On the asymptotic magnitude of subsets of Euclidean space

Magnitude is a canonical invariant of finite metric spaces which has its origins in category theory; it is analogous to cardinality of finite sets. Here, by approximating certain compact subsets of Euclidean space with finite subsets, the magnitudes of line segments, circles and Cantor sets are defined and calculated. It is observed that asymptotically these satisfy the inclusion-exclusion principle, relating them to intrinsic volumes of polyconvex sets.Comment: 23 pages. Version 2: updated to reflect more recent work, in particular, the approximation method is now known to calculate (rather than merely define) the magnitude; also minor alterations such as references adde

Apparent Fractality Emerging from Models of Random Distributions

The fractal properties of models of randomly placed $n$-dimensional spheres ($n$=1,2,3) are studied using standard techniques for calculating fractal dimensions in empirical data (the box counting and Minkowski-sausage techniques). Using analytical and numerical calculations it is shown that in the regime of low volume fraction occupied by the spheres, apparent fractal behavior is observed for a range of scales between physically relevant cut-offs. The width of this range, typically spanning between one and two orders of magnitude, is in very good agreement with the typical range observed in experimental measurements of fractals. The dimensions are not universal and depend on density. These observations are applicable to spatial, temporal and spectral random structures. Polydispersivity in sphere radii and impenetrability of the spheres (resulting in short range correlations) are also introduced and are found to have little effect on the scaling properties. We thus propose that apparent fractal behavior observed experimentally over a limited range may often have its origin in underlying randomness.Comment: 19 pages, 12 figures. More info available at http://www.fh.huji.ac.il/~dani

Oxygen exposure during red wine fermentation modifies tannin reactivity with poly-L-proline

Red wines injected with nitrogen or oxygen during fermentation were used to identify the effect of gas exposure on tannin structure and reactivity with poly-l-proline. Tannin was purified from wine after fermentation and after three years of bottle storage. Tannin from nitrogen-treated wine had a lower percentage of galloylation and were less pigmented than tannin from oxygen-exposed wine. Self-aggregation of tannin was measured by nanoparticle tracking analysis and a larger particle size was observed for the oxidized treatment. The interaction of tannin and poly-l-proline was measured by isothermal titration calorimetry, and involved more hydrogen bonding than hydrophobic interactions in the case of nitrogen-treated wine tannin. Conversely, oxidized tannin was more hydrophobic and the association with poly-l-proline was entropy-driven due to a change of solvation. The results show meaningful changes in the structure and reactivity of tannin as a result of oxygen exposure during fermentation, which may impact astringency perception.Aude A. Watrelot, Martin P. Day, Alex Schulkin, Robert J. Falconer, Paul Smith, Andrew L. Waterhouse, Keren A. Bindo