The Canopy and Shortest Path in a Self-Contacting Fractal Tree

Mandelbrot, Benoît B.; Frame, Michael

doi:10.1007/bf03024842

Cited by 27 publications

(20 citation statements)

References 3 publications

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“…, σ n ∈ {1, 2} is the set of the multiple points of Γ. Two situations can occur, depending on the angle θ (see [30]):…”

Section: Hausdorff Dimension Of γmentioning

confidence: 99%

A Transmission Problem Across a Fractal Self-Similar Interface

Achdou¹,

Deheuvels²

2016

Multiscale Model. Simul.

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We consider a transmission problem in which the interior domain has infinitely ramified structures. Transmission between the interior and exterior domains occurs only at the fractal component of the interface between the interior and exterior domains. We also consider the sequence of the transmission problems in which the interior domain is obtained by stopping the self-similar construction after a finite number of steps; the transmission condition is then posed on a prefractal approximation of the fractal interface. We prove the convergence in the sense of Mosco of the energy forms associated with these problems to the energy form of the limit problem. In particular, this implies the convergence of the solutions of the approximated problems to the solution of the problem with fractal interface. The proof relies in particular on an extension property. Emphasis is put on the geometry of the ramified domain. The convergence result is obtained when the fractal interface has no self-contact, and in a particular geometry with self-contacts, for which an extension result is proved.

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“…, σ n ∈ {1, 2} is the set of the multiple points of Γ. Two situations can occur, depending on the angle θ (see [30]):…”

Section: Hausdorff Dimension Of γmentioning

confidence: 99%

A Transmission Problem Across a Fractal Self-Similar Interface

Achdou¹,

Deheuvels²

2016

Multiscale Model. Simul.

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“…Note that Ω is symmetric with respect to the axis x 1 = 0. In [45] it was proved that, for any 0 < θ < π/2, there exists a unique positive number a * < 1/ √ 2, which does not depend on (α, β) , such that for 0 < a < a * , Γ f has "no self-contact". In this case the Hausdorff dimension of Γ f is d = − ln (2) / ln (a) > 1 and by [5], Ω possesses the W 1,2 -extension property of Sobolev functions.…”

Section: Preliminariesmentioning

confidence: 99%

Long-term behavior of reaction–diffusion equations with nonlocal boundary conditions on rough domains

Gal¹,

Warma²

2016

Z. Angew. Math. Phys.

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We investigate the long term behavior in terms of finite dimensional global and exponential attractors, as time goes to infinity, of solutions to a semilinear reaction-diffusion equation on non-smooth domains subject to nonlocal Robin boundary conditions, characterized by the presence of fractional diffusion on the boundary. Our results are of general character and apply to a large class of irregular domains, including domains whose boundary is Hölder continuous and domains which have fractal-like geometry. In addition to recovering most of the existing results on existence, regularity, uniqueness, stability, attractor existence, and dimension, for the well-known reactiondiffusion equation in smooth domains, the framework we develop also makes possible a number of new results for all diffusion models in other non-smooth settings.2010 Mathematics Subject Classification. 35J92, 35A15, 35B41,35K65.

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“…In general, fractal trees are compact connected subsets of R n (for some n ≥ 0) that exhibit some kind of branching pattern at arbitrary levels. The class of symmetric binary fractal trees were more recently studied by Mandelbrot and Frame [10] and the author [15], [16]. A symmetric binary fractal tree T (r, θ) is defined by two parameters, the scaling ratio r (a real number between 0 and 1) and the branching angle θ (an angle between 0 • and 180 • ).…”

Section: 2mentioning

confidence: 99%

“…A self-contacting tree has self-intersection but no actual branch crossings (see Figure 9). For a given branching angle θ, there is a unique scaling ratio r sc (θ) (or just r sc ) such that the corresponding tree is self-contacting [10]. The values of r sc as a function of θ have been completely determined [10].…”

Section: 2mentioning

confidence: 99%

Topological bar-codes of fractals: a new characterization of symmetric binary fractal trees

Taylor

2009

Convex and Fractal Geometry

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The goal of this paper is to provide foundations for a new way to classify and characterize fractals using methods of computational topology. The fractal dimension is a main characteristic of fractal-like objects, and has proved to be a very useful tool for applications. However, it does not fully characterize a fractal. We can obtain fractals with the same dimension that are quite different topologically. Motivated by techniques from shape theory and computational topology, we consider fractals along with their-hulls as ranges over the non-negative real numbers. In particular, we develop theory for the class of non-overlapping symmetric binary fractal trees that can be generalized to broader classes of fractals. We investigate various features of the-hulls of the trees, based on the holes in these hulls. We determine the hole sequence of these trees together with the persistence intervals of the holes as the 'topological bar-codes' of these fractals. We provide quantitative results for a selection of specific trees to illustrate the theory. Finally, we prove that for non-overlapping symmetric binary fractal trees, the growth rate of holes in-hulls is equal to the similarity dimension.

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The Canopy and Shortest Path in a Self-Contacting Fractal Tree

Cited by 27 publications

References 3 publications

A Transmission Problem Across a Fractal Self-Similar Interface

A Transmission Problem Across a Fractal Self-Similar Interface

Long-term behavior of reaction–diffusion equations with nonlocal boundary conditions on rough domains

Topological bar-codes of fractals: a new characterization of symmetric binary fractal trees

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