We consider a version of the Lipman-Zariski conjecture for logarithmic vector fields and $1$-forms on pairs. Let $(X,D)$ be pair consisting normal complex variety $X$ an effective Weil divisor $D$ such that sheaf (or dually reflexive $1$-forms) is locally free. prove in this case following holds: If dlt, then necessarily smooth $\lfloor D\rfloor $ snc. lc or are generated by closed forms, $(X,\...

best approximation results provide an approximate solution to the fixed point equation $tx=x$, when the non-self mapping $t$ has no fixed point. in particular, a well-known best approximation theorem, due to fan cite{5}, asserts that if $k$ is a nonempty compact convex subset of a hausdorff locally convex topological vector space $e$ and $t:krightarrow e$ is a continuous mapping, then there exi...