Random conformal snowflakes

In many problems of classical analysis extremal configurations appear to exhibit complicated fractal structure. This makes it much harder to describe extremals and to attack such problems. Many of these problems are related to the multifractal analysis of harmonic measure. We argue that, searching for extremals in such problems, one should work with random fractals rather than deterministic ones. We introduce a new class of fractals random conformal snowflakes and investigate its properties developing tools to estimate spectra and showing that extremals can be found in this class. As an application we significantly improve known estimates from below on the extremal behaviour of harmonic measure, showing how to constuct a rather simple snowflake, which has a spectrum quite close to the conjectured extremal value

On Littlewood's Constants

In two papers, Littlewood studied seemingly unrelated constants: (i) the best α such that for any polynomial f, of degree n, the areal integral of its spherical derivative is at most ·nα, and (ii) the extremal growth rate rβ of the length of Green's equipotentials for simply connected domains. These two constants are shown to coincide, thus greatly improving known estimates on α. 2000 Mathematics Subject Classification 30C50 (primary), 30C85, 30D35 (secondary

Harmonic measure and SLE

In this paper we rigorously compute the average multifractal spectrum of harmonic measure on the boundary of SLE clusters

Packing dimension of mean porous measures

We prove that the packing dimension of any mean porous Radon measure on $\mathbb R^d$ may be estimated from above by a function which depends on mean porosity. The upper bound tends to $d-1$ as mean porosity tends to its maximum value. This result was stated in \cite{BS}, and in a weaker form in \cite{JJ1}, but the proofs are not correct. Quite surprisingly, it turns out that mean porous measures are not necessarily approximable by mean porous sets. We verify this by constructing an example of a mean porous measure $\mu$ on $\mathbb R$ such that $\mu(A)=0$ for all mean porous sets $A\subset\mathbb R$.Comment: Revised versio

The Dutch Healthy Diet Index : development, evaluation, and application

The Dutch Healthy Diet index – Development, Evaluation, and Application Linde van Lee Abstract Background: Dietary indices evaluate the conformity of an individual’s diet with pre-defined standards. Generally, dietary guidelines are used for this purpose. As no index based on the current dietary guidelines was available in the Netherlands, the aim of the present thesis was to develop, evaluate, and apply a dietary index for use in the country. Methods and results: The Dutch Healthy Diet index (DHD-index) was developed on the basis of the 2006 Dutch dietary guidelines using data relating to 749 young adults who completed two 24-hour recalls in the Dutch national food consumption survey 2003. The index comprises ten components on physical activity, vegetables, fruit, dietary fibre, saturated fatty acids, trans fatty acids, consumption occasions with acidic drinks and foods, sodium, and alcohol. Scores for each component range between 0 (no adherence) and 10 (complete adherence) points. The DHD-index was inversely associated with energy intake and positively associated with most micronutrient intakes when adjusted for energy intake. We compared the DHD-index score based on two 24-hour recalls with the index based on the food frequency questionnaires (FFQ) of 121 adults from the European Food Consumption Validation study. We revealed an acceptable correlation (r=0.48) and absolute agreement between the indices based on the two methods. The prospective relationship with mortality outcomes was studied in 3593 of the Rotterdam Study participants who were followed for 20 years. The DHD-index per 10 points increment was associated with a 9% (95% CI 0.87-0.96) risk reduction for all-cause mortality, and non-significantly associated with risk reductions for cardiovascular disease, coronary heart disease, and stroke mortality. Among women, shared dinners were associated with lower DHD-index scores for that day than solo dinners in 1740 participants who contributed multiple 24-hour recalls in the Nutrition Questionnaires plus study. Among men and women, dinners shared with family members were associated with a higher DHD-index score on that day than dinners shared with others. Furthermore, in a subsample of 1235 participants in the Nutrition Questionnaires plus study, we evaluated the DHD-index based on the newly developed 34-item DHD-FFQ, a short questionnaire to assess diet quality in time-limited settings. The DHD-index based on the DHD-FFQ showed an acceptable correlation (r=0.56) with the index based on a 180-item FFQ, but showed a large variation in bias at individual level. Conclusions: The DHD-index based on an FFQ, on multiple 24-hour recalls, or on the DHD-FFQ was considered a valid tool to rank participants according to their diet quality. The DHD-index was therefore considered useful to monitor populations, study diet–disease associations, and identify subpopulations at risk of poor diet quality.</p

Global properties of Stochastic Loewner evolution driven by Levy processes

Standard Schramm-Loewner evolution (SLE) is driven by a continuous Brownian motion which then produces a trace, a continuous fractal curve connecting the singular points of the motion. If jumps are added to the driving function, the trace branches. In a recent publication [1] we introduced a generalized SLE driven by a superposition of a Brownian motion and a fractal set of jumps (technically a stable L\'evy process). We then discussed the small-scale properties of the resulting L\'evy-SLE growth process. Here we discuss the same model, but focus on the global scaling behavior which ensues as time goes to infinity. This limiting behavior is independent of the Brownian forcing and depends upon only a single parameter, $\alpha$, which defines the shape of the stable L\'evy distribution. We learn about this behavior by studying a Fokker-Planck equation which gives the probability distribution for endpoints of the trace as a function of time. As in the short-time case previously studied, we observe that the properties of this growth process change qualitatively and singularly at $\alpha =1$. We show both analytically and numerically that the growth continues indefinitely in the vertical direction for $\alpha > 1$, goes as $\log t$ for $\alpha = 1$, and saturates for $\alpha< 1$. The probability density has two different scales corresponding to directions along and perpendicular to the boundary. In the former case, the characteristic scale is $X(t) \sim t^{1/\alpha}$. In the latter case the scale is $Y(t) \sim A + B t^{1-1/\alpha}$ for $\alpha \neq 1$, and $Y(t) \sim \ln t$ for $\alpha = 1$. Scaling functions for the probability density are given for various limiting cases.Comment: Published versio

On Littlewood"s constants

In two papers, Littlewood studied seemingly unrelated constants: (i) the best α such that for any polynomial f, of degree n, the areal integral of its spherical derivative is at most const ·nα, and (ii) the extremal growth rate β of the length of Green's equipotentials for simply connected domains. These two constants are shown to coincide, thus greatly improving known estimates on α. © 2005 London Mathematical Society