Complex Oscillation and Removable Sets

Removable Sets for Subharmonic Functions

It is a classical result that a closed exceptional polar set is removable for subharmonic functions which are bounded above. Gardiner has shown that in the case of a compact exceptional set the above boundedness condition can be relaxed by imposing certain smoothness and Hausdorff measure conditions on the set. We give related results for a closed exceptional set, by replacing the smoothness an...

Complex Function Algebras and Removable Spaces

The compact Hausdorff space X has the Complex Stone-Weierstrass Property (CSWP) iff it satisfies the complex version of the Stone-Weierstrass Theorem. W. Rudin showed that all scattered spaces have the CSWP. We describe some techniques for proving that certain non-scattered spaces have the CSWP. In particular, if X is the product of a compact ordered space and a compact scattered space, then X ...

On Removable Sets for Sobolev Spaces in the Plane

The rapid points of a complex oscillation

By considering a counting-type argument on Brownian sample paths, we prove a result similar to that of Orey and Taylor on the exact Hausdorff dimension of the rapid points of Brownian motion. Because of the nature of the proof we can then apply the concepts to so-called complex oscillations (or algorithmically random Brownian motion), showing that their rapid points have the same dimension.

0n removable cycles in graphs and digraphs

In this paper we define the removable cycle that, if $Im$ is a class of graphs, $Gin Im$, the cycle $C$ in $G$ is called removable if $G-E(C)in Im$. The removable cycles in Eulerian graphs have been studied. We characterize Eulerian graphs which contain two edge-disjoint removable cycles, and the necessary and sufficient conditions for Eulerian graph to have removable cycles h...