In this article, we will construct a universal moduli space of stable parabolic vector bundles over the moduli space of marked Deligne--Mumford stable curves $\overline{M}_{g, n}$. The fiber of the moduli space over a marked smooth curve is the Mehta--Seshadri moduli space modulo the marked automorphism of the curve. The total space and the fibers over $\overline{M}_{g, n}$ will have good singularities.

For a class of forced non-autonomous dynamical systems, we prove the existence of invariant graphs and classify invariant graphs by supposing that the fibre maps, have negative Schwarzian derivative. Also, we investigate the existence of attracting basins that are intermingled.

Theory of Rota--Baxter operators on rings and algebras has been developed since 1960. In 2020, the notion of Rota--Baxter operator on a group was defined. Further, it was proved that one may define a skew left brace on any group endowed with a Rota--Baxter operator. Thus, a group endowed with a Rota--Baxter operator gives rise to a set-theoretical solution to the Yang--Baxter equation. We provide some general constructions of Rota--Baxter operators on a group. Given a map on a group, we study its extensions to a Rota--Baxter operator. We state the connection between Rota--Baxter operators on a group and Rota--Baxter operators on an associated Lie ring. We describe Rota--Baxter operators on sporadic simple groups.