Framework Of Measure Theory Research Articles

Derivation of Feynman–Kac and Bloch–Torrey Equations in a Trapping Medium

We derive rigorously the fractional counterpart of the Feynman–Kac equation for a transport problem with trapping events characterized by fat-tailed time distributions. Our starting point is a random walk model with an inherent time subordination process due to the difference between clock times and operational times. We pass to the hydrodynamic limit using weak convergence arguments. Due to the lack of regularity of physical data, we use the framework of measure theory. We finally derive a Bloch–Torrey type equation for nuclear magnetic resonance data in this subdiffusive context.

Measure-theoretic Lie brackets for nonsmooth vector fields

In this paper we prove a generalization of the classical notion of commutators of vector fields in the framework of measure theory, providing an extension of the set-valued Lie bracket introduced by Rampazzo-Sussmann for Lipschitz continuous vector fields. The study is motivated by some applications to control problems in the space of probability measures, modeling situations where the knowledge of the state is probabilistic, or in the framework of multi-agent systems, for which only a statistical description is available. Tools of optimal transportation theory are used.

4.3.3 Systems Engineering Measurement Primer – A Metrological Evaluation

AbstractThe paper presents the results of an investigation of the INCOSE Systems Engineering Measurement Primer from the framework of measurement theory. This approach to the Primer is outside the scope of the claims of the Primer, and so is not entirely fair to the authors of the Primer. This investigation is conducted in order to contribute to the development of measurement processes suitable for use in systems engineering and also to provide a theoretical framework for measurement of systems engineering processes.

The measurement of membership by comparisons

Suppose we want to use a particular algebraic operation (for example, the drastic product t-norm) for representing an operation on fuzzy sets (for example, the intersection). This algebraic operation will be used with membership degrees, i.e. most of the time, numbers between 0 and 1. If we want that the results of our calculations make sense, we must be sure that the algebraic operation is compatible with the nature of the membership degrees, which is determined by the technique used to measure them. But this technique must, in turn, be compatible with the structural properties of the knowledge we want to represent. This paper addresses these issues within the framework of measurement theory. We provide sound theoretical foundations for the measurement of membership on ordinal and interval scales. But we also show that the level of measurement (ordinal, interval or even ratio) is not critical for the choice of a particular algebraic operation. Within measurement theory, whatever the scale, only the max and the min can be used for representing the intersection and the union. We also present some results about the complementation.

Indistinguishability operators in measurement theory, Part I: Conversions of indistinguishability operators with respect to scales

For two given ordinal scales in a measurement process, the present paper investigates how an indistinguishability operator evaluated according to one of these ordinal scales can be converted to another indistinguishability operator w.r.t. the other ordinal scale, and establishes the mathematical base of these conversions under the framework of measurement theory [Krantz, D.H., Luce, R.D., Suppes, P., Tversky, A. (1971) Foundations of Measurement, Vol. 1 (Academic Press, San Diego)]. Additionally, this work exposes the rudimentary facts behind the studies in [“Fuzzy Numbers and Equality Relations”, Proc. FUZZ'IEEE 93 (1993) 1298–1301; “Fuzzy Sets and Vague Environments”, Fuzzy Sets and Systems 66 (1994) 207–221; “Fuzzy Control on the Basis of Equality Relations-with an Example from Idle Speed Control”, IEEE Transactions on Fuzzy Systems 3 (1995) 336–350; and “T-partitions of the Real Line Generated by Idempotent Shapes”, Fuzzy Sets and Systems 91 (1997) 177–184], and points out the measurement theoretic derivations of the results in these studies.

Decomposition and integral representation of Cauchy interactions associated with measures

Cauchy interactions between subbodies of a continuous body are introduced in the framework of Measure Theory, extending the class of previously admissible ones. A decomposition theorem into a volume and a surface interaction is proved, as well as characterizations of the single components. Finally, an extension result and a generalized balance law are given.

On the qualitative/necessity possibility measure. (I). Investigation in the framework of measurement theory

In this paper we try to investigate the qualitative possibility and/or necessity measures introduced by D. Dubois [Computer and Artificial Intelligence 5 (1986) 403–416] and D. Dubois and H. Prade [Artificial Intelligence 50 (1991) 223–239] as an extension of a qualitative (or subjective) probability studied by T. Fine [Theory of Probability, Academic Press, New York, 1973]. We particularly review some relationship between these concepts and the basic qualitative possibility distribution as introduced by R.R. Yager [IEEE Trans. Fuzzy Systems 1 (3) (1993) 184–193]. This will then be studied in the general framework of preference relations when the qualitative possibility/necessity measures are real valued, graded from 0 to 1. The justification of the proposal real valued relation will be carried out in the framework of measurement theory where a relationship between the difference measurement scale and our proposal will be established. Two kinds of relations will be proposed. The first one is based on some appropriate exponential form, and, the second one is based on a more general form of R-implication operators. Finally, we briefly investigate some heuristic approach concerning the problem of how to obtain a real valued qualitative possibility from a quantitative possibility distribution.