In this paper we discuss the fixed points of ASYMPTOTIC CONTRACTIONs and Boyd-Wong type CONTRACTIONs in uniform spaces equipped with an E-distance. A new version of Kirk's fixed point theorem is given for ASYMPTOTIC CONTRACTIONs and Boyd-Wong type CONTRACTIONs is investigated in uniform spaces.

In this article we obtain for the arithmetic progression of positive integers ki-r an ASYMPTOTIC formula, namely Pn i=1 (ki-r) ~ arknnnn-r/k+1/2/en (k>0) (r = 0, 1, . . ., k-1), which generalizes the Stirling’s formula for the sequence i of all positive integers. In this formula ar is a positive number which depend of r for a fixed k. We also obtain some information of the product Pn i=1 pi where pi is the i-th prime number.

Mathematical calculations are given concerning the evaluation of the negative ions photo detachment cross-section s, into a so-called saturation regime. The interaction between a negative ion particle beam and a laser beam is examined under theoretical aspects. A quantitative criterion Sis proposed to define the saturation threshold between the linear and the saturated domains, which are both present in this saturation regime. The ASYMPTOTIC behaviours extracted at the low and high energy limits are used to determine this threshold quantitative criterionS and to evaluate also the photo detachment cross-section s. The case of a symmetric gaussian photo detachment laser beam shape is examined according to the proposed formalism, which can be used either for the photo-detachment or photo-ionization processes, and could be potentially used into technological solutions for negative ion neutralisation processes (such as neutral beam injector) in the future fusion energy devices. Estimations onto the errors related to the use of this methodology are given.

In this paper, we improve divergence measure between populations that has been considered in [19, 24, 31]. Furthermore we use a version of this measure to test homogeneity of covariances, to test homogeneity of means, to test complete homogeneity in multivariate normal.