
GROUPS
.
138
6.2.
THE
MOD-2
COHOMOLOGY
OF
BRAID
GROUPS
.
138
6.3.
AN
APPLICATION:
SUPERPOSITION
OF
ALGEBRAIC
FUNCTIONS
.
140
6.4.
THE
INTEGER
COHOMOLOGY
OF
BRAID
GROUPS
.
140
6.5.
THE
COHOMOLOGY
OF
BRAID
GROUPS
WITH
TWISTED
COEFFICIENTS
.
141
6.6.
GENUS
OF
COVERINGS
ASSOCIATED
WITH
AN
ALGEBRAIC
FUNCTION,
AND
COMPLEXITY
OF
ALGORITHMS
FOR
COMPUTING
ROOTS
OF
POLYNOMIALS
.
142
4
CONTENTS
6.7.
THE
COHOMOLOGY
OF
BRIESKOM
BRAID
GROUPS
AND
COMPLEMENTS
OF
THE
DISCRIMINANTS
OF
SINGULARITIES
OF
THE
SERIES
C
AND
D
.
143
6.8.
THE
STABLE
COHOMOLOGY
OF
COMPLEMENTS
OF
LEVEL
BIFURCATION
SETS
.
143
6.9.
CHARACTERISTIC
CLASSES
OF
MILNOR
COHOMOLOGY
BUNDLES
.
146
6.10.
STABLE
IRREDUCIBILITY
OF
STRATA
OF
DISCRIMINANTS
.
147
CHAPTER
3.
BASIC
PROPERTIES
OF
MAPS
.
147
§
1.
STABLE
MAPS
AND
MAPS
OF
FINITE
MULTIPLICITY
.
148
1.1.
THE
LEFT-RIGHT
EQUIVALENCE
.
148
1.2.
STABILITY
.
149
1.3.
TRANSVERSALITY
.
151
1.4.
THE
THOM-BOARDMAN
CLASSES
.
153
1.5.
INFINITESIMAL
STABILITY
.
155
1.6.
THE
GROUPS
AND
X
.
156
1.7.
NORMAL
FORMS
OF
STABLE
GERMS
.
157
1.8.
EXAMPLES
.
157
1.9.
NICE
AND
SEMI-NICE
DIMENSIONS
.
160
1.10.
MAPS
OF
FINITE
MULTIPLICITY
.
161
1.11.
THE
NUMBER
OF
ROOTS
OF
A
SYSTEM
OF
EQUATIONS
.
162
1.12.
THE
INDEX
OF
A
SINGULAR
POINT
OF
A
REAL
GERM,
AND
POLYNOMIAL
VECTOR
FIELDS
.
163
§2.
FINITE
DETERMINACY
OF
MAP-GERMS,
AND
THEIR
VERSAL
DEFORMATIONS
165
2.1.
TANGENT
SPACES
AND
CODIMENSIONS
.
166
2.2.
FINITE
DETERMINACY
.
166
2.3.
VERSAL
DEFORMATIONS
.
167
2.4.
EXAMPLES
.
168
2.5.
GEOMETRIC
SUBGROUPS
.
170
2.6.
THE
ORDER
OF
A
SUFFICIENT
JET
.
174
2.7.
DETERMINACY
WITH
RESPECT
TO
TRANSFORMATIONS
OF
FINITE
SMOOTHNESS
.
178
§
3.
THE
TOPOLOGICAL
EQUIVALENCE
.
179
3.1.
THE
TOPOLOGICALLY
STABLE
MAPS
ARE
DENSE
.
179
3.2.
WHITNEY
STRATIFICATIONS
.
179
3.3.
THE
TOPOLOGICAL
CLASSIFICATION
OF
SMOOTH
MAP-GERMS
.
181
3.4.
TOPOLOGICAL
INVARIANTS
.
182
3.5.
TOPOLOGICAL
TRIVIALITY
AND
TOPOLOGICAL
VERSALITY
OF
DEFORMATIONS
OF
SEMI-QUASIHOMOGENEOUS
MAPS
.
183
CHAPTER
4.
THE
GLOBAL
THEORY
OF
SINGULARITIES
.
185
§
1.
THOM
POLYNOMIALS
FOR
MAPS
OF
SMOOTH
MANIFOLDS
.
186
1.1.
CYCLES
OF
SINGULARITIES
AND
TOPOLOGICAL
INVARIANTS
OF
MAPS
186
CONTENTS
5
1.2.
THOM
'
S
THEOREM
ON
THE
EXISTENCE
OF
THOM
POLYNOMIALS
.
187
1.3.
RESOLUTION
OF
THE
SINGULARITIES
OF
THE
CLOSURES
OF
THE
THOM-BOARDMAN
CLASSES
.
188
1.4.
THOM
POLYNOMIALS
FOR
SINGULARITIES
OF
FIRST
ORDER
.
189
1.5.
SURVEY
OF
RESULTS
ON
THOM
POLYNOMIALS
FOR
SINGULARITIES
OF
HIGHER
ORDER
.
190
§
2.
INTEGER
CHARACTERISTIC
CLASSES
AND
UNIVERSAL
COMPLEXES
OF
SINGULARITIES
.
191
2.1.
EXAMPLES:
THE
MASLOV
INDEX
AND
THE
FIRST
PONTRYAGIN
CLASS
192
2.2.
THE
UNIVERSAL
COMPLEX
OF
SINGULARITIES
OF
SMOOTH
FUNCTIONS
193
2.3.
COHOMOLOGY
OF
THE
COMPLEXES
OF
R
0
-INVARIANT
SINGULARITIES,
AND
INVARIANTS
OF
FOLIATIONS
.
196
2.4.
COMPUTATIONS
IN
COMPLEXES
OF
SINGULARITIES
OF
FUNCTIONS.
GEOMETRIC
CONSEQUENCES
.
197
2.5.
UNIVERSAL
COMPLEXES
OF
LAGRANGIAN
AND
LEGENDRIAN
SINGULARITIES
.
199
2.6.
ON
UNIVERSAL
COMPLEXES
OF
GENERAL
MAPS
OF
MANIFOLDS
.
201
§3.
MULTIPLE
POINTS
AND
MULTISINGULARITIES
.
201
3.1.
A
FORMULA
FOR
MULTIPLE
POINTS
OF
IMMERSIONS,
AND
EMBEDDING
OBSTRUCTIONS
FOR
MANIFOLDS
.
201
3.2.
TRIPLE
POINTS
OF
SINGULAR
SURFACES
.
202
3.3.
MULTIPLE
POINTS
OF
COMPLEX
MAPS
.
202
3.4.
SELF-INTERSECTIONS
OF
LAGRANGIAN
MANIFOLDS
.
203
3.5.
COMPLEXES
OF
MULTISINGULARITIES
.
203
3.6.
MULTISINGULARITIES
AND
MULTIPLICATION
IN
THE
COHOMOLOGY
OF
THE
TARGET
SPACE
OF
A
MAP
.
206
§4.
SPACES
OF
FUNCTIONS
WITH
CRITICAL
POINTS
OF
MILD
COMPLEXITY
.
207
4.1.
FUNCTIONS
WITH
SINGULARITIES
SIMPLER
THAN
A
3
.
207
4.2.
THE
GROUP
OF
CURVES
WITHOUT
HORIZONTAL
INFLEXIONAL
TANGENTS
.
208
4.3.
HOMOTOPY
PROPERTIES
OF
THE
COMPLEMENTS OF
UNFURLED
SWALLOWTAILS
.
211
§
5.
ELIMINATION
OF
SINGULARITIES
AND
SOLUTION
OF
DIFFERENTIAL
CONDITIONS
.
212
5.1.
CANCELLATION
OF
WHITNEY
UMBRELLAS
AND
CUSPS.
THE
IMMERSION
PROBLEM
.
212
5.2.
THE
SMALE-HIRSCH
THEOREM
.
213
5.3.
THE
W.H.E.
AND
H-PRINCIPLES
.
213
5.4.
THE
GROMOV-LEES
THEOREM
ON
LAGRANGIAN
IMMERSIONS
.
215
5.5.
ELIMINATION
OF
THOM-BOARDMAN
SINGULARITIES
.
215
5,6.
THE
SPACE
OF
FUNCTIONS
WITH
NO
A
3
SINGULARITIES
.
216
§6.
TANGENTIAL
SINGULARITIES
AND
VANISHING
INFLEXIONS
.
216
6.1.
THE
CALCULUS
OF
TANGENTIAL
SINGULARITIES
.
216
6.2.
VANISHING
INFLEXIONS:
THE
CASE
OF
PLANE
CURVES
.
217
6.3.
INFLEXIONS
THAT
VANISH
AT
A
MORSE
SINGULAR
POINT
.
218
6
CONTENTS
6.4.
INTEGRATION
WITH
RESPECT
TO
THE
EULER
CHARACTERISTIC,
AND
ITS
APPLICATIONS
.
219
REFERENCES
.
221
AUTHOR
INDEX
.
239
SUBJECT
INDEX
.
242 |
| adam_txt | |
| any_adam_object | 1 |
| any_adam_object_boolean | |
| author_GND | (DE-588)119540878 |
| building | Verbundindex |
| bvnumber | BV023625415 |
| classification_rvk | SK 350 |
| ctrlnum | (OCoLC)312762318 (DE-599)BVBBV023625415 |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| edition | 1. ed., 2. printing |
| format | Book |
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| id | DE-604.BV023625415 |
| illustrated | Illustrated |
| index_date | 2024-07-02T22:44:32Z |
| indexdate | 2024-08-13T00:22:48Z |
| institution | BVB |
| isbn | 3540637117 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016942601 |
| oclc_num | 312762318 |
| open_access_boolean | |
| owner | DE-521 DE-634 DE-703 DE-83 DE-188 |
| owner_facet | DE-521 DE-634 DE-703 DE-83 DE-188 |
| physical | 245 S. graph. Darst. |
| publishDate | 1998 |
| publishDateSearch | 1998 |
| publishDateSort | 1998 |
| publisher | Springer |
| record_format | marc |
| series | Encyclopaedia of mathematical sciences |
| series2 | Encyclopaedia of mathematical sciences |
| spelling | Singularity theory 1. V. I. Arnol'd ... 1. ed., 2. printing Berlin [u.a.] Springer 1998 245 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Encyclopaedia of mathematical sciences 6 Encyclopaedia of mathematical sciences ... Arnolʹd, V. I. 1937-2010 Sonstige (DE-588)119540878 oth (DE-604)BV017436526 1 1. Aufl. auch als Dynamical systems ; 6 Encyclopaedia of mathematical sciences 6 (DE-604)BV024126459 6 DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016942601&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Singularity theory Encyclopaedia of mathematical sciences |
| title | Singularity theory |
| title_auth | Singularity theory |
| title_exact_search | Singularity theory |
| title_exact_search_txtP | Singularity theory |
| title_full | Singularity theory 1. V. I. Arnol'd ... |
| title_fullStr | Singularity theory 1. V. I. Arnol'd ... |
| title_full_unstemmed | Singularity theory 1. V. I. Arnol'd ... |
| title_old | Dynamical systems ; 6 |
| title_short | Singularity theory |
| title_sort | singularity theory |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016942601&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV017436526 (DE-604)BV024126459 |
| work_keys_str_mv | AT arnolʹdvi singularitytheory1 |