Some possible definitions for the natural parametrization of SLE paths are proposed in terms of various limits. One of the definitions is used to give a new proof that the Hausdorff dimension of SLEκ paths is 1 + κ 8 for κ < 8.

We show that if n ≥ 1, Ω ⊂ R is a connected domain that is porous around a subset E ⊂ ∂Ω of finite and positive Hausdorff H-measure, and the harmonic measure ω is absolutely continuous with respect to H on E, then ω|E is concentrated on an n-rectifiable set.

We prove that the definition of supersimplicity in metric structures from [Ben05b] is equivalent to an a priori stronger variant. This stronger variant is then used to prove that if T is a supersimple Hausdorff cat then so is its theory of lovely pairs.

This paper looks at the use of Model Theoretic ideas to study certain non-Hausdorff spaces with a view to their application in Manin’s proposed theory of Real Multiplication, which seeks to establish a framework in which to prove Hilbert’s twelfth problem for real quadratic fields. We study morphisms between these non-Hausdorff spaces using Nonstandard Analysis, and as a consequece in certain s...

An ultrafilter U is Hausdorff if for any two functions f, g ∈ ω ω , f (U) = g(U) iff f ↾X = g↾X for some X ∈ U. We will show that it is consistent that there are no Hausdorff ultrafilters.

In this article, we will be interested in the extent to which the assumption of first countability in this theorem can be weakened. Recall that a Hausdorff topological space X is Fréchet if whenever x is a limit point of A ⊆ X, there is a sequence an (n < ω) of elements of A which converges to x. This is a natural weakening of first countability which has been extensively studied in the literat...

Throughout this paper, we denote byN and R the sets of positive integers and real numbers, respectively. Let D be a nonempty closed subset of a real Banach space X. A mapping T : D → D is said to be nonexpansive, if ‖Tx − Ty‖ ≤ ‖x − y‖, for all x, y ∈ D. Let N D and CB D denote the family of nonempty subsets and nonempty closed bounded subsets of D, respectively. The Hausdorff metric on CB D is...

Finitary endofunctors of locally presentable categories are proved to have equational presentations. Special attention is paid to the Hausdorff functor of non-empty compact subsets of a complete metric space.