Regular Structures

1. Patterns: From Chaos to Order.- 1.1. The search for regularity.- 1.2. Some regular structures.- 1.3. The mathematical study of regularity.- 2. A Pattern Formalism.- 2.1. The principle of atomism.- 2.2. The combinatory principle.- 2.3. The principle of observability.- 2.4. The principle of realism.- 3. Algebra of Regular Structures.- 3.1. Generator coordinates.- 3.2. Configuration coordinates.- 3.3. Connectors.- 3.4. Configuration homomorphisms.- 3.5. Configuration categories.- 3.6. Set operations in C (R).- 3.7. Operations on images.- 3.8. Homomorphisms for given global regularity.- 3.9. Representations by image isomorphisms.- 4. Some Topology of Image Algebras.- 4.1. A topology for configurations.- 4.2. A topology for images.- 4.3. Some examples.- 5. Metric Pattern Theory.- 5.1. Regularity controlled probabilities.- 5.2. Conditioning by regularity.- 5.3. Frozen patterns: finite G and n.- 5.4. Frozen patterns: infinite G and finite n..- 5.5. Quadratic energy function.- 5.6. Frozen patterns: infinite G and n.- 5.7. Asymptotically minimum energy.- 5.8. Asymptotics for large configurations.- 5.9. Spectral density matrix for ? = LINEAR(?).- 5.10. Factorization of the spectral density matrix.- 5.11. Representation of the random configurations.- 5.12. Spectral density matrix for ? = LATTICE(?).- 5.13. Factorization of the spectral density matrix in two dimensions.- 5.14. Representations of the random configurations in the two dimensional case.- 5.15. Laws of large numbers in pattern theory.- 5.16. Random dynamics for configurations.- 6. Patterns of Scientific Hypotheses.- 6.1. Hypotheses as regular structures.- 6.2. Patterns of statistical hypotheses.- 6.3. Generators for statistical hypotheses.- 6.4. Examples of configurations.- 6.5. Hypotheses as images.- 6.6. Imagealgebras of hypotheses.- 6.7. Conclusions.- 7. Synthesis of Social Patterns of Domination.- 7.1. Patterns in mathematical sociology.- 7.2. Domination regularity.- 7.3. Configuration dynamics.- 7.4. System in equilibrium.- 7.5. Large configurations — simulation results.- 7.6. Large configurations — analytical result.- 7.7. Further problems and extensions.- 7.8. Appendix.- 8. Taxonomic Patterns.- 8.1. A logic for taxonomic patterns.- 8.2. Logic of taxonomic affinity patterns.- 8.3. Synthesis of taxonomic affinity patterns.- 8.4. Analysis of affinity patterns.- 9. Patterns in Mathematical Semantics.- 9.1. Introduction.- 9.2. Introducing mathematical semantics.- 9.3. Formalization through regular structures.- 9.4. Two special image algebras.- 9.5. The choice of language type for the study.- 9.6. Semantic maps.- 9.7. Special semantic maps.- 9.8. Learning semantics.- 9.9. Abduction of semantic maps.- Outlook.- Notes.