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Tables of Contents for Complex Systems and Binary Networks
Chapter/Section Title
Page #
Page Count
I Randomness & Complexity in Pure Mathematics
2
20
G. J. Chaitin
1 Hilbert on the Axiomatic Method
2
3
2 Godel, Turing and Cantor's Diagonal Argument
5
4
3 The Halting Probability and Algorithmic Randomness
9
6
4 Randomness in Arithmetic
15
3
5 Experimental Mathematics
18
4
II The Berry Paradox
22
11
G. J. Chaitin
III Knots and Complex Systems
33
44
L. H. Kauffman
1 Introduction
33
1
2 Reidemeister Moves and the Involutory Quandle
34
12
3 Braids and Magmas
46
3
4 Link, Twist, Writhe and DNA
49
3
5 Kirby Calculus
52
3
6 Quantum Link Invariants
55
5
6.1 Dirac Brackets
55
1
6.2 Knot Amplitudes
56
3
6.3 Topological Quantum Field Theory - First Steps
59
1
7 Knot Logic
60
6
8 A Knot-Logical Dialogue Between Inquisitive Alpha and Her Knot Theorist Friend Beta
66
9
9 Coda
75
2
IV Towards a Theory of Landscapes
77
87
P. F. Stadler
1 Introduction
78
1
2 Configuration Spaces
79
27
2.1 An Example -- TSP
79
2
2.2 Groups
81
5
2.3 Graphs
86
7
2.4 Fourier Series on Graphs
93
2
2.5 Spectral Properties of the Adjacency Matrix
95
4
2.6 Examples of Configuration Spaces
99
7
3 Random Fields
106
15
3.1 Preliminaries
106
1
3.2 Karhunen-Loeve Decomposition
107
1
3.3 The Markov Property
108
1
3.4 Isotropy
109
4
3.5 Homogeneity
113
1
3.6 Superpositions of Random Fields
113
4
3.7 Transformations
117
4
4 Landscapes
121
23
4.1 Empirical Correlation Functions
121
4
4.2 Elementary Landscapes
125
1
4.3 Local Optima
126
5
4.4 Some Example Landscapes
131
5
4.5 Classification of Landscapes
136
5
4.6 Landscapes on Irregular Graphs
141
2
4.7 Empirical Anisotropy
143
1
5 Biological Landscape: RNA
144
11
5.1 RNA Secondary Structures
144
3
5.2 RNA Free Energy Landscapes
147
3
5.3 Combinatory Maps
150
5
6 Conclusions
155
9
V Coarsening Phenomena in One Dimension
164
19
B. Derrida
1 Introduction
165
1
2 Finite Temperature Dynamics of the Ising Chain
165
2
3 Zero Temperature Dynamics of the Ising Chain
167
3
4 Zero Temperature Dynamics of the Potts Model
170
4
5 Reaction Diffusion Models
174
1
6 The Spins Which Never Flip
175
5
7 Conclusion
180
3
VI Cosmology as a Problem in Critical Phenomena
183
L. Smolin
1 Introduction
184
5
1.1 Why Critical Phenomena May Be Important for Particle Physics and Cosmology
187
2
2 Spiral Galaxies as Self-organized Systems
189
8
3 What Is the Large-Scale Organization of the Universe?
197
5
3.1 Dark Matter and the Issue of XXX
198
1
3.2 Quasar Absorption Lines and the Universe at Earlier Times
199
2
3.3 The Issue of Homogeneity on Very Large Scales
201
1
4 The Problem of the Origin of the Large-Scale Structure
202
4
4.1 Understanding the Non-linear Stages of Galaxy Formation
204
1
4.2 Possibilities for Early Structure Formation
205
1
5 The Problem of the Parameters in Particle Physics and Cosmology
206
6
5.1 Cosmological Natural Selection
208
4
6 Critical Phenomena in Quantum Gravity and the Classical World
212
3
7 Variety, Complexity and Relativity
215