Abstract:This article demonstrates that a normal fuzzy number can be constructed from earthquake waveform data. According to the RandomnessFuzziness Consistency Principle, two independent laws of randomness in [α, β] and [β, γ] are necessary and sufficient to define a normal fuzzy number [α, β, γ]. In this article, we have shown how to construct normal fuzzy numbers using data from earthquake waveform and have studied the pattern of the membership curve of the waveform.

Presently in most of the urban areas all over the world, due to the exponential increase in traffic, vehicular pollution has become one of the key contributors to air pollution. As uncertainty prevails in the process of designating the level of pollution of a particular region, a fuzzy method can be applied to see the membership values of that region to a number of predefined clusters. Also, due to the existence of different pollutants in vehicular pollution, the data used to represent it are in the form of numerical vectors. In our work, we shall apply the fuzzy c-means technique of Bezdek on a dataset representing vehicular pollution to obtain the membership values of pollution due to vehicular emission of a city to one or more of some predefined clusters. We shall try also to see the benefits of adopting a fuzzy approach over the traditional way of determining the level of pollution of the particular region.

In fuzzy clustering, unlike hard clustering, depending on the membership value, a single object may belong exactly to one cluster or partially to more than one cluster. Out of a number of fuzzy clustering techniques Bezdek's Fuzzy C-Means and Gustafson-Kessel clustering techniques are well known where Euclidian distance and Mahalanobis distance are used respectively as a measure of similarity. We have applied these two fuzzy clustering techniques on a dataset of individual differences consisting of fifty feature vectors of dimension (feature) three. Based on some validity measures we have tried to see the performances of these two clustering techniques from three different aspects-first, by initializing the membership values of the feature vectors considering the values of the three features separately one at a time, secondly, by changing the number of the predefined clusters and thirdly, by changing the size of the dataset.

Abstract:A fuzzy real number [α, β, γ] is an interval around the real number β with the elements in the interval being partially present. Partial presence of an element in a fuzzy set is defined by the name membership function. According to the Randomness-Fuzziness Consistency Principle, two independent laws of randomness in [α, β] and [β, γ] are necessary and sufficient to define a normal fuzzy number [α, β, γ]. In this article, we have shown how to construct normal fuzzy number using daily temperature data.

In this article we are going to discuss imprecise randomness using the mathematics of partial presence. The mathematical explanations of imprecise randomness would actually be complete only if it is explained with reference to the Randomness-Impreciseness Consistency Principle. In this article, we have described imprecise randomness with reference to a numerical example of the two sample t-test.