Abstract Let K be a number field which is multiquadratic or quartic cyclic. We prove several results about the Kummer extensions of , namely concerning intersection between and cyclotomic . For G finitely generated subgroup × we consider cyclotomic-Kummer K ( ? </m:mro...

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(1.1) { Γ(α)Ψ(α,α−γ+1;ξ) }−1 exp(−ξv)v(1+v), v > 0, (1.2) respectively, where α > 0, β > 0, ξ > 0, −∞ < γ,λ < ∞, 1F1, and Ψ are confluent hypergeometric functions. These distributions are extensions of Gamma and Beta distributions, and for α < 1 (and certain values of λ and γ) yield bimodal distributions on finite and infinite ranges, respectively. These distributions are used (i) in the Bayesi...

In this paper we highlight the benefits of using genus 2 curves in public-key cryptography. Compared to the standardized genus 1 curves, or elliptic curves, arithmetic on genus 2 curves is typically more involved but allows us to work with moduli of half the size. We give a taxonomy of the best known techniques to realize genus 2 based cryptography, which includes fast formulas on the Kummer su...

In this paper, we propose extensions for the classical Kummer test, which is a very far-reaching criterion that provides sufficient and necessary conditions convergence divergence of series positive terms. Furthermore, present discuss some interesting consequences examples such as Olivier's theorem Raabe, Bertrand Gauss's test.