The Projective Heat Map

By: Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: Mathematical Surveys and Monographs ; v.219Publisher: Providence : American Mathematical Society, 2017Description: 1 Online-Ressource (210 p)Subject(s): Additional physical formats: 1470435144 | 9781470435141 | 1470440504 | 9781470440503 | Erscheint auch als: The projective heat map. Druck-Ausgabe Providence, Rhode Island : American Mathematical Society, 2017. x, 195 Seiten | Print version: The Projective Heat Map. Providence : American Mathematical Society,c2017MSC: MSC: 26A18 | 51M15 | 37B05 | 37E30RVK: RVK: SK 380 | SK 810 | SK 420LOC classification:
  • QA360.S39 2017
Online resources: Summary: This book introduces a simple dynamical model for a planar heat map that is invariant under projective transformations. The map is defined by iterating a polygon map, where one starts with a finite planar N-gon and produces a new N-gon by a prescribed geometric construction. One of the appeals of the topic of this book is the simplicity of the construction that yet leads to deep and far reaching mathematics. To construct the projective heat map, the author modifies the classical affine invariant midpoint map, which takes a polygon to a new polygon whose vertices are the midpoints of the originSummary: 3.2. Affine Patches3.3. Projective Transformations and Dualities; 3.4. The Cross Ratio; 3.5. The Hilbert Metric; 3.6. Projective Invariants of Polygons; 3.7. Duality and Relabeling; 3.8. The Gauss Group; Chapter 4. Elementary Algebraic Geometry; 4.1. Measure Zero Sets; 4.2. Rational Maps; 4.3. Homogeneous Polynomials; 4.4. Bezout's Theorem; 4.5. The Blow-up Construction; Chapter 5. The Pentagram Map; 5.1. The Pentagram Configuration Theorem; 5.2. The Pentagram Map in Coordinates; 5.3. The First Pentagram Invariant; 5.4. The Poincare Recurrence Theorem; 5.5. Recurrence of the Pentagram MapSummary: 3.2. Affine Patches3.3. Projective Transformations and Dualities; 3.4. The Cross Ratio; 3.5. The Hilbert Metric; 3.6. Projective Invariants of Polygons; 3.7. Duality and Relabeling; 3.8. The Gauss Group; Chapter 4. Elementary Algebraic Geometry; 4.1. Measure Zero Sets; 4.2. Rational Maps; 4.3. Homogeneous Polynomials; 4.4. Bezout's Theorem; 4.5. The Blow-up Construction; Chapter 5. The Pentagram Map; 5.1. The Pentagram Configuration Theorem; 5.2. The Pentagram Map in Coordinates; 5.3. The First Pentagram Invariant; 5.4. The Poincare Recurrence Theorem; 5.5. Recurrence of the Pentagram MapSummary: 8.1. Overview8.2. The Lower Bound; 8.3. The Upper Bound; Chapter 9. The Convex Case; 9.1. Flag Invariants of Convex Pentagons; 9.2. The Gauss Group Acting on the Unit Square; 9.3. A Positivity Criterion; 9.4. The End of the Proof; 9.5. The Action on the Boundary; 9.6. Discussion; Chapter 10. The Basic Domains; 10.1. The Space of Pentagons; 10.2. The Action of the Gauss Group; 10.3. Changing Coordinates; 10.4. Convex and Star Convex Classes; 10.5. The Semigroup; 10.6. A Global Point of View; Chapter 11. The Method of Positive Dominance; 11.1. The Divide and Conquer Algorithm; 11.2. PositivitySummary: 8.1. Overview8.2. The Lower Bound; 8.3. The Upper Bound; Chapter 9. The Convex Case; 9.1. Flag Invariants of Convex Pentagons; 9.2. The Gauss Group Acting on the Unit Square; 9.3. A Positivity Criterion; 9.4. The End of the Proof; 9.5. The Action on the Boundary; 9.6. Discussion; Chapter 10. The Basic Domains; 10.1. The Space of Pentagons; 10.2. The Action of the Gauss Group; 10.3. Changing Coordinates; 10.4. Convex and Star Convex Classes; 10.5. The Semigroup; 10.6. A Global Point of View; Chapter 11. The Method of Positive Dominance; 11.1. The Divide and Conquer Algorithm; 11.2. PositivitySummary: This book introduces a simple dynamical model for a planar heat map that is invariant under projective transformations. The map is defined by iterating a polygon map, where one starts with a finite planar N-gon and produces a new N-gon by a prescribed geometric construction. One of the appeals of the topic of this book is the simplicity of the construction that yet leads to deep and far reaching mathematics. To construct the projective heat map, the author modifies the classical affine invariant midpoint map, which takes a polygon to a new polygon whose vertices are the midpoints of the originPPN: PPN: 897975049Package identifier: Produktsigel: ZDB-4-NLEBK
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