Rough sets theory has been considered as a useful method to model the uncertainty and has been applied successfully in many fields. And every rough set is associated with some amount of fuzziness. On the other hand, rough sets theory has been generalized with coverings instead of classical partition. So it is necessary to consider the amount of fuzziness in generalized rough sets induced by a covering. In this paper, a measure of fuzziness in generalized rough sets induced by a covering is proposed. Moreover, some characterizations and properties of this measure are shown by examples, which is every useful in future research works of generalized rough sets induced by a covering.
Customer interaction in new service development has a positive impact on the performance of new services. In addition, prior studies recognize the importance of the fuzzy front-end stages of new service development. Yet, the researchers have not taken the next step to explore the relationship between these two key areas of service innovation. To address this critique of the literature, the process of customer interaction in the fuzzy front-end of new service development is investigated by conducting a rigorous qualitative field research involving 26 financial services firms. The findings suggest that the fuzzy front-end can be much less ‘fuzzy’ if customers are involved in the front-end stages of new service development.
This paper proposes a technique to deal with fuzziness in subjective evaluation data, and applies it to principal component analysis and correspondence analysis. In the existing method, or techniques developed directly from it, fuzzy sets are defined from some standpoint on a data space, and the fuzzy parameters of the statistical model are identified with linear programming or the method of least squares. In this paper, we try to map the variation in evaluation data into the parameter space while preserving information as much as possible, and thereby define fuzzy sets in the parameter space. Clearly, it is possible to use the obtained fuzzy model to derive things like the principal component scores from the extension principle. However, with a fuzzy model which uses the extension principle, the possibility distribution spreads out as the explanatory variable values increase. This does not necessarily make sense for subjective evaluations, such as a 5-level evaluation, for instance. Instead of doing so, we propose a method for explicitly expressing the vagueness of evaluation, using certain quantities related to the eigenvalues of a matrix which specifies the fuzzy parameter spread. As a numerical example, we present an analysis of subjective evaluation data on local environments.
A noncommutative space is considered, the position operators of which satisfy the commutativity relations of a Lie algebra. The basic tools for calculation on this space, including the product of the fields, inner product and the proper measure for integration are derived. Some general aspects of perturbative field theory calculations on this space are also discussed. One of the features of such models is that they are free from ultraviolet divergences (and hence free from UV/IR mixing as well), if the group is compact. The example of the group SO(3) or SU(2) is investigated in more detail. Copyright (C) EPLA, 2007.