In this work, we discuss the law of α-migrativity as applied to fuzzy implication functions in a meaningful way. A generalisation of this law leads us to Pexider-type functional equations connected with the law of importation, viz., the generalised law of importation I(C(x,α),y)=I(x,J(α,y)) (GLI) and the generalised cross-law of importation I(C(x,α),y)=J(x,I(α,y)) (CLI), where C is a generalised conjunction. In this article we investigate only (GLI). We begin by showing that the satisfaction of law of importation by the pairs (C, I) and/or (C, J) does not necessarily lead to the satisfaction of (GLI). Hence, we study the conditions under which these three laws are related.

High dimensional data analysis gives rise to many challenges. One such that has come to gain a lot of attention recently is the concentration of distances (CoD) phenomenon, which is the inability of distance functions to distinguish points well in high dimensions. CoD affects almost every machine learning and data analysis algorithm in high dimensions. In this work, we present a novel efficient and effective empirical index that not only illustrates whether a distance function tends to concentrate for a given data set, but also enables us to measure the rate of concentration and allows us to compare different distance functions vis-a-vis their rate of concentration. As opposed to existing empirical indices, the proposed empirical measure uses only the internal characteristics of a given data set and hence is applicable on real data sets, which was hitherto not possible.