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Inhaltsverzeichnis

Frontmatter

Introduction

Abstract
This is the third and final volume of the Lectures in Pattern Theory. Its two first chapters describe the science-theoretic principles on which pattern theory rests. Chapter 3 is devoted to the algebraic study of regularity while Chapter 5 contains new results in metric pattern theory. Some brief remarks on topological image algebras can be found in Chapter 4.
Ulf Grenander

Chapter 1. Patterns: From Chaos to Order

Abstract
The search for regularity is a dominant theme in man’s attempt to understand the world around him. Any such attempt is based on an assumption, tacitly made or explicit, that phenomena in nature and in the man-made world are governed by laws that result in order and structure.
Ulf Grenander

Chapter 2. A Pattern Formalism

Abstract
We shall now examine the basic concepts and relations of pattern theory as expressed in a formalism whose terminology and notation will be discussed in this chapter.
Ulf Grenander

Chapter 3. Algebra of Regular Structures

Abstract
In Volume I we introduced the objects of pattern theory: generators, configurations, and images, as well as pattern relations: similarities, combinatory relations, and deformations. We are now going to deepen our understanding of these concepts with the emphasis on their algebraic properties.
Ulf Grenander

Chapter 4. Some Topology of Image Algebras

Abstract
Combinatory regularity is algebraic in character and can be studied from the perspective of partial universal algebra. At the same time it supports other mathematical structures, for example, measures, and, as we shall see below, topologies. By this we mean at the moment notions of neighborhood, convergence, and continuity, not the topologies that characterize global regularity in terms of the connection type.
Ulf Grenander

Chapter 5. Metric Pattern Theory

Abstract
Given a set of laws R for the regular structure, they induce natural probability measures over the configuration space T(R) and associated image algebras. This topic — metric pattern theory — was introduced in Section 2.10 of Volume I and we shall pursue it further in this chapter, extend the results to great generality and deepen some of them. When doing this we shall concentrate our attention on the configurations and neglect the corresponding questions for images; see Notes A. A reader can therefore in the present chapter think of the identification rule R as EQUAL, treating images as identical to configuration. Important advances have been made in metric pattern theory after the appearance of Volume I, some of which are contained in two reports by Hwang and Thrift, see Bibliography; much of this chapter is devoted to presenting their results.
Ulf Grenander

Chapter 6. Patterns of Scientific Hypotheses

Abstract
The question of how scientific hypotheses are formed is an intriguing problem that has puzzled psychologists and philosophers of science for a long time. More recently it has been studied from the point of view of artificial intelligence (see Notes A) and this may eventually lead to a better understanding of how humans create hypotheses. The way hypotheses are suggested in the sciences may have more in common with the mental processes of artistic creation than with the strict schemes of the computer programs used in artificial intelligence.
Ulf Grenander

Chapter 7. Synthesis of Social Patterns of Domination

Abstract
Sociology studies systems of interacting individuals or groups of individuals, and mathematical sociology creates the mathematical tools needed for such studies. We suggest that pattern theory — the mathematical theory of regular structures — can be used to create such tools.
Ulf Grenander

Chapter 8. Taxonomic Patterns

Abstract
When Carolus Linnaeus writes that the major activities of the taxonomist are divisio et denominatio — separation into classes, families, orders, and naming them — he emphasizes distinctness or dissimilarity of specimens. In abstract terms if the specimens are represented as elements x of some space X, x € X, we look for a way of decomposing X into subsets Xυ; υ = 1,2,…; such that the elements in Xυ are quite different from those of Xµ when υ ≠ µ, see Notes A.
Ulf Grenander

Chapter 9. Patterns in Mathematical Semantics

Abstract
In this chapter we shall introduce mathematical semantics as the pattern theoretic study of mappings between image algebras and formal languages.
Ulf Grenander

Backmatter

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