Additive and Polynomial Representations
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.
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Chapter 3 Extensive Measurement
Chapter 4 Difference Measurement
Chapter 5 Probability Representations
Chapter 6 Additive Conjoint Measurement
Chapter 7 Polynomial Conjoint Measurement
Chapter 8 Conditional Expected Utility
a o b a o c A U B A X A A1 X A2 additive conjoint structure additive representation algebra algebra of sets Archimedean axiom Archimedean property assume assumption axiomatization axioms of Definition b o c binary operation binary relation Chapter components concatenation conditional decision conditional probability construct countable defined denote dimensional analysis dimensionally invariant dimensions elements empirical Equation equivalence classes essential maximum example exists extensive measurement extensive structure finite formulate gambles hence holds hypothesis implies independent indifference curves inequalities interval scale Lemma monotonicity multiplicative order preserving ordered ring pair physical quantities polynomial proof of Theorem prove qualitative probability ratio scale real numbers real-valued function representation theorem restricted solvability result Section semigroup sign dependence similar solution strictly increasing subset Suppose theory transformations variables vector velocity weak order yields