We introduce the geometry of pseudo-totally umbilical lightlike submanifold $M$ a semi-Riemannian manifold $\bar{M}$. In line with above, we give complete classification $1$-lightlike submanifolds, such as hypersurfaces and half-lightlike submanifolds. Furthermore, screen distributions are also investigated, for any whose umbilical. Closely linked to above show, under some geometric conditions,...

Veech groups are discrete subgroups of $\mathrm{SL}(2,\mathbb{R})$ which play an important role in the theory translation surfaces. For a special class surfaces called origamis or square-tiled surfaces, their finite index $\mathrm{SL}(2,\mathbb{Z})$. We show that each stratum space contains infinitely many whose group is totally non-congruence group, i.e., it surjects to $\mathrm{SL} (2,\mathbb...

The dangers of opioid overuse have been known for at least 150 years, yet in 2021 we are the midst an epidemic with addiction and overdose deaths a crisis level worldwide, especially US. Opioids like morphine were used cautiously severe pain by most physicians until massive change attitude occurred late 1980s. idea was spread, through aggressive marketing, that opioids oxycodone could be admini...