On the Minkowski Measurability of Fractals

Falconer, K. J.

doi:10.2307/2160708

Cited by 49 publications

(79 citation statements)

References 5 publications

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“…We remark that the s-sets defined with respect to the Hausdorff measure play an essential role in the theoretical aspect of the study of fractal sets in R 2 , see [3]. Furthermore, in the case of nondegeneracy of the Minkowski content, the s-sets defined in Definition 1.11 appear also as Minkowski measurability, which is an important concept in the study of measurable fractals and eigenfrequencies of fractal strings and fractal drums (Weyl-Berry conjecture), see [3,8], and [9].…”

Section: Introduction and Statement Of The Main Problemmentioning

confidence: 99%

“…Furthermore, in the case of nondegeneracy of the Minkowski content, the s-sets defined in Definition 1.11 appear also as Minkowski measurability, which is an important concept in the study of measurable fractals and eigenfrequencies of fractal strings and fractal drums (Weyl-Berry conjecture), see [3,8], and [9]. Minkowski measurability has been studied in the context of dynamical systems, see [21].…”

Section: Introduction and Statement Of The Main Problemmentioning

confidence: 99%

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Rectifiable oscillations in second-order linear differential equations

Kwong

Pašić

Wong

2008

Journal of Differential Equations

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We study the linear differential equation (P ): y (x) + f (x)y(x) = 0, on I = (0, 1), where the coefficient f (x) is strictly positive and continuous on I , and satisfies the Hartman-Wintner condition at x = 0. The four main results of the paper are: (i) a criterion for rectifiable oscillations of (P ), characterized by the integrability of 4 √ f (x) on I ; (ii) a stability result for rectifiable and unrectifiable oscillations of (P ), in terms of a perturbation on f (x); (iii) the s-dimensional fractal oscillations (for which we assume also f (x) ∼ cx −α when x → 0, α > 2, and s = max{1, 3/2 − 2/α}); and (iv) the co-existence of rectifiable and unrectifiable oscillations in the absence of the Hartman-Wintner condition on f (x). Explicit examples related to the above results are given.

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Section: Introduction and Statement Of The Main Problemmentioning

confidence: 99%

Section: Introduction and Statement Of The Main Problemmentioning

confidence: 99%

Rectifiable oscillations in second-order linear differential equations

Kwong

Pašić

Wong

2008

Journal of Differential Equations

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“…In [12,13], we proved the MWB conjecture in the case n = 1 and established in the process a connection with the Riemann zeta function. More recently, Falconer [4] gave a simplified proof of the result of [13] characterizing the Minkowski measurable sets when n = 1. This result is key to the proof in [13] of the MWB conjecture in dimension 1.…”

Section: Introductionmentioning

confidence: 99%

“…However, we show that though in both cases we have N(A) -<f>(A) as3 r mptotically a constant times A D/2 , the two constants are not the same. Our second family of examples, discussed in Sections 5 and 6, can be used to disprove both (1)(2) and (1)(2)(3)(4). Both families of examples are given in dimension n = 2, and so a simple Cartesian product construction can be used to get counterexamples in all higher dimensions.…”

Section: Introductionmentioning

confidence: 99%

Counterexamples to the modified Weyl–Berry conjecture on fractal drums

Lapidus¹,

Pomerance

1996

Math. Proc. Camb. Phil. Soc.

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Let Ω be a non-empty open set in ℝn with finite ‘volume’ (n-dimensional Lebesgue measure). Let be the Laplacian operator. Consider the eigenvalue problem (with Dirichlet boundary conditions):where λ ∈ ℝ and u is a non-zero member of (the closure in the Sobolev space H1(Ω) of the set of smooth functions with compact support contained in Ω). It is well known that the values of λ∈ℝ for which (1·1) has a non-zero solution are positive and form a discrete set. Moreover, for each λ, the associated eigenspace is finite dimensional. Let the spectrum of (1·1) be denoted where 0 < λ1 ≤ λ2 ≤ … and where the multiplicity of each λ in the sequence is the dimension of the associated eigenspace. Let

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“…This characterization is very useful from a computational point of view, and the name of box dimension follows from it. We refer the reader to [16] for the proof and other properties of the Minkowski dimension and content, see also [15] for some criteria about Minkowski measurability.…”

Section: And the Upper Minkowski Content Ismentioning

confidence: 99%

Eigenvalue distribution of second-order dynamic equations on time scales considered as fractals

Amster¹,

Nápoli²,

Pinasco³

2008

Journal of Mathematical Analysis and Applications

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Let T ⊂ [a, b] be a time scale with a, b ∈ T. In this paper we study the asymptotic distribution of eigenvalues of the following linear problem −u = λu σ , with mixed boundary conditions αu(a) + βu (a) = 0 = γ u(ρ(b)) + δu (ρ(b)). It is known that there exists a sequence of simple eigenvalues {λ k } k ; we consider the spectral counting function N(λ, T) = #{k: λ k λ}, and we seek for its asymptotic expansion as a power of λ. Let d be the Minkowski (or box) dimension of T, which gives the order of growth of the number K(T, ε) of intervals of length ε needed to cover T, namely K(T, ε) ≈ ε d . We prove an upper bound of N(λ) which involves the Minkowski dimension, N(λ, T) Cλ d/2 , where C is a positive constant depending only on the Minkowski content of T (roughly speaking, its d-volume, although the Minkowski content is not a measure). We also consider certain limiting cases (d = 0, infinite Minkowski content), and we show a family of self similar fractal sets where N(λ, T) admits two-side estimates.

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On the Minkowski Measurability of Fractals

Cited by 49 publications

References 5 publications

Rectifiable oscillations in second-order linear differential equations

Rectifiable oscillations in second-order linear differential equations

Counterexamples to the modified Weyl–Berry conjecture on fractal drums

Eigenvalue distribution of second-order dynamic equations on time scales considered as fractals

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