Algebraic Geometry Santa Cruz 1995, Part 1 by Kollar J., Lazarsfeld R., Morrison D. (eds.)

By Kollar J., Lazarsfeld R., Morrison D. (eds.)

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Mörters, D. Preiss, Tangent measure distributions of fractal measures. Math. Ann. 312(1), 53–93 (1998) 20. D. Preiss, Geometry of measures in Rn : distribution, rectifiability, and densities. Ann. Math. (2) 125(3), 537–643 (1987) 21. U. Zähle, Self-similar random measures and carrying dimension, in Proceedings of the Conference: Topology and Measure, V, Binz, 1987. Wissenschaftliche Beiträge der ErnstMoritz-Arndt Universität Greifswald (DDR) (Greifswald, 1988), pp. 84–87 The Shape of Anisotropic Fractals: Scaling of Minkowski Functionals Philipp Schönhöfer and Klaus Mecke Abstract The shape of fractals can be characterized by intrinsic volumes, socalled Minkowski functionals, which share with the common d-dimensional volume of spatial structures the property of being additive.

J. D. Howroyd, Projection theorems for box and packing dimensions. Math. Proc. Camb. Philos. Soc. 119, 287–295 (1996) 20. J. D. Howroyd, Packing dimensions of projections and dimension profiles. Math. Proc. Camb. Philos. Soc. 121, 269–286 (1997) 21. J. Falconer, X. Jin, Exact dimensionality and projections of random self-similar measures and sets. J. Lond. Math. Soc. (2) 90, 388–412 (2014) 22. J. Falconer, X. 1093/imrn/rnv103 (2015). 1882 23. J. Falconer, P. Mattila, The packing dimension of projections and sections of measures.

Here, we study the effects of anisotropy on the scaling behavior beyond the fractal dimension by applying tensorial functionals. It can be shown that Minkowski tensors of anisotropic prefractals scale with additional subdimensions. In addition, for anisotropic pre-fractals even scalar Minkowski functionals exhibit multiple edge subterms which merge for the isotropic case. Keywords Integral geometry • Fractals • Anisotropy • Minkowski functionals • DLA 1 Introduction The concept of fractal dimension is a standard method of characterizing complex structures and processes [14].

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