Contributors

Foreword

Preface

1 The Treatment of Uncertainty in the Physical Sciences

DENNIS H. ROUVRAY

I. General Introduction

II. The Quest for Certainty

III. Probabilistic Panaceas

IV. Quantum Indeterminacy

V. Chaotic Phenomena

VI. The Approach to Uncertainty

VII. Multivalued Logics

VIII. Possibilistic Paradigms

IX. Concluding Remarks

References

2 From Classical Mathematics to Fuzzy Mathematics: Emergence of a New Paradigm for Theoretical Science

GEORGE J. KLIR

I. Introduction

II. Types of Uncertainty

III. Fuzzy Sets and Fuzzy Logic: An Overview

A. Basic Concepts of Fuzzy Sets

B. Fuzzy Numbers and Fuzzy Arithmetic

C. Fuzzy Systems

D. Fuzzy Relations

E. Fuzzy Logic

F. Fuzzy Logic and Possibility Theory

IV. Scientific Paradigms and Paradigm Shifts

V. From Classical Sets to Fuzzy Sets: A Grand Paradigm Shift

VI. Stages in the Paradigm Shift

VII. Conclusions

Acknowledgment

References

3 Fuzzy Restrictions and Inherent Uncertainties in Chirality Studies

KURT MISLOW

I. Introduction

II. Manifestations of Chirality and the Choice of Models

A. Cryptochirality

B. Fuzzy Set Theory and Chemistry

C. Chirality as a Primitive Fuzzy Concept

D. On Quantifying the Chirality of Geometrical Objects

III. The Homochirality Problem

A. Homochirality Classes

B. Ruch's Model

C. Chiral and Achiral Enantiomerization Pathways

D. Pseudoscalar Properties and Chiral Zeroes

E. Homochirality Classes of Topological Constructions

Acknowledgments

References

4 Fuzzy Classical Structures in Genuine Quantum Systems

ANTON AMANN

I. Introduction

II. Strange States of Molecules

III. Chemical Concepts Are Fuzzy Classical Concepts

IV. Statistical versus Individual Formalisms of Quantum Mechanics

V. The Decomposition of a Nonpure State into Pure States Is Not Unique

VI. Decompositions of a Thermal State into Continuously Many Pure States

VII. Chemical versus Quantum-Mechanical Point of View

VIII. Effective Thermal States

IX. Stochastic Dynamics on the Level of Pure States

X. A Canonical Decomposition of Thermal States into Pure States

XI. Fuzzy Classical Observables and Large Deviation Theory

XII. The Structure of Single Molecules

XIII. Concluding Remarks

Acknowledgments

References

5 Fuzzy Measures of Molecular Shape and Size

PAUL G. MEZEY

I. Introduction

II. A Brief Review of Some Fuzzy Set Concepts Relevant to the Molecular Shape Problem

III. A Generalization of the Hausdorff Metric for Fuzzy Sets

IV. Fuzzy Symmetry Deficiency Measures, Fuzzy Chirality Measures, and Fuzzy Symmetry Groups Based on the Mass of Fuzzy Sets and Fuzzy Hausdorff-Type Metrics

V. Another Fuzzy Symmetry Approach: Syntopy and Syntopy Groups

VI. A Third Fuzzy Symmetry Approach: Symmorphy and Fuzzy Symmorphy Groups Based on Fuzzy Hausdorff-Type Metrics

VII. Proof of the Metric Properties of the Symmetric Scaling-Nesting Dissimilarity Measure

VIII. Chirality Measures and Symmetry Deficiency Measures for Continua Using the SNDSM Metric

IX. A Fuzzy Scaling-Nesting Similarity Measure and the Fuzzy Scaling-Nesting Dissimilarity Metric

X. Fuzzy Measures of Chirality and Symmetry Deficiency, Fuzzy Symmetry Groups, and Fuzzy Symmorphy Groups Based on the Fuzzy Scaling-Nesting Similarity Measure

XI. The Center of Mass of a Fuzzy Set, the Center of Molecular Electron Density, and Fuzzy Central Measures of Symmetry Deficiency

XII. The Fuzzy Average of Crisp Sets, the Fuzzy Average of Fuzzy Sets, the Crisp Average of Crisp Sets, the Crisp Average of Fuzzy Sets, and Related Fuzzy Symmetry Measures

XIII. Two Generalizations of the ZPA Folding-Unfolding Continuous Symmetry Measures for Continua Using the SNDSM Metric and the Hausdorff Metric

XIV. Fuzzy Set Generalizations of ZPA Folding-Unfolding Continuous Symmetry Measures Based on the Fuzzy FSNDSM Metric and Fuzzy Hausdorff-Type Metrics

XV. The Chiral Racemization Path Problem in n-Dimensions and Mislow's Label Paradox

XVI. Some Developments in the Computation of Properties of Fuzzy Electron Densities

Appendix

Acknowledgment

References

6 Linguistic Variables in the Molecular Recognition Problem

JURGEN BRICKMANN

I. Introduction

II. Transformation of Molecular Scenarios to a Three-Dimensional World

A. Molecular Surfaces

B. Electrostatic Maps

C. Local Hydrophobicity

D. Topographical Properties of Molecular Surfaces

E. Surface Flexibility

III. Fuzzy Logic Strategies and Molecular Recognition

A. Fuzzy Logic and Linguistic Variables

B. Segmentation of Molecular Surfaces with Linguistic Variables

C. Application: Topographical Analysis of the Molecular Surfaces of the Proteins Trypsin and Trypsinogen

IV. Matching of Molecular Surfaces with Fuzzy Logic Strategies

A. Rough Matching of Surface Patches

B. Fine Matching of Surface Patches

V. Conclusions

Acknowledgments

References

7 The Use of Fuzzy Graphs in Chemical Structure Research

JUN XU

I. Introduction

II. Fuzzy Graph Theory

A. Independent Spin Coupling Networks

B. Cluster Centers

C. Fuzzy Graph Pattern Recognition for ISNet

III. Fuzzy Graph Theory Applications in Computer-Assisted Biopolymer NMR Assignment

A. ISNet Generation

B. Integration of ISNets

C. Sequence-Specific Assignments

IV. Structure Elucidation Research Based upon Multiple Spectra

A. Rules in Multiple Spectral Knowledge Bases

B. Structure Deduction from Multiple Spectra

V. Summary

Acknowledgments

References

8 Fuzzy Logic in Computer-Aided Structure Elucidation

IVAN P. BANGOV

I. Why Is Fuzzy Logic Necessary?

II. Computer-Aided Structure Elucidation Strategies

III. Fuzzy Sets, Fuzzy Logic, and Fuzzy Graphs

IV. A Novel Strategy for Computer-Aided Structure Elucidation

A. Determination of the Atom Kind and Atom Valence Attributes

B. Determination of the Assigned Signal Parameters Attributes

C. Determination of the Hybridization / a-Environment Attributes

D. Determination of the List of -Type BSs (SSs) Associated with Each Atom Type (SV) Attribute

E. Guided Structure Generation

Acknowledgments

References

9 Fuzzy Hierarchical Classification Methods in Analytical Chemistry

DAN-DUMITRU DUMITRESCU

I. Introduction

II. Fuzzy Partition of a Fuzzy Class

III. Cluster Substructure in a Fuzzy Class

A. The Generalized Fuzzy n-Means Algorithm

B. Hard n-Means Algorithm

C. Adaptive Distances in Fuzzy Clustering

D. Linear Clusters

E. Clusters with a Degree of Linearity

F. Principal Components of a Fuzzy Class

G. Cluster Validity

IV. Fuzzy Divisive Hierarchical Clustering

A. Polarization Degree of a Fuzzy Partition

B. Fuzzy Divisive Hierarchical Clustering

V. Fuzzy Cross-Classification

A. One-Level Cross-Classification

B. Hierarchical Cross-Classification

VI. Fuzzy Hierarchical Classification Techniques in Analytical Chemistry

A. Selectivity Control in Acrylonitrile Electroreduction

B. Classification of Mineral Waters

C. Provenance of Archaeological Artifacts

D. Optimal Choice of Solvent Systems

E. Classification of Roman Pottery

F. Cross-Classification of Therapeutic Muds

References

Index