There are three types of parabolic subgroups in $Sp(2,\mathbf{R})$. In this paper we show that the Eisenstein series with respect to the Siegel parabolic subgroup corresponds to the Eisenstein series with respect to the Jacobi parabolic subgroup by theta correspondences.

As an analogue of the classical uncertainty inequality on the Euclidean space, we shall obtain a generalization on the Sturm-Liouville hypergroups $(\mathbf{R}_{+},*(A))$. Especially, we shall obtain a condition on $A$ under which the discrete part of the Plancherel formula vanishes.

For any subset, which is not always finite, of $\mathbf{C}^{n+1}-\{\mathbf{0}\}$ in subgeneral position, we introduce a weight function and a constant like as Nochka weight function and Nochka constant which was introduced for a finite subset of $\mathbf{C}^{n+1}-\{\mathbf{0}\}$ in subgeneral position.

H.Choda proved that every element in the closed unit ball of a von Neumann algebra is average of two extreme points of the ball. Here, we prove the strict generalisation of Choda’s result to arbitrary $JBW^{*}$-triples.