Results 251 to 260 of about 144,663 (264)

The Hausdorf dimension of the Apollonian packing of circles

Journal of Physics A: Mathematical and General, 1994
Summary: We formulate the problem of determining the Hausdorff dimension, \(d_f\), of the Apollonian packing of circles as an eigenvalue problem of a linear integral equation. We show that solving a finite-dimensional approximation to this infinite-order matrix equation and extrapolating the results provides a fast algorithm for obtaining high ...
P. B. Thomas, Deepak Dhar
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The packing dimension of projections and sections of measures

Mathematical Proceedings of the Cambridge Philosophical Society, 1996
AbstractWe show that for a probability measure μ on ℝnfor almost all m–dimensional subspaces V, provided dimH μ≤m. Here projv denotes orthogonal projection onto V, and dimH and dimp denote the Hausdorff and packing dimension of a measure. In the case dimH μ > m we show that at μ-almost all points x the slices of μ by almost all (n − m)-planes Vx ...
Pertti Mattila, Kenneth J. Falconer
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Projection theorems for box and packing dimensions

Mathematical Proceedings of the Cambridge Philosophical Society, 1996
AbstractWe show that if E is an analytic subset of ℝn thenfor almost all m–dimensional subspaces V of ℝn, where projvE is the orthogonal projection of E onto V and dimp denotes packing dimension. The same inequality holds for lower and upper box counting dimensions, and these inequalities are the best possible ones.
Kenneth J. Falconer, J. D. Howroyd
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Hausdorff, Similarity, and Packing Dimensions

2020
In this chapter we consider three fractal dimensions of a geometric object \({\varOmega } \subset {\mathbb {R}}^E\):
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On Hausdorff and packing dimension of product spaces

Mathematical Proceedings of the Cambridge Philosophical Society, 1996
AbstractWe show that for arbitrary metric spaces X and Y the following dimension inequalities hold:where ‘dim’ denotes Hausdorff dimension and ‘Dim’ denotes packing dimension. The main idea of the proof is to use modified constructions of the Hausdorff and packing measure to deduce appropriate inequalities for the measure of X × Y.
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THE FRACTAL DIMENSION OF THE APOLLONIAN SPHERE PACKING

Fractals, 1994
The fractal dimension of the Apollonian sphere packing has been computed numerically up to six trusty decimal digits. Based on the 31 944 875 541 924 spheres of radius greater than 2−19 contained in the Apollonian packing of the unit sphere, we obtained an estimate of 2.4739465, where the last digit is questionable.
M. Borkovec, W. De Paris, Ronald Peikert
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Packing dimension, intersection measures, and isometries

Mathematical Proceedings of the Cambridge Philosophical Society, 1997
Let \(\mu\) and \(\nu\) be Radon probability measures on \(\mathbb{R}^n\). If \(f:\mathbb{R}^n\to \mathbb{R}^n\) is some isometry then \(f_\#\nu\) denotes the image measure of \(\nu\) with respect to \(f\). The intersection measure \(\mu\cap f_\#\nu\) of \(\mu\) and \(f_\#\nu\) is defined as follows: Let \(f=\tau_z\circ g\) be the representation of \(f\
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A relationship between packing and topological dimensions

Mathematika, 1998
The relationship between the topological dimension of a separable metric space and the Hausdorff dimensions of its homeomorphic images has been known for some time. In this note we consider topological and packing dimensions, and show that if \(X\) is a separable metric space, then \[ \dim_{\mathcal T}(X)= \min\{\dim_{\mathcal P}(X'): X'\text{ is ...
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The residual set dimension of the Apollonian packing

Mathematika, 1973
In this paper we show that, for the Apollonian or osculatory packing C0 of a curvilinear triangle T, the dimension d(C0, T) of the residual set is equal to the exponent of the packing e(Co, T) = S. Since we have [5, 6] exhibited constructible sequences λ(K) and μ(K) such that λ(K) < S < μ(K), and μ(K)–λ(K) → 0 as κ → 0, we have thus effectively ...
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Concerning the packing dimension of intersection measures

Mathematical Proceedings of the Cambridge Philosophical Society, 1997
Summary: For a Radon probability measure \(\mu\) on \(\mathbb{R}^n\) we can use the Hausdorff dimension \(\dim_{\text{H}}\) and the packing dimension \(\dim_{\text{p}}\) to define lower indices \[ \dim_{\text{H}}\mu= \inf\{\dim_{\text{H}}A:A\text{ is a Borel set and }\mu(A)>0\} \] and \[ \dim_{\text{p}}\mu= \inf\{\dim_{\text{p}}A:A\text{ is a Borel set
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