In this paper, we determine the symmetrised density of a nonsingular doubly noncentral matrix variate beta type I and II distributions under different definitions.

We develop a general classification theory for Brumer’s dihedral quintic polynomials by means of Kummer theory arising from certain elliptic curves. We also give a similar theory for cubic polynomials.

We establish an isomorphism between the quintic cyclic polynomials discovered by Hashimoto-Tsunogai and those arising from Kummer theory for certain algebraic tori. This enables us to solve the isomorphism problem for Hashimoto-Tsunogai polynomials and also Brumer’s quintic polynomials.

Using the representation of Eq(2) on the non-commutative space zz ∗−qz∗z = σ; q < 1, σ > 0 summation formulas for the product of two, three and four q-Kummer functions are derived.

Forschungen zur Bau- und Ausstattungsgeschichte von Kloster Bronnbach. Hrsg. HelmutFlachenecker, Stefan Kummer Monika Schaupp. Veröffentlichungen des Historischen Vereins Wertheim 9. (Verlag e.V.) 2014. 327 S., Abb.

We compare the Kummer flat (resp. etale) cohomology with coefficients in smooth commutative group schemes, finite schemes and logarithmic multiplicative of Kato. will be particularly interested case algebraic tori topology. also make some computations for certain special cases base log scheme.