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Fractals and chaos
Published in A. W. Jayawardena, Environmental and Hydrological Systems Modelling, 2013
Mathematically, the packing dimension is another concept that can be used to define the dimension of a subset of a metric space. In some sense, it is the dual of the Hausdorff dimension since the packing dimension is constructed by packing small open balls inside the given subset, whereas the Hausdorff dimension is constructed by covering the given subset by such small open balls.
Hausdorff and packing dimensions of homogeneous product Moran sets
Published in Dynamical Systems, 2023
We prove Theorem 1.2 for the Hausdorff dimension case in Section 2 and packing dimension case in Section 3, respectively. Without loss of generality, we always assume that , . Throughout the following context, a rectangle always means a rectangle whose sides parallel to the coordinate axes. We always use to denote the cardinality of a set E, and for , write the maximal integer less than x, the minimal integer greater that x. We use convertion that for b<a throughout the paper. It is worthy to mention that the concept of packing dimension, introduced by Tricot [11], is a natural duality to Hausdorff (covering) dimension. See [4,5] for necessary background of the Hausdorff dimension and the packing dimension .
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.
A theoretical parametric analysis to optimize the bed depth of packed bed solar air collector
Published in International Journal of Green Energy, 2022
Vimal Kumar Chouksey, Anil Singh Yadav, Sheela Raha, Vipin Shrivastava, S. P. Shrivas
The value of product of absorptivity and extinction coefficient used in the work is 0.12. The packing dimension has following relation with packing volume and packed surface area