Existence and Properties of ℎ-Sets

2002 ◽  
Vol 9 (1) ◽  
pp. 13-32
Author(s):  
Michele Bricchi
Keyword(s):  
Compact Support ◽  
Regular Measure ◽  
Self Similar ◽  
Suitable Measure

Abstract In this note we shall consider the following problem: which conditions should satisfy a function ℎ : (0, 1) → ℝ in order to guarantee the existence of a (regular) measure μ in with compact support and for some positive constants 𝑐2, and 𝑐2 independent of γ ∈ Γ and 𝑟 ∈ (0,1)? The theory of self-similar fractals provides outstanding examples of sets fulfilling (♡) with ℎ(𝑟) = 𝑟𝑑, 0 ≤ 𝑑 ≤ 𝑛, and a suitable measure μ. Analogously, we shall rely on some recent techniques for the construction of pseudo self-similar fractals in order to deal with our more general task.

In-homogenous self-similar measures and their Fourier transforms

2008 ◽  
Vol 144 (2) ◽  
pp. 465-493 ◽  
Author(s):  
L. OLSEN ◽  
N. SNIGIREVA
Keyword(s):  
Probability Measure ◽  
Asymptotic Behaviour ◽  
Compact Support ◽  
Fourier Transforms ◽  
Probability Vector ◽  
Similar Measure ◽  
Special Cases ◽  
Non Linear ◽  
Self Similar ◽  
Image Position

AbstractLetSj: ℝd→ ℝdforj= 1, . . .,Nbe contracting similarities. Also, let (p1,. . .,pN,p) be a probability vector and let ν be a probability measure on ℝdwith compact support. We show that there exists a unique probability measure μ such thatThe measure μ is called an in-homogenous self-similar measure. In this paper we study the asymptotic behaviour of the Fourier transforms of in-homogenous self-similar measures. Finally, we present a number of applications of our results. In particular, non-linear self-similar measures introduced and investigated by Glickenstein and Strichartz are special cases of in-homogenous self-similar measures, and as an application of our main results we obtain simple proofs of generalizations of Glickenstein and Strichartz's results on the asymptotic behaviour of the Fourier transforms of non-linear self-similar measures.

scholarly journals Weyl asymptotics for Hanoi attractors

2017 ◽  
Vol 29 (5) ◽  
pp. 1003-1021 ◽  
Author(s):  
Patricia Alonso Ruiz ◽  
Uta R. Freiberg
Keyword(s):  
Asymptotic Behavior ◽  
Dirichlet Form ◽  
Counting Function ◽  
Quantum Graph ◽  
Weyl Asymptotics ◽  
Self Similar ◽  
Resistance Form ◽  
Classical Construction ◽  
Suitable Measure

AbstractThis paper studies the asymptotic behavior of the eigenvalue counting function of the Laplacian on some weakly self-similar fractals called Hanoi attractors. A resistance form is constructed and equipped with a suitable measure in order to obtain a Dirichlet form and its associated Laplacian. Hereby, the classical construction for p.c.f. self-similar fractals has to be modified by combining discrete and quantum graph methods.

scholarly journals Self-similar solutions of the p-Laplace heat equation: the case when p > 2

2009 ◽  
Vol 139 (1) ◽  
pp. 1-43 ◽  
Author(s):  
Marie Françoise Bidaut-Véron
Keyword(s):  
Heat Equation ◽  
Compact Support ◽  
The Self ◽  
Uniqueness Of Solutions ◽  
Existence Of Positive Solutions ◽  
Complete Study ◽  
Particular Solution ◽  
Self Similar ◽  
Image Position ◽  
Similar Solutions

We study the self-similar solutions of the equationin ℝN, when p > 2. We make a complete study of the existence and possible uniqueness of solutions of the formof any sign, regular or singular at x = 0. Among them we find solutions with an expanding compact support or a shrinking hole (for t > 0), or a spreading compact support or a focusing hole (for t < 0). When t < 0, we show the existence of positive solutions oscillating around the particular solution $\smash{U(x,t)=C_{N,p}(|x|^{p}/(-t))^{1/(p-2)}}$.

Self-similar collapse with non-radial motions

2006 ◽  
Vol 20 ◽  
pp. 1-4
Author(s):  
A. Nusser
Keyword(s):  
Self Similar

Gluing Solutions

2017 ◽  
Author(s):  
Philip Isett
Keyword(s):  
Stress Tensor ◽  
Compact Support ◽  
Smooth Solution ◽  
Continuous Solution ◽  
Total Momentum ◽  
Frame Of Reference ◽  
Galilean Transformation ◽  
Hölder Continuous ◽  
Pressure Form ◽  
Original Frame

This chapter deals with the gluing of solutions and the relevant theorem (Theorem 12.1), which states the condition for a Hölder continuous solution to exist. By taking a Galilean transformation if necessary, the solution can be assumed to have zero total momentum. The cut off velocity and pressure form a smooth solution to the Euler-Reynolds equations with compact support when coupled to a smooth stress tensor. The proof of Theorem (12.1) proceeds by iterating Lemma (10.1) just as in the proof of Theorem (10.1). Applying another Galilean transformation to return to the original frame of reference, the theorem is obtained.

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