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Regular Structures U. Grenander

Regular Structures By U. Grenander

Regular Structures by U. Grenander

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Regular Structures Summary

Regular Structures: Lectures in Pattern Theory Volume III by U. Grenander

Most of the material in this book has been presented in lectures at Brown University, either in courses taught in the Division of Applied Mathematics or in the author's Re- search Seminar in Pattern Theory. I would like to thank the several members of the Division of Applied Mathematics that have participated in the discussions and in particular w. Freiberger, S. Geman, C.-R. Hwang, D. McClure and P. Thrift. I would also like to thank F. John, J. P. LaSalle, and L. Sirovich for accepting the manuscript for the Series Applied Mathematical Sciences published by Springer-Verlag. The research reported here has been supported by the National Science Foundation, Office of Naval Research and the Air Force Office of Scientific Research. I am grateful for the active interest and help given in various ways by Dr. Eamon Barrett, Dr. Kent Curtis, Dr. Robert Grafton and Dr. I. Shimi of these agencies. I also thank C.-R. Hwang and P. Thrift for help with proofreading. I am indebted to Mrs. E. Fonseca for her careful pre- paration of the manuscript, to Miss E. Addison for helping me with the many diagrams, and to Mrs. K. MacDougall for the final typing of the manuscript. Ulf Grenander Providence, Rhode Island October 1980 v TABLE OF CONTENTS Page INTRODUCTION ...

Table of Contents

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. Image algebras 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.

Additional information

NLS9780387905600
9780387905600
038790560X
Regular Structures: Lectures in Pattern Theory Volume III by U. Grenander
U. Grenander
Applied Mathematical Sciences
New
Paperback
Springer-Verlag New York Inc.
1981-05-04
570
N/A
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