We use Ulam's method to provide rigorous approximation of diffusion coefficients for uniformly expanding maps. An algorithm is provided and its implementation is illustrated using Lanford's map.

We present a general setting in which the formula describing the linear response of the physical measure of a perturbed system can be obtained. In this general setting we obtain an algorithm to rigorously compute the linear response. We apply our results to expanding circle maps. In particular, we present examples where we compute, up to a pre-specified error in the L ∞norm, the response of expanding circle maps under stochastic and deterministic perturbations. Moreover, we present an example where we compute, up to a pre-specified error in the L 1 -norm, the response of the intermittent family at the boundary; i.e., when the unperturbed system is the doubling map.

Despite the well-known design principles of vascular systems, it is unclear whether the vascular arterial tree obeys some scaling constraints during normal growth and ageing in a given species. Based on the micro-computed tomography measurements of coronary arterial trees in mice at different ages (one week to more than eight months), we show a constant exponent of 3/4, but age-dependent scaling coefficients in a length-volume scaling law (L c ¼ K length-volume Á V 3=4 c ; L c is the crown length, V c is the crown volume, K length -volume is the age-dependent scaling coefficient) during normal growth and ageing. The constant 3/4 exponent represents the self-similar fractal-like branching pattern (i.e. basic mechanism to regulate the development of vascular trees within a species), whereas the agedependent scaling coefficients characterize the structural growth or resorption of vascular trees during normal growth or ageing, respectively. This study enhances the understanding of age-associated changes in vascular structure and function.

We fabricated a new sea urchin-like zirconium(iv) oxide which exhibits efficient phosphate sequestration with remarkable selectivity, a large capacity and recyclability .

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