Krantz / Luce / Suppes Additive and Polynomial Representations

1. Auflage 2014
ISBN: 978-1-4832-5830-0
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 608 Seiten, Web PDF

Additive and Polynomial Representations deals with major representation theorems in which the qualitative structure is reflected as some polynomial function of one or more numerical functions defined on the basic entities. Examples are additive expressions of a single measure (such as the probability of disjoint events being the sum of their probabilities), and additive expressions of two measures (such as the logarithm of momentum being the sum of log mass and log velocity terms). The book describes the three basic procedures of fundamental measurement as the mathematical pivot, as the utilization of constructive methods, and as a series of isomorphism theorems leading to consistent numerical solutions. The text also explains the counting of units in relation to an empirical relational structure which contains a concatenation operation. The book notes some special variants which arise in connection with relativity and thermodynamics. The text cites examples from physics and psychology for which additive conjoint measurement provides a possible method of fundamental measurement. The book will greatly benefit mathematicians, econometricians, and academicians in advanced mathematics or physics.

Autoren/Hrsg.

Krantz, David H.

Luce, R Duncan

Suppes, Patrick

Inhaltsverzeichnis

1;Front Cover;1
2;Additive and Polynomial Representations;4
3;Copyright Page;5
4;Table of Contents;8
5;Dedication;6
6;Summary of Contents;7
7;Preface;18
7.1;Mathematical Background;20
7.2;Selecting Among the Chapters;21
8;Acknowledgments;22
9;Notational Conventions;24
10;Chapter 1. Introduction;28
10.1;1.1 THREE BASIC PROCEDURES OF FUNDAMENTAL MEASUREMENT;28
10.2;1.2 THE PROBLEM OF FOUNDATIONS;33
10.3;1.3 ILLUSTRATIONS OF MEASUREMENT STRUCTURES;40
10.4;1.4 CHOOSING AN AXIOM SYSTEM;48
10.5;1.5 EMPIRICAL TESTING OF A THEORY OF MEASUREMENT;53
10.6;1.6 ROLES OF THEORIES OF MEASUREMENTIN THE SCIENCES;58
10.7;1.7 PLAN OF THE BOOK;60
10.8;EXERCISES4;62
11;Chapter 2. Construction of Numerical Functions;65
11.1;2.1 REAL-VALUED FUNCTIONS ON SIMPLY ORDERED SETS;65
11.2;2.2 ADDITIVE FUNCTIONS ON ORDERED ALGEBRAIC STRUCTURES;70
11.3;2.3 FINITE SETS OF HOMOGENEOUS LINEAR INEQUALITIES;86
11.4;EXERCISES;96
12;Chapter 3. Extensive Measurement;98
12.1;3.1 INTRODUCTION;98
12.2;3.2 NECESSARY AND SUFFICIENT CONDITIONS;99
12.3;3.3 PROOFS;104
12.4;3.4 SUFFICIENT CONDITIONS WHEN THE CONCATENATION OPERATION IS NOT CLOSED;108
12.5;3.7 ESSENTIAL MAXIMA IN EXTENSIVE STRUCTURES;119
12.6;3.8 PROOFS;123
12.7;3.9 ALTERNATIVE NUMERICAL REPRESENTATIONS;126
12.8;3.10 CONSTRUCTIVE METHODS;129
12.9;3.11 PROOFS 8;133
12.10;3.12 CONDITIONALLY CONNECTED EXTENSIVE STRUCTURES;138
12.11;3.13 PROOFS;144
12.12;3.14 EXTENSIVE MEASUREMENT IN THE SOCIAL SCIENCES;150
12.13;3.15 LIMITATIONS OF EXTENSIVE MEASUREMENT;157
12.14;EXERCISES;159
13;Chapter 4. Difference Measurement;163
13.1;4.1 INTRODUCTION;163
13.2;4.2 POSITIVE-DIFFERENCE STRUCTURES;172
13.3;4.3 PROOF OF THEOREM 1;175
13.4;4.4 ALGEBRAIC-DIFFERENCE STRUCTURES;177
13.5;4.5 PROOFS;184
13.6;4.6 CROSS-MODALITY ORDERING;191
13.7;4.7 PROOF OF THEOREM 4;193
13.8;4.8 FINITE, EQUALLY SPACED DIFFERENCE STRUCTURES;194
13.9;4.9 PROOFS;195
13.10;4.10 ABSOLUTE-DIFFERENCE STRUCTURES;197
13.11;4.11 PROOFS;201
13.12;4.12 STRONGLY CONDITIONAL DIFFERENCE STRUCTURES;204
13.13;4.13 PROOFS;211
13.14;EXERCISES;222
14;Chapter 5. Probability Representations;226
14.1;5.1 INTRODUCTION;226
14.2;5.2 A REPRESENTATION BY UNCONDITIONALPROBABILITY;229
14.3;5.3 PROOFS;238
14.4;5.4 MODIFICATIONS OF THE AXIOM SYSTEM;241
14.5;5.5 PROOFS;244
14.6;5.6 A REPRESENTATION BY CONDITIONAL PROBABILITY;247
14.7;5.7 PROOFS;255
14.8;5.8 INDEPENDENT EVENTS;265
14.9;5.9 PROOF OF THEOREM 10;268
14.10;EXERCISES;270
15;Chapter 6. Additive Conjoint Measurement;272
15.1;6.1 SEVERAL NOTIONS OF INDEPENDENCE;272
15.2;6.2 ADDITIVE REPRESENTATION OF TWO COMPONENTS;277
15.3;6.3 PROOFS;288
15.4;6.4 EMPIRICAL EXAMPLES;294
15.5;6.5 MODIFICATIONS OF THE THEORY;298
15.6;6.6 PROOFS;306
15.7;6.7 INDIFFERENCE CURVES AND UNIFORM FAMILIES OF FUNCTIONS;310
15.8;6.8 PROOFS;315
15.9;6.9 BISYMMETRIC STRUCTURES;320
15.10;6.10 PROOFS;325
15.11;6.11 ADDITIVE REPRESENTATION OF n COMPONENTS;328
15.12;6.12 PROOFS;333
15.13;6.13 CONCLUDING REMARKS;338
15.14;EXERCISES;339
16;Chapter 7. Polynomial Conjoint Measurement;343
16.1;7.1 INTRODUCTION;343
16.2;7.2 DECOMPOSABLE STRUCTURES;344
16.3;7.3 POLYNOMIAL MODELS;348
16.4;7.4 DIAGNOSTIC ORDINAL PROPERTIES;356
16.5;7.5 SUFFICIENT CONDITIONS FOR THREE-VARIABLESIMPLE POLYNOMIALS;374
16.6;7.6 PROOFS;383
16.7;EXERCISES;393
17;Chapter 8. Conditional Expected Utility;396
17.1;8.1 INTRODUCTION;396
17.2;8.2 A FORMULATION OF THE PROBLEM;399
17.3;8.3 PROOFS;409
17.4;8.4 TOPICS IN UTILITY AND SUBJECTIVE PROBABILITY;418
17.5;8.5 PROOFS;428
17.6;8.6 OTHER FORMULATIONS OF RISKY AND UNCERTAIN DECISIONS;433
17.7;8.7 CONCLUDING REMARKS;444
17.8;EXERCISES;447
18;Chapter g Measurement Inequalities;450
18.1;9.1 INTRODUCTION;450
18.2;9.2 FINITE LINEAR STRUCTURES;454
18.3;9.3 PROOF OF THEOREM 1;460
18.4;9.4 APPLICATIONS;461
18.5;9.5 POLYNOMIAL STRUCTURES;474
18.6;9.6 PROOFS;477
18.7;EXERCISES;479
19;Chapter l0. Dimensional Analysis and Numerical Laws;481
19.1;10.1 INTRODUCTION;481
19.2;10.2 THE ALGEBRA OF PHYSICAL QUANTITIES;486
19.3;10.3 THE PI THEOREM OF DIMENSIONAL ANALYSIS;491
19.4;10.4 PROOFS;494
19.5;10.5 EXAMPLES OF DIMENSIONAL ANALYSIS;498
19.6;10.6 BINARY LAWS AND UNIVERSAL CONSTANTS;507
19.7;10.7 TRINARY LAWS AND DERIVED MEASURES;510
19.8;10.8 PROOFS;520
19.9;10.9 EMBEDDING PHYSICAL ATTRIBUTES IN A STRUCTURE OF PHYSICAL QUANTITIES;526
19.10;10.10 WHY ARE NUMERICAL LAWS DIMENSIONALLY INVARIANT?;531
19.11;10.11 PROOFS;540
19.12;10.12 INTERVAL SCALES IN DIMENSIONAL ANALYSIS;542
19.13;10.13 PROOFS;547
19.14;10.14 PHYSICAL QUANTITIES IN MECHANICS AND GENERALIZATIONS OF DIMENSIONAL INVARIANCE;550
19.15;10.15 CONCLUDING REMARKS;562
19.16;EXERCISES;563
19.17;DIMENSIONS AND UNITS OF PHYSICAL QUANTITIES;566
20;Answers and Hints to Selected Exercises;572
21;References;578
22;Author Index;598
23;Subject Index;604