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Plaid Model: (AMS-198) [Minkštas viršelis]

  • Formatas: Paperback / softback, 280 pages, aukštis x plotis: 235x155 mm, 103 b/w illus.
  • Serija: Annals of Mathematics Studies
  • Išleidimo metai: 19-Feb-2019
  • Leidėjas: Princeton University Press
  • ISBN-10: 0691181381
  • ISBN-13: 9780691181387
Kitos knygos pagal šią temą:
Plaid Model: (AMS-198)Didesnis paveikslėlis
  • Formatas: Paperback / softback, 280 pages, aukštis x plotis: 235x155 mm, 103 b/w illus.
  • Serija: Annals of Mathematics Studies
  • Išleidimo metai: 19-Feb-2019
  • Leidėjas: Princeton University Press
  • ISBN-10: 0691181381
  • ISBN-13: 9780691181387
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9780691181387.html
Raktažodžiai:

Outer billiards provides a toy model for planetary motion and exhibits intricate and mysterious behavior even for seemingly simple examples. It is a dynamical system in which a particle in the plane moves around the outside of a convex shape according to a scheme that is reminiscent of ordinary billiards. The Plaid Model, which is a self-contained sequel to Richard Schwartz’s Outer Billiards on Kites, provides a combinatorial model for orbits of outer billiards on kites.

Schwartz relates these orbits to such topics as polytope exchange transformations, renormalization, continued fractions, corner percolation, and the Truchet tile system. The combinatorial model, called “the plaid model,” has a self-similar structure that blends geometry and elementary number theory. The results were discovered through computer experimentation and it seems that the conclusions would be extremely difficult to reach through traditional mathematics.

The book includes an extensive computer program that allows readers to explore the materials interactively and each theorem is accompanied by a computer demonstration.

Recenzijos

"[ An] enjoyable excursion into a new field."---Meghan De Witt, MAA Reviews

Preface xi
Introduction 1(12)
0.1 Part 1: The Plaid Model and its Properties
5(1)
0.2 Part 2: The Plaid PET
5(1)
0.3 Part 3: The Graph PET
6(1)
0.4 Part 4: Plaid-Graph Correspondence
7(2)
0.5 Part 5: The Distribution of Orbits
9(1)
0.6 Companion Program
10(3)
Part 1: The Plaid Model 13(78)
Chapter 1 Definition of the Plaid Model
15(10)
1.1
Chapter Overview
15(1)
1.2 Basic Quantities and Notation
15(1)
1.3 Six Families of Lines
15(2)
1.4 Capacity, Mass, and Sign
17(1)
1.5 Light Points
18(3)
1.6 Transverse Directions for the Light Points
21(2)
1.7 Main Definition
23(2)
Chapter 2 Properties of the Model
25(10)
2.1
Chapter Overview
25(1)
2.2 Symmetries
25(3)
2.3 The Number of Intersection Points
28(2)
2.4 The Meaning of Capacity
30(2)
2.5 A Subtle Symmetry
32(3)
Chapter 3 Using the Model
35(10)
3.1
Chapter Overview
35(1)
3.2 The Big Polygon
35(2)
3.3 Hierarchical Information
37(2)
3.4 A Subdivision Algorithm
39(3)
3.5 Comparing Different Parameters
42(3)
Chapter 4 Particles and Spacetime Diagrams
45(12)
4.1
Chapter Overview
45(1)
4.2 Remote Adjacency
46(1)
4.3 Horizontal Particles
46(2)
4.4 Vertical Particles
48(3)
4.5 Spacetime Diagrams and Their Symmetries
51(2)
4.6 The Bad Tile Lemma
53(4)
Chapter 5 Three-Dimensional Interpretation
57(14)
5.1
Chapter Overview
57(1)
5.2 Stacking the Blocks
57(1)
5.3 Pixelated Spacetime Diagrams
58(2)
5.4 Tile Compatibility
60(3)
5.5 Spacetime Plaid Surfaces
63(3)
5.6 Discussion and Speculation
66(5)
Chapter 6 Pixellation and Curve Turning
71(10)
6.1
Chapter Overview
71(1)
6.2 Orienting the Worldlines
71(2)
6.3 The Sparseness of Worldlines
73(1)
6.4 Curve Turning Theorem: Vertical Case
74(2)
6.5 Curve Turning Theorem: Horizontal Case
76(2)
6.6 Two Applications
78(3)
Chapter 7 Connection to the Truchet Tile System
81(10)
7.1
Chapter Overview
81(1)
7.2 Truchet Tilings
81(2)
7.3 The Truchet Comparison Theorem
83(2)
7.4 The Fundamental Surface
85(1)
7.5 A Result from Elementary Number Theory
86(2)
7.6 Proof of the Truchet Comparison Theorem
88(3)
Part 2: The Plaid Pet 91(42)
Chapter 8 The Plaid Master Picture Theorem
93(10)
8.1
Chapter Overview
93(1)
8.2 The Spaces
94(1)
8.3 The Checkerboard Partition
94(4)
8.4 The Classifying Map
98(2)
8.5 The Main Result
100(3)
Chapter 9 The Segment Lemma
103(8)
9.1
Chapter Overview
103(2)
9.2 The Anchor Point
105(1)
9.3 A Computational Tool
105(2)
9.4 The Vertical Case
107(1)
9.5 The Horizontal Case
108(3)
Chapter 10 The Vertical Lemma
111(8)
10.1
Chapter Overview
111(1)
10.2 Using Symmetry
112(1)
10.3 Translating the Picture
113(1)
10.4 Some Useful Formulas
113(2)
10.5 The Undirected Result
115(2)
10.6 Determining the Directions
117(2)
Chapter 11 The Horizontal Lemma
119(6)
11.1
Chapter Overview
119(1)
11.2 Using Symmetry
120(1)
11.3 Translating the Picture
121(1)
11.4 Two Easy Technical Lemmas
122(1)
11.5 The Undirected Result
123(1)
11.6 Determining the Directions
123(2)
Chapter 12 Proof of the Main Result
125(8)
12.1
Chapter Overview
125(1)
12.2 Prism Structure
125(2)
12.3 Some Extra Symmetry
127(1)
12.4 The Vertical Case
128(2)
12.5 The Horizontal Case
130(3)
Part 3: The Graph Pet 133(38)
Chapter 13 Graph Master Picture Theorem
135(8)
13.1
Chapter Overview
135(1)
13.2 Special Orbits
135(1)
13.3 The Arithmetic Graph
136(1)
13.4 A Preliminary Result
137(2)
13.5 The PET Structure
139(2)
13.6 The Fundamental Polytopes
141(2)
Chapter 14 Pinwheels and Quarter Turns
143(10)
14.1
Chapter Overview
143(1)
14.2 The Pinwheel Map
143(2)
14.3 Outer Billiards and the Pinwheel Map
145(1)
14.4 Quarter Turn Compositions
146(1)
14.5 The Pinwheel Map as a QTC
147(4)
14.6 The Case of Kites
151(2)
Chapter 15 Quarter Turn Compositions and PETS
153(8)
15.1
Chapter Overview
153(1)
15.2 A Result from Linear Algebra
154(1)
15.3 The Map
154(2)
15.4 Compactifying Shears
156(1)
15.5 Compactifying Quarter Turn Maps
156(3)
15.6 The End of the Proof
159(2)
Chapter 16 The Nature of the Compactification
161(10)
16.1
Chapter Overview
161(1)
16.2 The Singular Directions
162(1)
16.3 The First Parallelotope
163(2)
16.4 The Second Parallelotope
165(1)
16.5 The General Master Picture Theorem
166(1)
16.6 Structure of the PET
167(1)
16.7 The Case of Kites
168(3)
Part 4: The Plaid-Graph Correspondence 171(40)
Chapter 17 The Orbit Equivalence Theorem
173(12)
17.1
Chapter Overview
173(1)
17.2 The Prisms
174(1)
17.3 The Map
175(1)
17.4 Characterizing the Image
176(1)
17.5 The Clean Partition
177(1)
17.6 The Main Proof
178(2)
17.7 Computational Techniques
180(2)
17.8 The Calculations
182(3)
Chapter 18 The Quasi-Isomorphism Theorem
185(10)
18.1
Chapter Overview
185(1)
18.2 The Canonical Affine Transformation
186(1)
18.3 The Graph Grid
187(1)
18.4 The Intertwining Lemma
188(1)
18.5 The Correspondence of Orbits
189(3)
18.6 The End of the Proof
192(1)
18.7 The Projection Theorem
193(1)
18.8 Renormalization Interpretation
194(1)
Chapter 19 Geometry of the Graph Grid
195(4)
19.1
Chapter Overview
195(1)
19.2 The Grid Geometry Lemma
195(2)
19.3 The Graph Reconstruction Lemma
197(2)
Chapter 20 The Intertwining Lemma
199(12)
20.1
Chapter Overview
199(1)
20.2 A Resume of Transformations
200(1)
20.3 Injectivity of the Map
201(1)
20.4 Calculating a Single Point
201(1)
20.5 Dissecting the Set
202(2)
20.6 The Induction Step
204(2)
20.7 Discussion
206(1)
20.8 The Diagonal Case
206(5)
Part 5: The Distribution Of Orbits 211(54)
Chapter 21 Existence of Infinite Orbits
213(6)
21.1
Chapter Overview
213(1)
21.2 Definedness Criterion
214(1)
21.3 Spacetime Diagrams Revisited
214(1)
21.4 Taking a Limit
215(1)
21.5 Associated Paths
216(1)
21.6 Sketch of an Alternate Proof
217(2)
Chapter 22 Existence of Many Large Orbits
219(8)
22.1
Chapter Overview
219(1)
22.2 Equidistribution Properties
220(1)
22.3 The Ubiquity Lemma
221(1)
22.4 The Rectangle Lemma
222(1)
22.5 Proof of the Main Result
222(1)
22.6 The Continued Fraction Length
223(2)
22.7 The End of the Proof
225(2)
Chapter 23 Infinite Orbits Revisited
227(12)
23.1
Chapter Overview
227(1)
23.2 The Approximating Sequence
227(2)
23.3 The Copy Theorem
229(2)
23.4 The End of the Proof
231(1)
23.5 The Copy Letnma
232(2)
23.6 Proof of the Box Theorem
234(1)
23.7 Proof of the Copy Theorem
235(2)
23.8 Hidden Symmetries
237(2)
Chapter 24 Some Elementary Number Theory
239(6)
24.1
Chapter Overview
239(1)
24.2 A Structural Result
239(3)
24.3 Unfinished Business
242(3)
Chapter 25 The Weak and Strong Case
245(8)
25.1
Chapter Overview
245(1)
25.2 The First Two Statements
245(1)
25.3 A Technical Lemma
246(1)
25.4 The Mass and Capacity Sequences
247(1)
25.5 Vertical Intersection Points
248(1)
25.6 A Matching Criterion
249(1)
25.7 Verifying the Matching Criterion
250(3)
Chapter 26 The Core Case
253(12)
26.1
Chapter Overview
253(1)
26.2 The First Two Statements
254(1)
26.3 Geometric and Arithmetic Alignment
254(1)
26.4 Geometric Alignment
255(1)
26.5 Alignment of the Capacity Sequences
256(1)
26.6 A Technical Lemma
257(3)
26.7 The Mass Sequences: Central Case
260(2)
26.8 The Mass Sequences: Peripheral Case
262(1)
26.9 The End of the Proof
263(2)
References 265(2)
Index 267
Richard Evan Schwartz is the Chancellors Professor of Mathematics at Brown University. He is the author of Spherical CR Geometry and Dehn Surgery and Outer Billiards on Kites (both Princeton).