The problem of representing intersection and union in fuzzy set theory is considered. There are various proposals in the literature to model these concepts. The possibility of using continuous triangular norms and conorms (including min and max) are taken up in a measurement--theoretic setting. The conditions are laid out to arrive at cardinal scales on which addition and multiplication are meaningful and critically discussed. These conditions must either be accepted on normative grounds or must be empirically verified before the modeling process in order to see which operations are meaningful. It is emphasized that the Archimedean axiom and the existence of natural bounds are crucial in arriving at ratio and absolute scale representations. Keywords: Membership functions, measurement theory, operators, relations. 1 Introduction and Preview When Zadeh [45] introduced the concept of a fuzzy set he suggested to use the functions min and max to model set theoretic intersection an...

Epistasis is defined as the influence of the genotype at one locus on the effect of a mutation at another locus. As such it plays a crucial role in a variety of evolutionary phenomena such as speciation, population bottle necks and the evolution of genetic architecture (i.e. the evolution of dominance, canalization and genetic correlations). In mathematical population genetics, however, epistasis is often represented as a mere noise term in an additive model of gene effects. In this paper it is argued that epistasis needs to be scaled in a way that is more directly related to the mechanisms of evolutionary change. A review of general measurement theory shows that the scaling of a quantitative concepts has to reflect the empirical relationships among the objects. To apply these ideas to epistatic mutation effects it is proposed to scale AxA epistatic effects as the change in the magnitude of the additive effect of a mutation at one locus due to a mutation at a second locus. It is shown ...

This self-contained paper is part of a series [FF1] seeking to understand groups of homeomorphisms of manifolds in analogy with the theory of Lie groups and their discrete subgroups. In this paper we consider groups which act on R with restrictions on the fixed point set of each element. One result is a topological characterization of affine groups in Diff 2 (R) as those groups whose elements have at most one fixed point. 1

A near-identity nilpotent pseudogroup of order m ≥ 1 is a family f1,...,fn: (−1,1) → R of C 2 functions for which: |fi − id | C 1 < ǫ for some small positive real number ǫ < 1/10 m+1 and commutators of the functions fi of order at least m equal the identity. We present a classification of near-identity nilpotent pseudogroups: our results are similar to those of Plante, Thurston, Farb and Franks. As an application, we classify certain foliations of nilpotent manifolds.

Since the time of Savage (1954) it has been accepted that subjective expected utility (SEU) embodies the concept of rational individual behavior under uncertainty. If, however, one alters the domain formulation in two ways, by distinguishing gains from losses and by adding a binary operation of joint receipt, then equally rational arguments lead in the case of binary mixed gambles to predictions quite different from those of SEU. A question, raised but not really answered, is whether there is a rational argument for choosing

This paper reports on a construction of the real numbers in the ProofPower implementation of the HOL logic. Since the original construction was implemented, some major improvements to the ideas have been discovered. The improvements involve some entertaining mathematics: it turns out that the Dedekind cuts provide many routes one can travel to get from the ring of integers, Z, to the eld of real numbers, R.

Abstract. In this paper, we study the maximal bounded Z-filtrations of a complex semisimple Lie algebra L. Specifically, we show that if L is simple of classical type An, Bn, Cn or Dn, then these filtrations correspond uniquely to a precise set of linear functionals on its root space. We obtain partial, but not definitive, results in this direction for the remaining exceptional algebras. Maximal bounded filtrations were first introduced in the context of classifying the maximal graded subalgebras of affine Kac-Moody algebras, and the maximal graded subalgebras of loop toroidal Lie algebras. Indeed, our main results complete this classification in most cases. Finally, we briefly discuss the analogous question for bounded filtrations with respect to other Archimedean ordered groups. 1.

The foundations of measurement theory were invesUgated by modeltheoreUc methods. The purpose was to establish a firm ba~is for <;leneral system theory. One major result was the formulatton of the concepts of scale, scale transformation and the proof of the existence, for an arbitrary scale, of the group of scale transformatiqns which "leave the scale form invariantII. As an illustration of the appUcabiUty of these concepts and because of its intrinsic interest an expo~ition was <;liven of the theory of measurement for extensive quantities. A novel formulation of the usual axioms was developed which made them ~lementary formulae in the first order predicate calculus. Thus it was P9ssible to show that this theory is model-complete 0 Then A. Robinson I f! tlleorems were applied to show that this theory is negation complete. In the formulation of the existence theorem for scale transformation 1 groups no restriction was placed on the empirical 'E-model Qr the numerical 'E-model and in the other illustrative examples it wa $ shown how different choices of these 'E-models lead to different scale transformation groups. An example of a theory of a non-extensive quantity was presented but the methods used before would not Yield model-completeness for this theory and thus an interesting unsolved problem remains.

In this paper I examine Field’s account of the applicability of mathematics from a measurementtheoretic perspective. Within this context, I object to Field’s instrumentalism, arguing that it depends on an incomplete analysis of applicability. I show in particular that, once the missing piece of analysis is provided, the role played by numerical entities in basic empirical theories must be revised: such revision implies that instrumentalism should be rejected and mathematical entities be regarded not merely as useful tools but also as conceptual schemata by means of which we can articulate our understanding of experience.

Abstract. This paper reports on a construction of the real numbers in the ProofPower implementation of the HOL logic. Since the original construction was implemented, some major improvements to the ideas have been discovered. The improvements involve some entertaining mathematics: it turns out that the Dedekind cuts provide many routes one can travel to get from the ring of integers, Z, to the field of real numbers, R. The traditional stop-over on the way is the field of rational numbers, Q. This paper shows that going via certain rings of algebraic numbers can provide a pleasant alternative to the more well-trodden track.