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Constraint Theory: Multidimensional Mathematical Model Management / Edition 1

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Overview

The enormous potential of digital computation to manage new complex systems is impeded by exponential increases in complexity. As the model's dimensionality increases from hundreds to thousands of variables, and as submodels constructed by diverse technical teams are integrated into the total model, the model is likely to become inconsistent and even more likely, the computational requests on the model become unallowable. This text analyzes the way constraint theory employs bipartite graphs and constraint matrices to detect and correct these well-posed problems. It also presents the process of locating the "kernel of constraint", literally trillions of times faster than a random search, determining consistency and compatibility within seconds. This text is an invaluable reference for all engineers, mathematicians, and managers concerned with modeling.

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Editorial Reviews

From the Publisher
Dr. George Friedman is indisputably the father of the very powerful methods of Constraint Theory. Indeed, in 1970 it received major commendation by the IEEE: Dr. Friedman's pioneering research was awarded the highly prestigious W.R.G. Baker prize as the most significant published research contribution among thousands of published research papers. This speaks volumes for Dr. Friedman's contribution. —Cornelius T. Leondes, UCLA

Friedman's accomplishment represents engineering at its finest. The choice of subject matter must be honored by the experience gained in his many years of high-level service in the company now known as Northrop-Grumman. The credibility of the theory rests upon the formal proofs which are interspersed among the illuminating hypothetical dialog sequences between manager and analyst, which bring out distinctions that the organization must face, en route to accepting Friedman's work as essential to achieve quality control in developing and applying large models. —John N. Warfield

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Product Details

Table of Contents

Chapter 1 Motivations 1

What Is Constraint Theory And Why Is It Important?

1.1 Trends and Problems in System Technologies

1.2 An Example of Low Dimension

1.3 The Manager and Analyst Continue their Dialogue

1.4 Preliminary Conclusions

1.5 A Little Window into Future Chapters

1.6 Problems for the Curious Reader

Chapter 2 The Four-Fold Way 25

How to Perceive Complex Mathematical Models and Well-Posed Problems

2.1 Prologue: The Manager and Analyst Discuss the Origins of Multidimensional Models and Well-Posedness

2.2 The First View: Set Theoretic

2.3 The Second View: Family of Submodels

2.4 The Third View: The Bipartite Graph

2.5 The Fourth View: The Constraint Matrix

2.6 Model Consistency and Computational Allow Ability

2.7 The Manager and Analyst Continue their Dialogue

2.8 Chapter Summary

2.9 Problems for the Interested Student

Chapter 3 General Results 49

From Protomath to Math to Metamath

3.1 Language and Mathematics

3.2 Most General Trustworthy Results

3.3 Classes of Relations

3.4 Manager and Analyst Revisited

3.5 Chapter Summary

3.6 Problems for the General Student

Chapter 4 Regular Relations 61

Searching for the Kernels of Constraint

4.1 Cognitive Barriers to Circuits

4.2 Node, Knot and Basic Nodal Square Sanctification

4.3 Useful Properties of Bapartite Graphs

4.4 Cornering the Culprit Kernels; Ten Easy Pieces

4.5 Continuing the Pursuit Inside the Circuit Clusters (CC)

4.6 Locating the BNSs in Minutes, Not University Lifetimes

4.7 Comparison of Computational Complexity; Trillions and Trillions of Times Faster

4.8 Zero Constraint All Along the Computational Path

4.9 Recapitulation of Computational Flow

4.10 General Procedure for Determining Consistency and Allow Ability in a Model of Regular Relations

4.11 Summary of Chapter and Constraint Theory Toolkit

4.12 Queries for the Regular Student

Chapter 5 Discrete and Interval Relations 111

The Diminished Utility of Metamodels

5.1 Metamodel Issues and Perspectives

5.2 The General Taxonomy and Primary Property of Discrete Relations

5.3 Bolean Relations

5.4 Topological Implications

5.5 Allowability of Discrete Computations

5.6 Inequality Relations

5.7 Summary

5.8 Problems for the Discrete Student

Chapter 6 The Logical Structure of Constraint Theory 129

A Compact Summary

6.1 Overview

6.2 Postulates and Philosophical Assumptions

6.3 Definitions

6.4 Theorems

6.5 Graphs of the Logical Structure of Constraint Theory

6.6 Completeness

Chapter 7 Examples of Constraint Theory Applied to Real-World Problems 137

7.1 Apologies Not Required

7.2 Cost as an Independent Variable (CAIV)

7.3 The Kinematics of Free-Fall Weapons

7.4 The Deflection of an Earth-Threatening Asteroid Employing Mass Drivers

Chapter 8 Manager and Analyst Meet Again 157

Gists and Schizophrenia

Appendices

Appendix A Computational Request Disappointments; Results of the USC Allowability Project 163

Appendix B Graph Theory Overview 171

Appendix C The Logic of "IF" and "IF and Only IF" 175

Appendix D Vector Spaces Applied to Graph Theory 177

References 181

Index 183

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