Let $R$ be a smooth Riemannian manifold of finite volume, $\Delta$ its Laplace (-Beltrami) operator. Canonical direct-sum decompositions of certain subspaces of the Wiener and Royden algebras of $R$ are found, and for biharmonic functions (those for which $\Delta \Delta u=0$) the decompositions are related to the values of the functions and their Laplacians on appropriate ideal boundaries.

Compactifications of biframes are defined, and characterized internally by means of strong inclusions. The existing description of the compact, zero-dimensional coreflection of a biframe is used to characterize all zero-dimensional compactifications, and a criterion identifying them by their strong inclusions is given. In contrast to the above, two sufficient conditions and several examples show that the existence of smallest biframe compactifications differs significantly from the corresponding...

We prove the following theorem: Given a⊆ω and $1\le \alpha <{\omega }_1^{CK}$, if for some $\eta <{\aleph }_1$ and all u ∈ WO of length η, a is ${\Sigma }_{\alpha }^0\left(u\right)$, then a is ${\Sigma }_{\alpha }^0$.We use this result to give a new, forcing-free, proof of Leo Harrington’s theorem: ${\Sigma }_1^1$-Turing-determinacy implies the existence of $0^{}$.