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Measures of Noncompactness
Published in Eberhard Malkowsky, Vladimir Rakočević, Advanced Functional Analysis, 2019
Eberhard Malkowsky, Vladimir Rakočević
Now we establish a relation between the Hausdorff measure of noncompactness and the Hausdorff distance. In particular, (7.42) shows that if (X, d) is a complete metric space, then the map χ:(MX,dH)→R is uniformly continuous.
General properties of fractals
Published in Xie Heping, Fractals in Rock Mechanics, 2020
Generally, one calls the Hausdorff-Besicovitch dimension fractal dimension. Hence, the measure of fractal sets should be the Hausdorff measure. For this reason, we can see that a fractal set (or fractal) must be required to satisfy two conditions: the first condition is that the measure of the set must be the Hausdorff measure, and the second one is D > DT.
Dimensions in infinite iterated function systems consisting of bi-Lipschitz mappings
Published in Dynamical Systems, 2020
For any , we use , and to denote, respectively, the Hausdorff dimension, packing dimension, box dimension, s-dimensional Hausdorff measure, d-dimensional Lebesgue measure, Euclidean diameter, and interior of E. For any set A, we let denote its cardinality. A set is said to be open if it is open in the relative Euclidean topology of X.
Global existence for a nonlocal model for adhesive contact
Published in Applicable Analysis, 2018
Elena Bonetti, Giovanna Bonfanti, Riccarda Rossi
Combining the previous constitutive relations with the balance laws, we obtain the following boundary value problem where all integrals on involve the Lebesgue measure, which coincides with the Hausdorff measure on by the flatness requirement, cf. (2.1) ahead.
Disintegration property of coherent upper conditional previsions with respect to Hausdorff outer measures for unbounded random variables
Published in International Journal of General Systems, 2021
Let Ω be a set with positive and finite Hausdorff measure in its Hausdorff dimension t. Let be a partition of Ω and let be the subclass of of sets B with positive upper coherent probability , thus is a non-null partition if .