Entire functions with Julia sets of positive measure

Let f be a transcendental entire function for which the set of critical and asymptotic values is bounded. The Denjoy-Carleman-Ahlfors theorem implies that if the set of all z for which |f(z)|>R has N components for some R>0, then the order of f is at least N/2. More precisely, we have log log M(r,f) > (N/2) log r - O(1), where M(r,f) denotes the maximum modulus of f. We show that if f does not grow much faster than this, then the escaping set and the Julia set of f have positive Lebesgue measure. However, as soon as the order of f exceeds N/2, this need not be true. The proof requires a sharpened form of an estimate of Tsuji related to the Denjoy-Carleman-Ahlfors theorem.Comment: 17 page

On the dimension of graphs of Weierstrass-type functions with rapidly growing frequencies

We determine the Hausdorff and box dimension of the fractal graphs for a general class of Weierstrass-type functions of the form $f(x) = \sum_{n=1}^\infty a_n \, g(b_n x + \theta_n)$, where $g$ is a periodic Lipschitz real function and $a_{n+1}/a_n \to 0$, $b_{n+1}/b_n \to \infty$ as $n \to \infty$. Moreover, for any $H, B \in [1, 2]$, $H \leq B$ we provide examples of such functions with \dim_H(\graph f) = \underline{\dim}_B(\graph f) = H, \bar{\dim}_B(\graph f) = B.Comment: 18 page

Escape rate and Hausdorff measure for entire functions

The escaping set of an entire function is the set of points that tend to infinity under iteration. We consider subsets of the escaping set defined in terms of escape rates and obtain upper and lower bounds for the Hausdorff measure of these sets with respect to certain gauge functions.Comment: 24 pages; some errors corrected, proof of Theorem 2 shortene

Higher antioxidant and lower cadmium concentrations and lower incidence of pesticide residues in organically grown crops: a systematic literature review and meta-analyses

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Boundaries of univalent Baker domains

Let $f$ be a transcendental entire function and let $U$ be a univalent Baker domain of $f$. We prove a new result about the boundary behaviour of conformal maps and use this to show that the non-escaping boundary points of $U$ form a set of harmonic measure zero with respect to $U$. This leads to a new sufficient condition for the escaping set of $f$ to be connected, and also a new general result on Eremenko's conjecture

The green halo: Mechanisms and limits of the eco-label effect

Consumers believe that “eco-labeled” products taste better, which, at least in part, may be an effect of the label. The purpose of the current series of experiments was to examine some mechanisms and limits of this eco-label effect. In Experiment 1, an eco-label effect of similar magnitude was found for taste ratings of both conventional and organic bananas. Experiment 2 showed eco-label effects for a wider range of judgmental dimensions (i.e., health, calories, vitamins/minerals, mental performance, and willingness to pay) and the effect was about the same in magnitude for judgments of grapes and raisins. Experiment 3, with water as the tasted product, found no eco-label effect on judgments of taste, calories and vitamins/minerals, but an effect on willingness to pay, judgments of health benefits and judgments of mental performance benefits. Experiments 2 and 3 also included questionnaires on social desirability traits, schizotypal traits and pro-environmental consumer traits. The last was the strongest predictor of the eco-label effect amongst the three. In all, the eco-label effect is a robust phenomenon, but depends on interactions between product type and judgmental dimension. Implications for several accounts of the effect are discussed

Higher antioxidant and lower cadmium concentrations and lower incidence of pesticide residues in organically grown crops: a systematic literature review and meta-analyses

Demand for organic foods is partially driven by consumers' perceptions that they are more nutritious. However, scientific opinion is divided on whether there are significant nutritional differences between organic and non-organic foods, and two recent reviews have concluded that there are no differences. In the present study, we carried out meta-analyses based on 343 peer-reviewed publications that indicate statistically significant and meaningful differences in composition between organic and non-organic crops/crop-based foods. Most importantly, the concentrations of a range of antioxidants such as polyphenolics were found to be substantially higher in organic crops/crop-based foods, with those of phenolic acids, flavanones, stilbenes, flavones, flavonols and anthocyanins being an estimated 19 (95% CI 5, 33)%, 69 (95% CI 13, 125)%, 28 (95% CI 12, 44)%, 26 (95% CI 3, 48)%, 50 (95% CI 28, 72)% and 51 (95% CI 17, 86)% higher, respectively. Many of these compounds have previously been linked to a reduced risk of chronic diseases, including CVD and neurodegenerative diseases and certain cancers, in dietary intervention and epidemiological studies. Additionally, the frequency of occurrence of pesticide residues was found to be four times higher in conventional crops, which also contained significantly higher concentrations of the toxic metal Cd. Significant differences were also detected for some other (e.g. minerals and vitamins) compounds. There is evidence that higher antioxidant concentrations and lower Cd concentrations are linked to specific agronomic practices (e.g. non-use of mineral N and P fertilisers, respectively) prescribed in organic farming systems. In conclusion, organic crops, on average, have higher concentrations of antioxidants, lower concentrations of Cd and a lower incidence of pesticide residues than the non-organic comparators across regions and production season

Designing sustainable landuse in a 1.5 °C world: the complexities of projecting multiple ecosystem services from land

Land provides a range of critical services for humanity (including the provision of food, water and energy). It also provides many services that are often socially valuable but may not have a market value. Demand projections for land-based services, accounting for the significant requirement for negative emissions needed to meet a 1.5 °C pathway, may exceed what can be sustainably supplied. It is therefore critical to explore how to optimise land use (and if necessary, limit demand), so societies can continue to benefit from all services into the future. Unlike the energy or the transport sectors, however, there is limited understanding or consensus over what ‘optimal’ land use might look like (from a science perspective), or how to bring it about (from a governance perspective)