In this paper, we give a different proof of the fact that the odd dimensional quantum spheres are groupoid ${{C}^{*}}$-algebras. We show that the ${{C}^{*}}$-algebra $C\left( S_{q}^{2\ell +1} \right)$ is generated by an inverse semigroup $T$ of partial isometries. We show that the groupoid ${{\mathcal{G}}_{tight}}$ associated with the inverse semigroup $T$ by Exel is exactly the same as the groupoid considered by Sheu.

We provide a novel proof of the existence of regulator indecomposables in the cycle group $C{{H}^{2}}\left( X,\,1 \right)$, where $X$ is a sufficiently general product of two elliptic curves. In particular, the nature of our proof provides an illustration of Beilinson rigidity.

Let $F$ be a local non-archimedean field of characteristic zero. We prove that a representation of $GL\left( n,\,F \right)$ obtained from irreducible parabolic induction of supercuspidal representations is distinguished by an orthogonal group only if the inducing data is distinguished by appropriate orthogonal groups. As a corollary, we get that an irreducible representation induced from supercuspidals that is distinguished by an orthogonal group is metic.

We establish asymptotics and uniqueness (up to translation) of travelling waves for delayed $2\text{D}$ lattice equations with non-monotone birth functions. First, with the help of Ikehara’s Theorem, the a priori asymptotic behavior of travelling wave is exactly derived. Then, based on the obtained asymptotic behavior, the uniqueness of the traveling waves is proved. These results complement earlier results in the literature.