Carathéodory's Theorem and moduli of local connectivity
We give a constructive proof of the Carathéodory Theorem by means of the concept of a modulus of local connectivity and the extremal distance of the separating curves of an annulus.
math.CV↗
arXiv subjects
Timothy H. McNicholl
Publications and source records attributed to Timothy H. McNicholl.
Asymptotic density and the Ershov hierarchy
We classify the asymptotic densities of the $Δ^0_2$ sets according to their level in the Ershov hierarchy. In particular, it is shown that for $n \geq 2$, a real $r \in [0,1]$ is the density of an $n$-c.e.\ set if and only if it is a difference of left-$Π_2^0$ reals. Further, we show that the densities of the $ω$-c.e.\ sets coincide with the densities of the $Δ^0_2$ sets, and there are $ω$-c.e.\ sets whose density is not the density of an $n$-c.e. set for any $n \in ω$.
math.LO↗
Computing boundary extensions of conformal maps
Let $ϕ$ be a conformal map of the unit disk onto a domain $D$, and suppose $ϕ$ has a boundary extension. We show that arbitrarily good approximations of the boundary extension of $ϕ$ can be computed from sufficiently good approximations of $ϕ$ and sufficient local connectivity information for the boundary of $D$.
math.CV↗
Computing conformal maps onto circular domains
We show that, given a non-degenerate, finitely connected domain $D$, its boundary, and the number of its boundary components, it is possible to compute a conformal mapping of $D$ onto a circular domain \emph{without} prior knowledge of the circular domain. We do so by computing a suitable bound on the error in the Koebe construction (but, again, without knowing the circular domain in advance). As a scientifically sound model of computation with continuous data, we use Type-Two Effectivity.
math.CV↗
Computing links and accessing arcs
Sufficient conditions are given for the computation of accessing arcs and arcs that links boundary components of multiply connected domains. The existence of a not-computably-accessible but computable point on a computably compact arc is also demonstrated.
math.LO↗
An operator-theoretic existence proof of solutions to planar Dirichlét problems
By using some elementary techniques from operator theory, we prove constructively prove the existence of solutions to Dirichlét problems for planar Jordan domains with at least two boundary curves. An iterative method is thus obtained, and explicit bounds on the error in the resulting approximations are given. Finally, a closed form for the solution is given. No amount of differentiability of the boundary is assumed.
math.CV↗
Effective local connectivity properties
We investigate, and prove equivalent, effective versions of local connectivity and uniformly local arcwise connectivity for connected and computably compact subspaces of Euclidean space. We also prove that Euclidean continua that are computably compact and effectively locally connected are computably arcwise connected.
math.LO↗
An effective Caratheodory Theorem
By means of the property of effective local connectivity, the computability of finding the Carathéodory extension of a conformal map of a Jordan domain onto the unit disk is demonstrated.
math.CV↗
The power of backtracking and the confinement of length
We show that there is a point on a computable arc that does not belong to any computable rectifiable curve. We also show that there is a point on a computable rectifiable curve with computable length that does not belong to any computable arc.
math.LO↗