Comparing the Hausdorff and packing measures of sets of small dimension in metric spaces [PDF]
Monatshefte für Mathematik, 2010 The author gives a new estimate for the ratio of \(s\)-dimensional Hausdorff measure \({\mathcal H}^s\) and (radius-based) packing measure \({\mathcal P}^s\) of a set in any metric space. The estimate is that the infimum of the ratio \(c(s,X)\) satisfies \(c(s,X) \geq 1+(2-\frac{3}{2^{1/s}})^s\), where ...
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Biomolecular Topology: Modelling and Analysis. [PDF]
Acta Math Sin Engl Ser, 2022 Liu J, Xia KL, Wu J, Yau SS, Wei GW.
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Complete positivity and distance-avoiding sets. [PDF]
Math Program, 2022 DeCorte E, Filho FMO, Vallentin F.
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On entropy and Hausdorff dimension of measures defined through a non-homogeneous Markov process
, 2003 In this work we study the Hausdorff dimension of measures whose weight distribution satisfies a markov non-homogeneous property. We prove, in particular, that the Hausdorff dimensions of this kind of measures coincide with their lower R\'enyi dimensions (
Batakis, Athanasios
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The fractal brain: scale-invariance in structure and dynamics. [PDF]
Cereb Cortex, 2023 Grosu GF +8 more
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A multiscale generative model to understand disorder in domain boundaries. [PDF]
Sci Adv, 2023 Dan J +6 more
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Exact Hausdorff and packing measures of linear Cantor sets with overlaps
, 2012 41 pages, 5 ...
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The density theorem and Hausdorff inequality for packing measure in general metric spaces
Illinois Journal of Mathematics, 1995 There are two basic mechanisms (diameter method and radius method) for extending the definition of Euclidean packing measure to general metric spaces. Although the diameter method has received the most attention in the literature, we show that it is generally inferior to the radius method in preserving the desirable properties of Euclidean packing ...
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Neural network-assisted automated image registration for MRI-guided adaptive brachytherapy in cervical cancer. [PDF]
Z Med Phys, 2022 Ecker S +8 more
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Hausdorff measures and packing premeasures
Hausdorff measures and packing premeasuresWe estimate the HausdorR measures and the packing premeasures of symmetric generalized Cantor sets in the d-dimensional Euclidean space R^d. Two simple estimations will be obtained. Let φ_1 and φ_2 be two measure functions. Suppose limt_(t→0) φ_2(t)/φ_1(t) = 0, limt_(t→0) φ_2(t)/t^d = ∞, and φ_1(t)/t^d is strictly decreasing as t increases. Then we can
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