Invariance and Structural Dependence
(Sprache: Englisch)
This book deals with the notices of structural dependence
and independence, which are used in many applications of
mathematics to science. For instance,a physical law states
that one physical aspect is structurally dependent on one or
more other...
and independence, which are used in many applications of
mathematics to science. For instance,a physical law states
that one physical aspect is structurally dependent on one or
more other...
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Klappentext zu „Invariance and Structural Dependence “
This book deals with the notices of structural dependenceand independence, which are used in many applications of
mathematics to science. For instance,a physical law states
that one physical aspect is structurally dependent on one or
more other aspects. Structural dependence is closely related
to the mathematical idea of functional dependence. However,
functional dependence expresses a kind of complete
dependence while structural dependence includes partial
dependence. Further, structural dependence is primarily
thought of as a relation holding between aspects rather than
between their measures. In this book, the traditional way of
treating aspects within measurement theory is modified. An
aspect is not viewed as a set-theoretical structure but as a
function which has sets as arguments and set-theoretical
structures as values. This way of regardingaspects is
illustrated with an application to social choice and group
decision theory. Structural dependence is connected with the
idea of concomitant variations and the mathematical notion
of invariance. A distinction between dependence in the sense
of determination and in the sense of relevance is drawn.
Structural dependence both in the sense of determination and
of relevance can be found in degrees from complete
determination and relevance to complete undetermination and
irrelevance. The possibility of different "scales" of
determination and relevance is considered. One chapter of
the book is devoted to a study of the dependence relation
between the social ordering and the individual orderings in
social choice and group decision theory.
Inhaltsverzeichnis zu „Invariance and Structural Dependence “
1. Problem Area and Basic Formal Apparatus.- 1. The Concept of Dependence in Applied Mathematics; a First Account.- 1.0 Introduction.- 1.1 Determination and relevance.- 1.2 Partial determination.- 1.3 Structural dependence.- 1.4 Dependence and concomitant variations.- 1.5 Supervenience and dependence.- 1.6 Invariance and dependence.- 1.7 Independence of primitive symbols.- 1.8 Relations as functions.- 1.9 Notions of independence in modern measurement and decision theory.- 1.10 Applications of structural dependence.- 1.11 Summing up.- 2. Basic Formal Concepts and Terminology.- 2.0 Introduction.- 2.1 Relations and functions.- 2.2 Properties of binary relations.- 2.3 Order relations.- 2.4 Two lemmas on weak orders.- 2.5 Semiorders.- 2.6 Correspondences.- 2.7 Invariance.- 2.8 Relational structures.- 2.9 Isomorphisms and homomorphisms.- 2.10 Congruence relations.- 2.11 Lattices.- 2. An Informal Presentation of the Main Themes.- 3. Relationals.- 3.0 Introduction.- 3.1 The fundamentals of relational.- 3.2 Formal properties of relationals.- 3.3 Some examples.- 3.4 Finitary systems of relationals.- 3.5 Historical and bibliographical remarks.- 4. Subordination, Uncorrelation and Derivation.- 4.0 Introduction.- 4.1 Isomorphism preservation and transitions.- 4.2 Subordination and definability.- 4.3 Uncorrelation.- 4.4 The dependence between R and its regionalization R*.- 4.5 Equality and decision methods for relationals.- 4.6 Derived and derivable relationals.- 4.7 Stability of transitions.- 4.8 The structural character of transitions and subordination.- 4.9 Significance.- 5. An Example: Social Choice.- 5.0 Introduction.- 5.1 The notion of dependence in social choice theory.- 5.2 Preference relationals and collective choice rules.- 5.3 Isomorphism preservation, subordination and social choice.- 5.4 Relative effectiveness, derivability and social choice.- 5.5 Stability and background for collective choice rules.- 5.6 Structural dependence and aggregation; a preliminary remark.-
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6. Conformity and Measures.- 6.0 Introduction.- 6.1 Equality preservation and independent realizability.- 6.2 Congruence relational, conformity and import.- 6.3 Homomorphic representations.- 6.4 Measures.- 6.5 Numerical measures and representations.- 6.6 Connections between relational defined by measures.- 3. Formal Treatment of Basic Topics.- 7. Transitions Between Systems of Relationals.- 7.0 Introduction.- 7.1 Relational systems.- 7.2 Transitions and subordination.- 7.3 Transitions and uncorrelation.- 7.4 Concatenation and transition.- 7.5 Significance.- 7.6 Stability and monotonicity of s-functions.- 8. The Structure of Subordination.- 8.0 Introduction.- 8.1 Subalternation and rank.- 8.2 The lattice of subalternation.- 8.3 Correlation and collaterally.- 8.4 Semiranks.- 8.5 On equality preservation and independent realizability of structures.- 9. Isomorphic Mappings and Invariance.- 9.0 Introduction.- 9.1 Mappings.- 9.2 Isomorphic mappings.- 9.3 Automorphic mapping invariance.- 9.4 Global isomorphic mappings and global subordination.- Final remarks.- References.
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Bibliographische Angaben
- Autor: Jan Odelstad
- XII, 245 Seiten, 1 Schwarz-Weiß-Abbildungen, Maße: 17 x 24,4 cm, Kartoniert (TB), Englisch
- Verlag: Springer Berlin
- ISBN-10: 354055260X
- ISBN-13: 9783540552604
- Erscheinungsdatum: 15.04.1992
Sprache:
Englisch
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