Meanders : Sturm global attractors, seaweed Lie algebras and classical Yang-Baxter equation
This unique book’s subject is meanders (connected, oriented, non-self-intersecting planar curves intersecting the horizontal line transversely) in the context of dynamical systems. By interpreting the transverse intersection points as vertices and the arches arising from these curves as directed edg...
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| Main Author | |
|---|---|
| Format | eBook Book |
| Language | English |
| Published |
Berlin
De Gruyter
2017
Walter de Gruyter GmbH |
| Edition | 1 |
| Subjects | |
| Online Access | Get full text |
| ISBN | 9783110531473 311053147X |
| DOI | 10.1515/9783110533026 |
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| Abstract | This unique book’s subject is meanders (connected, oriented, non-self-intersecting planar curves intersecting the horizontal line transversely) in the context of dynamical systems. By interpreting the transverse intersection points as vertices and the arches arising from these curves as directed edges, meanders are introduced from the graphtheoretical perspective. Supplementing the rigorous results, mathematical methods, constructions, and examples of meanders with a large number of insightful figures, issues such as connectivity and the number of connected components of meanders are studied in detail with the aid of collapse and multiple collapse, forks, and chambers. Moreover, the author introduces a large class of Morse meanders by utilizing the right and left one-shift maps, and presents connections to Sturm global attractors, seaweed and Frobenius Lie algebras, and the classical Yang-Baxter equation. Contents Seaweed Meanders Meanders Morse Meanders and Sturm Global Attractors Right and Left One-Shifts Connection Graphs of Type I, II, III and IV Meanders and the Temperley-Lieb Algebra Representations of Seaweed Lie Algebras CYBE and Seaweed Meanders |
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| AbstractList | This unique book's subject is meanders (connected, oriented, non-self-intersecting planar curves intersecting the horizontal line transversely) in the context of dynamical systems. By interpreting the transverse intersection points as vertices and the arches arising from these curves as directed edges, meanders are introduced from the graphtheoretical perspective. Supplementing the rigorous results, mathematical methods, constructions, and examples of meanders with a large number of insightful figures, issues such as connectivity and the number of connected components of meanders are studied in detail with the aid of collapse and multiple collapse, forks, and chambers. Moreover, the author introduces a large class of Morse meanders by utilizing the right and left one-shift maps, and presents connections to Sturm global attractors, seaweed and Frobenius Lie algebras, and the classical Yang-Baxter equation. Contents Seaweed Meanders Meanders Morse Meanders and Sturm Global Attractors Right and Left One-Shifts Connection Graphs of Type I, II, III and IV Meanders and the Temperley-Lieb Algebra Representations of Seaweed Lie Algebras CYBE and Seaweed Meanders |
| Author | Karnauhova, Anna |
| Author_xml | – sequence: 1 fullname: Karnauhova, Anna |
| BackLink | https://cir.nii.ac.jp/crid/1130282269096778880$$DView record in CiNii |
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| DOI | 10.1515/9783110533026 |
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| Discipline | Mathematics |
| EISBN | 3110533022 9783110533026 3110531712 9783110531718 |
| Edition | 1 1st edition. |
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| ISBN | 9783110531473 311053147X |
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| Language | English |
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| Notes | "Doctoral dissertation at the Free University of Berlin, 2016."--T.p. verso Includes bibliographical references (p. [137]-138) |
| OCLC | 986177886 987934923 |
| PQID | EBC4851893 |
| PageCount | 146 |
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| PublicationDecade | 2010 |
| PublicationPlace | Berlin |
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| PublicationYear | 2017 |
| Publisher | De Gruyter Walter de Gruyter GmbH |
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| Snippet | This unique book’s subject is meanders (connected, oriented, non-self-intersecting planar curves intersecting the horizontal line transversely) in the context... This unique book's subject is meanders (connected, oriented, non-self-intersecting planar curves intersecting the horizontal line transversely) in the context... |
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| SubjectTerms | Attractors (Mathematics) Attraktor Curves, Algebraic Dynamisches System Lie algebras Lie-Algebra MATHEMATICS / Geometry / General Morse-Theorie Yang-Baxter equation Yang-Baxter-Gleichung |
| TableOfContents | 8.1 The Origin of Yang-Baxter Equation -- 8.2 YBE and the Braid Relation -- 8.3 Classical Yang-Baxter Equation (CYBE) -- 8.3.1 Frobenius Lie Algebras and Constant Solutions of CYBE -- Summary in German (Zusammenfassung) -- Bibliography Intro -- Contents -- Preface -- 1. Seaweed Meanders -- 1.1 Seaweed Meanders -- 1.1.1 Collapse Procedure for Seaweed Meanders -- 1.1.2 Collapse: Results on Connectivity -- 1.2 Proof of the Obstruction-Theorem -- 1.2.1 Obstruction-Theorem for Seaweed Meanders: Four Upper Blocks -- 1.2.2 Obstruction-Theorem for Seaweed Meanders with at Least Four Upper Blocks -- 2. Meanders -- 2.1 Collapse of (Closed) Meanders -- 2.2 Results and Examples for Connected Meanders -- 3. Morse Meanders and Sturm Global Attractors -- 3.1 Sturm Global Attractors -- 3.1.1 CW Complexes: Sturm Global Attractors -- 3.2 Morse Meanders -- 3.2.1 Definition of Morse Meanders -- 3.2.2 Raising the Morse Indices -- 4. Right and Left One-Shifts -- 4.1 Morse Meanders of Type I -- 4.2 Morse Meanders of Type II -- 4.3 Morse Meanders of Type III and IV -- 4.4 Sets of Morse Meanders of Type I, II, III, and IV are Disjoint and Consist of Pitchforkable Elements -- 4.5 Detecting Morse Meanders - Discussion -- 5. Connection Graphs of Type I, II, III and IV -- 5.1 Suspensions of Connection Graphs -- 5.2 Construction of Connection Graphs of Type I, II, III and IV -- 5.2.1 Connection Graphs of Type I -- 5.2.2 Isomorphism Class of Type I Connection Graphs -- 5.2.3 Connection Graphs of Type II -- 5.2.4 Isomorphism Class of Type II Connection Graphs -- 5.2.5 Connection Graphs of Type III and IV -- 5.3 Connection Graphs of Different Types Are Non-Isomorphic -- 6. Meanders and the Temperley-Lieb Algebra -- 6.1 From Open Meanders to Closed Meanders -- 6.2 From Blocks to Strand Diagrams -- 6.3 Temperley-Lieb Algebra -- 6.4 Left and Right One-Shifts in Terms of Temperley-Lieb Algebra -- 7. Representations of Seaweed Lie Algebras -- 7.1 Index of a Lie Algebra -- 7.2 Seaweed Lie Algebras -- 7.3 Graph Representations of Seaweed Lie Algebras -- 8. CYBE and Seaweed Meanders 1. Seaweed Meanders -- 6. Meanders and the Temperley–Lieb Algebra -- Contents -- 7. Seaweed Lie Algebras and Seaweed Meanders -- Preface -- 5. Connection Graphs of Type I, II, III and IV -- 3. Morse Meanders and Sturm Global Attractors -- 8. Classical Yang–Baxter Equation and Seaweed Meanders -- 2. Meanders -- Frontmatter -- 4. Right and Left One-Shifts -- Bibliography Summary in German (Zusammenfassung) -- |
| Title | Meanders : Sturm global attractors, seaweed Lie algebras and classical Yang-Baxter equation |
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