Exact Hausdorff and Packing measure of certain Cantor sets, not necessarily self-similar or homogeneous

Zuberman, Leandro

doi:10.1016/j.jmaa.2019.01.036

Cited by 6 publications

(4 citation statements)

References 16 publications

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“…1.14-1.15]; for more recent related results, see e.g. [4] and [43]. For n > 1 , it appears that the exact value of the Hausdorff measure of even the simplest IFS (30)…”

Section: Proposition 33mentioning

confidence: 96%

“…Let 0 < h < diam (Γ) , and suppose that m ∉ Hull(Γ m � ) for each m � ∈ L h (Γ) such that m ′ 1 ≠ m . Then the quadrature rule defined by (43) for the integral (39) with = m satisfies the error estimate where Proof For 0 ≤ t < t m , we have from Theorem 4.3 and Eq. ( 43) that and the result follows by applying Proposition 4.2 to each term in the sum, and recalling (22) to see that…”

Section: Single Integralsmentioning

confidence: 99%

“…• The rule Q h Γ,t,m (43) for single integrals of the singular integrand Φ t ; • The rule Q h Γ,Γ,t (48) for double integrals of the singular integral Φ t ; • The singularity-subtraction rule Q h Γ,Γ,Φ (61) for double integrals of the singular Helmholtz fundamental solution Φ ; this rule reduces to (60) for small wavenumbers.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Numerical quadrature for singular integrals on fractals

Gibbs

Hewett²,

Moiola³

2022

Numer Algor

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We present and analyse numerical quadrature rules for evaluating regular and singular integrals on self-similar fractal sets. The integration domain $$\Gamma \subset \mathbb {R}^n$$ Γ ⊂ R n is assumed to be the compact attractor of an iterated function system of contracting similarities satisfying the open set condition. Integration is with respect to any “invariant” (also known as “balanced” or “self-similar”) measure supported on $$\Gamma$$ Γ , including in particular the Hausdorff measure $$\mathcal {H}^d$$ H d restricted to $$\Gamma$$ Γ , where d is the Hausdorff dimension of $$\Gamma$$ Γ . Both single and double integrals are considered. Our focus is on composite quadrature rules in which integrals over $$\Gamma$$ Γ are decomposed into sums of integrals over suitable partitions of $$\Gamma$$ Γ into self-similar subsets. For certain singular integrands of logarithmic or algebraic type, we show how in the context of such a partitioning the invariance property of the measure can be exploited to express the singular integral exactly in terms of regular integrals. For the evaluation of these regular integrals, we adopt a composite barycentre rule, which for sufficiently regular integrands exhibits second-order convergence with respect to the maximum diameter of the subsets. As an application we show how this approach, combined with a singularity-subtraction technique, can be used to accurately evaluate the singular double integrals that arise in Hausdorff-measure Galerkin boundary element methods for acoustic wave scattering by fractal screens.

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“…1.14-1.15]; for more recent related results, see e.g. [4] and [43]. For n > 1 , it appears that the exact value of the Hausdorff measure of even the simplest IFS (30)…”

Section: Proposition 33mentioning

confidence: 96%

Section: Single Integralsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Numerical quadrature for singular integrals on fractals

Gibbs

Hewett²,

Moiola³

2022

Numer Algor

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“…Domains nucleate from the surface or an interface and the nucleation centers repel each other. The periodicity and deviations from a periodic array stem from the mistakes in distribution of nucleation sites, which would often form Cantor sets leading to phenomena like period doubling and local periodicity disorder [64][65][66]. Finally, we mention that, in free crystals, periodicity of domain walls, period doubling, and disorder can be induced by the sideways movement of domains in terms of front propagation [55,60,67,68].…”

mentioning

confidence: 97%

A Ferroelastic Film at the Edge of Chaos

Lloveras

2019

Physics

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Very sensitive responses to external forces are found near phase transitions. However, transition dynamics and preequilibrium phenomena are difficult to detect and control. We have observed that the equilibrium domain structure following a phase transition in ferroelectric and ferroelastic BaTiO 3 is attained by halving of the domain periodicity multiple times. The process is reversible, with periodicity doubling as temperature is increased. This observation is reminiscent of the period-doubling cascades generally observed during bifurcation phenomena, and, thus, it conforms to the "spatial chaos" regime earlier proposed by Jensen and Bak [Phys. Scr. T 9, 64 (1985)] for systems with competing spatial modulations.

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A Hausdorff-measure boundary element method for acoustic scattering by fractal screens

Caetano,

Chandler-Wilde,

Gibbs

et al. 2024

Numer. Math.

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Sound-soft fractal screens can scatter acoustic waves even when they have zero surface measure. To solve such scattering problems we make what appears to be the first application of the boundary element method (BEM) where each BEM basis function is supported in a fractal set, and the integration involved in the formation of the BEM matrix is with respect to a non-integer order Hausdorff measure rather than the usual (Lebesgue) surface measure. Using recent results on function spaces on fractals, we prove convergence of the Galerkin formulation of this “Hausdorff BEM” for acoustic scattering in $$\mathbb {R}^{n+1}$$ R n + 1 ($$n=1,2$$ n = 1 , 2 ) when the scatterer, assumed to be a compact subset of $$\mathbb {R}^n\times \{0\}$$ R n × { 0 } , is a d-set for some $$d\in (n-1,n]$$ d ∈ ( n - 1 , n ] , so that, in particular, the scatterer has Hausdorff dimension d. For a class of fractals that are attractors of iterated function systems, we prove convergence rates for the Hausdorff BEM and superconvergence for smooth antilinear functionals, under certain natural regularity assumptions on the solution of the underlying boundary integral equation. We also propose numerical quadrature routines for the implementation of our Hausdorff BEM, along with a fully discrete convergence analysis, via numerical (Hausdorff measure) integration estimates and inverse estimates on fractals, estimating the discrete condition numbers. Finally, we show numerical experiments that support the sharpness of our theoretical results, and our solution regularity assumptions, including results for scattering in $$\mathbb {R}^2$$ R 2 by Cantor sets, and in $$\mathbb {R}^3$$ R 3 by Cantor dusts.

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Exact Hausdorff and Packing measure of certain Cantor sets, not necessarily self-similar or homogeneous

Abstract: We compute the exact Hausdorff and Packing measures of linear Cantor sets which might not be self-similar or homogeneous . The calculation is based on the local behavior of the natural probability measure supported on the sets.2000 Mathematics Subject Classification. 28A78,28A80.

Cited by 6 publications

References 16 publications

Numerical quadrature for singular integrals on fractals

Numerical quadrature for singular integrals on fractals

A Ferroelastic Film at the Edge of Chaos

A Hausdorff-measure boundary element method for acoustic scattering by fractal screens

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