On Weierstra{\ss} semigroups at one and two points and their corresponding Poincar\'e series

The aim of this paper is to introduce and investigate the Poincar\'e series associated with the Weierstra{\ss} semigroup of one and two rational points at a (not necessarily irreducible) non-singular projective algebraic curve defined over a finite field, as well as to describe their functional equations in the case of an affine complete intersection.Comment: Beginning of Section 3 and Subsection 3.1 were modifie

Patterns on the numerical duplication by their admissibility degree

We develop the theory of patterns on numerical semigroups in terms of the admissibility degree. We prove that the Arf pattern induces every strongly admissible pattern, and determine all patterns equivalent to the Arf pattern. We study patterns on the numerical duplication $S \Join^d E$ when $d \gg0$. We also provide a definition of patterns on rings

Strain Engineered Electrically Pumped SiGeSn Microring Lasers on Si

SiGeSn holds great promise for enabling fully group-IV integrated photonics operating at wavelengths extending in the mid-infrared range. Here, we demonstrate an electrically pumped GeSn microring laser based on SiGeSn/GeSn heterostructures. The ring shape allows for enhanced strain relaxation, leading to enhanced optical properties, and better guiding of the carriers into the optically active region. We have engineered a partial undercut of the ring to further promote strain relaxation while maintaining adequate heat sinking. Lasing is measured up to 90 K, with a 75 K T0. Scaling of the threshold current density as the inverse of the outer circumference is linked to optical losses at the etched surface, limiting device performance. Modeling is consistent with experiments across the range of explored inner and outer radii. These results will guide additional device optimization, aiming at improving electrical injection and using stressors to increase the bandgap directness of the active material