Computational homology:
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
2004
|
| Series: | Applied mathematical sciences
157 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 465 - 469 |
| Physical Description: | XVII, 480 S. Ill., graph. Darst. |
| ISBN: | 0387408533 |
Staff View
MARC
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| 100 | 1 | |a Kaczynski, Tomasz |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Computational homology |c Tomasz Kaczynski ; Konstantin Mischaikow ; Marian Mrozek |
| 264 | 1 | |a New York [u.a.] |b Springer |c 2004 | |
| 300 | |a XVII, 480 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 490 | 1 | |a Applied mathematical sciences |v 157 | |
| 500 | |a Literaturverz. S. 465 - 469 | ||
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| 650 | 4 | |a Homology theory | |
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| 700 | 1 | |a Mischaikow, Konstantin |e Verfasser |4 aut | |
| 700 | 1 | |a Mrozek, Marian |e Verfasser |4 aut | |
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Record in the Search Index
| _version_ | 1804130254574321664 |
|---|---|
| adam_text | Contents
Preface
........................................................
VII
Part I Homology
1
Preview
................................................... 3
1.1
Analyzing Images
....................................... 3
1.2
Nonlinear Dynamics
..................................... 13
1.3
Graphs
................................................. 17
1.4
Topological and Algebraic Boundaries
...................... 19
1.5
Keeping Track of Directions
.............................. 24
1.6
Mod
2
Homology of Graphs
............................... 26
2
Cubical Homology
......................................... 39
2.1
Cubical Sets
............................................ 39
2.1.1
Elementary Cubes
................................. 40
2.1.2
Cubical Sets
...................................... 42
2.1.3
Elementary Cells
.................................. 44
2.2
The Algebra of Cubical Sets
.............................. 47
2.2.1
Cubical Chains
.................................... 47
2.2.2
Cubical Chains in a Cubical Set
..................... 53
2.2.3
The Boundary Operator
............................ 54
2.2.4
Homology of Cubical Sets
.......................... 60
2.3
Connected Components and Hq(X)
........................ 66
2.4
Elementary Collapses
.................................... 70
2.5
Acyclic Cubical Spaces
................................... 79
2.6
Homology of Abstract Chain Complexes
.................... 85
2.7
Reduced Homology
...................................... 88
2.8
Bibliographical Remarks
................................. 91
XIV Contents
3 Computing Homology
Groups
............................. 93
3.1 Matrix Algebra
over Z
................................... 94
3.2
Row Echelon Form
......................................107
3.3
Smith Normal Form
.....................................117
3.4
Structure of Abelian Groups
..............................125
3.5
Computing Homology Groups
.............................132
3.6
Computing Homology of Cubical Sets
......................134
3.7
Preboundary of a Cycle- Algebraic Approach
...............139
3.8
Bibliographical Remarks
.................................141
4
Chain Maps and Reduction Algorithms
....................143
4.1
Chain Maps
............................................143
4.2
Chain Homotopy
........................................149
4.3
Internal Elementary Reductions
...........................155
4.3.1
Elementary Collapses Revisited
.....................155
4.3.2
Generalization of Elementary Collapses
..............157
4.4
CCR Algorithm
.........................................165
4.5
Bibliographical Remarks
.................................171
5
Preview of Maps
..........................................173
5.1
Rational Functions and Interval Arithmetic
.................174
5.2
Maps on an Interval
.....................................176
5.3
Constructing Chain Selectors
.............................185
5.4
Maps of
Г1
.............................................189
6
Homology of Maps
........................................199
6.1
Representable Sets
.......................................199
6.2
Cubical Multivalued Maps
................................206
6.3
Chain Selectors
.........................................210
6.4
Homology of Continuous Maps
............................215
6.4.1
Cubical Representations
............................216
6.4.2
Rescaling
.........................................222
6.5
Homotopy
Invariance
....................................231
6.6
Bibliographical Remarks
.................................234
7
Computing Homology of Maps
............................235
7.1
Producing Multivalued Representation
.....................236
7.2
Chain Selector Algorithm
.................................240
7.3
Computing Homology of Maps
............................242
7.4
Geometric Preboundary Algorithm (optional section)
........244
7.5
Bibliographical Remarks
.................................253
Contents
XV
Part II Extensions
8
Prospects in Digital Image Processing
.....................257
8.1
Images and Cubical Sets
.................................257
8.2
Patterns from Cahn-Hilliard
..............................259
8.3
Complicated Time-Dependent Patterns
.....................266
8.4
Size Function
...........................................269
8.5
Bibliographical Remarks
.................................277
9
Homological Algebra
......................................279
9.1
Relative Homology
......................................279
9.1.1
Relative Homology Groups
.........................279
9.1.2
Maps in Relative Homology
.........................286
9.2
Exact Sequences
.........................................289
9.3
The Connecting Homomorphism
..........................292
9.4
Mayer-Vietoris Sequence
.................................299
9.5
Weak Boundaries
........................................303
9.6
Bibliographical Remarks
.................................306
10
Nonlinear Dynamics
.......................................307
10.1
Maps and Symbolic Dynamics
.............................308
10.2
Differential Equations and Flows
..........................318
10.3
Ważewski
Principle
......................................320
10.4
Fixed-Point Theorems
...................................324
10.4.1
Fixed Points in the Unit, Ball
.......................324
10.4.2
The Lefschetz Fixed-Point Theorem
.................326
10.5
Degree Theory
..........................................332
10.5.1
Degree on Spheres
.................................333
10.5.2
Topological Degree
................................336
10.6
Complicated Dynamics
...................................342
10.6.1
Index Pairs and Index Map
.........................343
10.6.2
Topological Conjugacy
.............................357
10.7
Computing Chaotic Dynamics
.............................361
10.8
Bibliographical Remarks
.................................375
11
Homology of Topological Polyhedra
........................377
11.1
Simplicial Homology
.....................................378
11.2
Comparison of Cubical and Simplicial Complexes
............385
11.3
Homology Functor
.......................................388
11.3.1
Category of Cubical Sets
...........................389
11.3.2
Connected Simple Systems
.........................390
11.4
Bibliographical Remarks
.................................393
XVI Contents
Part III Tools from Topology and Algebra
12
Topology
..................................................397
12.1
Norms and Metrics in Rd
.................................397
12.2
Topology
...............................................402
12.3
Continuous Maps
........................................407
12.4
Connectedness
..........................................411
12.5
Limits and Compactness
.................................415
13
Algebra
....................................................419
13.1
Abelian Groups
.........................................419
13.1.1
Algebraic Operations
..............................419
13.1.2
Groups
...........................................420
13.1.3
Cyclic Groups and Torsion Subgroup
................422
13.1.4
Quotient Groups
..................................424
13.1.5
Direct Sums
......................................426
13.2
Fields and Vector Spaces
.................................427
13.2.1
Fields
............................................427
13.2.2
Vector Spaces
.....................................429
13.2.3
Linear Combinations and Bases
.....................430
13.3
Homomorphisms
........................................433
13.3.1
Homomorphisms of Groups
.........................433
13.3.2
Linear Maps
......................................437
13.3.3
Matrix Algebra
...................................438
13.4
Free Abelian Groups
.....................................441
13.4.1
Bases in Groups
...................................441
13.4.2
Subgroups of Free Groups
..........................446
13.4.3
Homomorphisms of Free Groups
.....................447
14
Syntax of Algorithms
......................................451
14.1
Overview
...............................................451
14.2
Data Structures
.........................................453
14.2.1
Elementary Data Types
............................453
14.2.2
Lists
.............................................454
14.2.3
Arrays
...........................................455
14.2.4
Vectors and Matrices
..............................456
14.2.5
Sets
.............................................457
14.2.6
Hashes
...........................................458
14.3
Compound Statements
...................................459
14.3.1
Conditional Statements
............................459
14.3.2
Loop Statements
..................................459
14.3.3
Keywords break and next
.........................460
14.4
Function and Operator Overloading
........................461
14.5
Analysis of Algorithms
...................................462
Contents XVII
References
.....................................................465
Symbol Index
..................................................471
Subject Index
.................................................475
|
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| id | DE-604.BV017448467 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T19:18:07Z |
| institution | BVB |
| isbn | 0387408533 |
| language | English |
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| physical | XVII, 480 S. Ill., graph. Darst. |
| publishDate | 2004 |
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| publisher | Springer |
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| spelling | Kaczynski, Tomasz Verfasser aut Computational homology Tomasz Kaczynski ; Konstantin Mischaikow ; Marian Mrozek New York [u.a.] Springer 2004 XVII, 480 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Applied mathematical sciences 157 Literaturverz. S. 465 - 469 Homologie Homology theory Homologie (DE-588)4141951-0 gnd rswk-swf Homologie (DE-588)4141951-0 s DE-604 Mischaikow, Konstantin Verfasser aut Mrozek, Marian Verfasser aut Applied mathematical sciences 157 (DE-604)BV000005274 157 Digitalisierung UB Augsburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010511038&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kaczynski, Tomasz Mischaikow, Konstantin Mrozek, Marian Computational homology Applied mathematical sciences Homologie Homology theory Homologie (DE-588)4141951-0 gnd |
| subject_GND | (DE-588)4141951-0 |
| title | Computational homology |
| title_auth | Computational homology |
| title_exact_search | Computational homology |
| title_full | Computational homology Tomasz Kaczynski ; Konstantin Mischaikow ; Marian Mrozek |
| title_fullStr | Computational homology Tomasz Kaczynski ; Konstantin Mischaikow ; Marian Mrozek |
| title_full_unstemmed | Computational homology Tomasz Kaczynski ; Konstantin Mischaikow ; Marian Mrozek |
| title_short | Computational homology |
| title_sort | computational homology |
| topic | Homologie Homology theory Homologie (DE-588)4141951-0 gnd |
| topic_facet | Homologie Homology theory |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010511038&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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