Applications of algebraic topology: graphs and networks, the Picard-Lefschetz theory and Feynman integrals
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
1975
|
| Series: | Applied mathematical sciences
16 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | VIII, 189 S. graph. Darst. |
| ISBN: | 038790137X 354090137X |
Staff View
MARC
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| 245 | 1 | 0 | |a Applications of algebraic topology |b graphs and networks, the Picard-Lefschetz theory and Feynman integrals |c S. Lefschetz |
| 264 | 1 | |a New York [u.a.] |b Springer |c 1975 | |
| 300 | |a VIII, 189 S. |b graph. Darst. | ||
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| 650 | 7 | |a Topologia Algebrica |2 larpcal | |
| 650 | 4 | |a Topologie algébrique | |
| 650 | 4 | |a Algebraic topology | |
| 650 | 4 | |a Electric networks | |
| 650 | 4 | |a Feynman integrals | |
| 650 | 4 | |a Graph theory | |
| 650 | 4 | |a Picard-Lefschetz theory | |
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Record in the Search Index
| _version_ | 1804116404794818560 |
|---|---|
| adam_text | TABLE OF CONTENTS
PART I
APPLICATION OF CLASSICAL TOPOLOGY
TO GRAPHS AND NETWORKS
Page
INTRODUCTION 4
CHAPTER I. A RESUME OF LINEAR ALGEBRA 5
1. Matrices 5
2. Vector and Vector Spaces 7
3. Column Vectors and Row Vectors 10
4. Application to Linear Equations XI
CHAPTER II. DUALITY IN VECTOR SPACES 13
1. General Remarks on Duality 13
2. Questions of Nomenclature 14
3. Linear Functions on Vector Spaces.
Multiplication 15
4. Linear Transformations. Duality 16
5. Vector Space Sequence of Walter Mayer 18
CHAPTER III. TOPOLOGICAL PRELIMINARIES 22
1. First Intuitive Notions of Topology 22
2. Affine and Euclidean Spaces 25
3. Continuity, Mapping, Homeomorphism 26
4. General Sets and Their Combinations 27
5. Some Important Subsets of a Space 28
6. Connectedness 29
7. Theorem of Jordan-Schoenflies 30
CHAPTER IV. GRAPHS. GEOMETRIC STRUCTURE 34
1. Structure of Graphs 34
2. Subdivision. Characteristic Betti Number 37
CHAPTER V. GRAPH ALGEBRA 43
1. Preliminaries 43
2. Dimensional Calculations 46
3. Space Duality. Co-theory 48
CHAPTER VI. ELECTRICAL NETWORKS 51
1. Kirchoff s Laws 51
2. Different Types of Elements in the Branches 53
3. A Structural Property 54
4. Differential Equations of an
Electrical Network 56
CHAPTER VII. COMPLEXES 61
1. Complexes 61
2. Subdivision 64
3. Complex Algebra 66
4. Subdivision Invariance 68
CHAPTER VIII. SURFACES 71
1. Definition of Surfaces 71
2. Orientable and Nonorientable Surfaces 72
3. Cuts . 76
4. A Property of the Sphere 7 9
Page
5. Reduction of Orientable Surfaces
to a Normal Form 8 3
6. Reduction of Nonorientable Surfaces
to a Normal Form 84
7. Duality in Surfaces 86
CHAPTER IX. PLANAR GRAPHS 89
1. Preliminaries 89
2. Statement and Solution of the
Spherical Graph Problem 90
3. Generalization 95
4. Direct Characterization of Planar
Graphs by Kuratowski 96
5. Reciprocal Networks 103
6. Duality of Electrical Networks 104
PART II
THE PICARD-LEFSCHETZ THEORY
AND FEYNMAN INTEGRALS
INTRODUCTION 113
CHAPTER I. TOPOLOGICAL AND ALGEBRAIC CONSIDERATIONS 119
1. Complex Analytic and Projective Spaces 119
2. Application to Complex Projective
n-space ^n 119
3. Algebraic Varieties 121
4. A Resume of Standard Notions of
Algebraic Topology 124
5. Homotopy. Simplicial Mappings 128
6. Singular Theory 129
7. The Poincare Group of Paths 130
8. Intersection Properties for Orientable
M2n Complex 131
9. Real Manifolds 133
CHAPTER II. THE PICARD-LEFSCHETZ THEORY 135
1. Genesis of the Problem 135
2. Method 136
3. Construction of the Lacets of
Surface $z 138
4. Cycles of $ . Variations of Integrals
Taken On $ ^4q
5. An Alternate Proof of the
Picard-Lefschetz Theorem 140
6. The A,-manifold M. Its Cycles and
Their Relation to Variations 146
CHAPTER III. EXTENSION TO HIGHER VARIETIES 149
1. Preliminary Remarks 149
2. First Application 150
3. Extension to Multiple Integrals 151
Page
4. The 2-Cycles of an Algebraic Surface 152
CHAPTER IV. FEYNMAN INTEGRALS 154
1. On Graphs 154
2. Algebraic Properties 156
3. Feynman Graphs I60
4. Feynman Integrals 162
5. Singularities 163
6. Polar Loci 164
7. More General Singularities 168
8. On the Loop-Complex 170
9. Some Complements 170
10. Examples 171
11. Calculation of an Integral 174
12. A Final Observation 175
CHAPTER V. FEYNMAN INTEGRALS. B. 177
1. Introduction 177
2. General Theory 177
3. Relative Theory 178
4. Application to Graphs 178
5. On Certain Transformations 180
BIBLIOGRAPHY 181
SUBJECT INDEX PART I 183
SUBJECT INDEX PART II 187
|
| any_adam_object | 1 |
| author | Lefschetz, Solomon 1884-1972 |
| author_GND | (DE-588)117714895 |
| author_facet | Lefschetz, Solomon 1884-1972 |
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| building | Verbundindex |
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| classification_rvk | SK 300 |
| ctrlnum | (OCoLC)1256925 (DE-599)BVBBV001962003 |
| dewey-full | 514/.2 510/.8 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 514 - Topology 510 - Mathematics |
| dewey-raw | 514/.2 510/.8 |
| dewey-search | 514/.2 510/.8 |
| dewey-sort | 3514 12 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| indexdate | 2024-07-09T15:37:59Z |
| institution | BVB |
| isbn | 038790137X 354090137X |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-001279457 |
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| physical | VIII, 189 S. graph. Darst. |
| publishDate | 1975 |
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| publisher | Springer |
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| series | Applied mathematical sciences |
| series2 | Applied mathematical sciences |
| spelling | Lefschetz, Solomon 1884-1972 Verfasser (DE-588)117714895 aut Applications of algebraic topology graphs and networks, the Picard-Lefschetz theory and Feynman integrals S. Lefschetz New York [u.a.] Springer 1975 VIII, 189 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Applied mathematical sciences 16 Algebraïsche topologie gtt Feynman, Intégrales de Feynman-integralen gtt Grafen gtt Graphes, Théorie des Netwerktheorie gtt Réseaux électriques (Circuits) Topologia Algebrica larpcal Topologie algébrique Algebraic topology Electric networks Feynman integrals Graph theory Picard-Lefschetz theory Graphentheorie (DE-588)4113782-6 gnd rswk-swf Topologie (DE-588)4060425-1 gnd rswk-swf Algebraische Topologie (DE-588)4120861-4 gnd rswk-swf Topologie (DE-588)4060425-1 s Graphentheorie (DE-588)4113782-6 s DE-604 Algebraische Topologie (DE-588)4120861-4 s Applied mathematical sciences 16 (DE-604)BV000005274 16 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001279457&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Lefschetz, Solomon 1884-1972 Applications of algebraic topology graphs and networks, the Picard-Lefschetz theory and Feynman integrals Applied mathematical sciences Algebraïsche topologie gtt Feynman, Intégrales de Feynman-integralen gtt Grafen gtt Graphes, Théorie des Netwerktheorie gtt Réseaux électriques (Circuits) Topologia Algebrica larpcal Topologie algébrique Algebraic topology Electric networks Feynman integrals Graph theory Picard-Lefschetz theory Graphentheorie (DE-588)4113782-6 gnd Topologie (DE-588)4060425-1 gnd Algebraische Topologie (DE-588)4120861-4 gnd |
| subject_GND | (DE-588)4113782-6 (DE-588)4060425-1 (DE-588)4120861-4 |
| title | Applications of algebraic topology graphs and networks, the Picard-Lefschetz theory and Feynman integrals |
| title_auth | Applications of algebraic topology graphs and networks, the Picard-Lefschetz theory and Feynman integrals |
| title_exact_search | Applications of algebraic topology graphs and networks, the Picard-Lefschetz theory and Feynman integrals |
| title_full | Applications of algebraic topology graphs and networks, the Picard-Lefschetz theory and Feynman integrals S. Lefschetz |
| title_fullStr | Applications of algebraic topology graphs and networks, the Picard-Lefschetz theory and Feynman integrals S. Lefschetz |
| title_full_unstemmed | Applications of algebraic topology graphs and networks, the Picard-Lefschetz theory and Feynman integrals S. Lefschetz |
| title_short | Applications of algebraic topology |
| title_sort | applications of algebraic topology graphs and networks the picard lefschetz theory and feynman integrals |
| title_sub | graphs and networks, the Picard-Lefschetz theory and Feynman integrals |
| topic | Algebraïsche topologie gtt Feynman, Intégrales de Feynman-integralen gtt Grafen gtt Graphes, Théorie des Netwerktheorie gtt Réseaux électriques (Circuits) Topologia Algebrica larpcal Topologie algébrique Algebraic topology Electric networks Feynman integrals Graph theory Picard-Lefschetz theory Graphentheorie (DE-588)4113782-6 gnd Topologie (DE-588)4060425-1 gnd Algebraische Topologie (DE-588)4120861-4 gnd |
| topic_facet | Algebraïsche topologie Feynman, Intégrales de Feynman-integralen Grafen Graphes, Théorie des Netwerktheorie Réseaux électriques (Circuits) Topologia Algebrica Topologie algébrique Algebraic topology Electric networks Feynman integrals Graph theory Picard-Lefschetz theory Graphentheorie Topologie Algebraische Topologie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001279457&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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