In this paper we consider the computational complexity of uniformizing a domain with a given computable boundary. We give nontrivial upper and lower bounds in two settings: when the approximation of the boundary is given either as a list of pixels, or by a Turing machine.

If$n$is a non-negative integer, then denote by ∂^{-$n$}$H$^{∞}the space of all complex-valued functions$f$defined on $\mathbb{D}$ such that$f$,$f$^{(1)},$f$^{(2)},...,$f$^{($n$)}belong to$H$^{∞}, with the norm $$\|f\|=\sum_{j=0}^{n}\frac{1}{j!}\|f^{(j)}\|_{\infty}.$$ We prove bounds on the solution in the corona problem for ∂^{-$n$}$H$^{∞}. As corollaries, we obtain estimates in the corona theorem also for some other subalgebras of the Hardy space$H$^{∞}.

We prove the vanishing of the Koszul homology group$H$_{μ}(Kos($M$)_{μ}), where μ is the minimal number of generators of M. We give a counterexample that the Koszul complex of a module is not always acyclic and show its relationship with the homology of commutative rings.

The purpose of this erratum is to point out a mistake in a previous article by the authors and to explain how this affects the results of that article.

In this paper, we construct some unirational Calabi–Yau threefolds in characteristic 3. We adopt the method by Schoen, but we use quasi-elliptic surfaces instead of elliptic surfaces. We find new examples which do not admit a lifting to characteristic zero.