Knot points of typical continuous functions

It is well known that most continuous functions are nowhere differentiable. Furthermore, in terms of Dini derivatives, most continuous functions are nondifferentiable in the strongest possible sense except in a small set of points. In this paper, we completely characterise families S of sets of points for which most continuous functions have the property that such small set of points belongs to S. The proof uses a topological zero-one law and the Banach-Mazur game.Comment: 24 page

Differentiability of Lipschitz Functions in Lebesgue Null Sets

We show that if n>1 then there exists a Lebesgue null set in R^n containing a point of differentiability of each Lipschitz function mapping from R^n to R^(n-1); in combination with the work of others, this completes the investigation of when the classical Rademacher theorem admits a converse. Avoidance of sigma-porous sets, arising as irregular points of Lipschitz functions, plays a key role in the proof.Comment: 33 pages. Corrected minor misprints and added more detail to the proofs of Lemma 3.2 and Lemma 8.

Diabetic microvascular complications as simple indicators of risk for cardiovascular outcomes and heart failure

No abstract available

Boundary structure and size in terms of interior and exterior harmonic measures in higher dimensions

In this work we introduce the use of powerful tools from geometric measure theory (GMT) to study problems related to the size and structure of sets of mutual absolute continuity for the harmonic measure $\omega^+$ of a domain $\Omega=\Omega^+\subset\mathbb{R}^n$ and the harmonic measure $\omega^-$ of $\Omega^-$, $\Omega^-=$int$(\Omega^c)$, in dimension $n\ge 3$