Bijections of geodesic lamination space preserving left Hausdorff convergence

We construct a geodesic lamination on the hyperbolic disk and a dynamical system defined on this lamination. We prove that this dynamical system is a geometrical realization of the symbolic dynamical system that arises from the following Pisot substitution: $1\rightarrow 12, \dotsc, (n-1) \rightarrow 1n, n\rightarrow 1$.

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A MODEL FOR CRACK PROPAGATION BASED ON VISCOUS APPROXIMATION

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202511005647 ◽

2011 ◽

Vol 21 (10) ◽

pp. 2019-2047 ◽

Cited By ~ 30

Author(s):

GIULIANO LAZZARONI ◽

RODICA TOADER

Keyword(s):

Crack Propagation ◽

Energy Release Rate ◽

Energy Release ◽

Release Rate ◽

A Priori ◽

Brittle Materials ◽

Continuity Property ◽

Vanishing Viscosity ◽

Hausdorff Convergence ◽

Linearized Elasticity

In the setting of antiplane linearized elasticity, we show the existence of quasistatic evolutions of cracks in brittle materials by using a vanishing viscosity approach, thus taking into account local minimization. The main feature of our model is that the path followed by the crack need not be prescribed a priori: indeed, it is found as the limit (in the sense of Hausdorff convergence) of curves obtained by an incremental procedure. The result is based on a continuity property for the energy release rate in a suitable class of admissible cracks.

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The regularity series of a convergence space II

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700022590 ◽

1976 ◽

Vol 15 (2) ◽

pp. 223-243 ◽

Cited By ~ 6

Author(s):

D.C. Kent ◽

G.D. Richardson

Keyword(s):

Symmetric Space ◽

Convergence Space ◽

Hausdorff Convergence

This study is a continuation of an earlier paper on the regularity series of a convergence space. The notions of a R-Hausdorff series and the T3-modification of a convergence space are introduced, and their relationship with the regularity series is studied. The concept of a symmetric space is shown to be useful in studying T3-compactifications. Several examples are given; one being a Hausdorff convergence space with an arbitrarily large regularity series.

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A Denoised Version of Some Kinds of Set Convergence

Advanced Nonlinear Studies ◽

10.1515/ans-2005-0301 ◽

2005 ◽

Vol 5 (3) ◽

Author(s):

Filomena A. Lops

Keyword(s):

Hausdorff Measure ◽

Metric Space ◽

Lower Semicontinuity ◽

Set Convergence ◽

Minimizing Sequence ◽

Locally Compact ◽

Compact Metric Space ◽

Compactness Result ◽

Hausdorff Convergence ◽

Kuratowski Convergence

AbstractThe aim of this paper consists of introducing on a locally compact and σ-compact metric space a notion of set convergence, which generalizes the Hausdorff convergence, the local Hausdorff convergence and the Kuratowski convergence. We analyze the connections beetwen the three new notions: and. in particular, we prove a compactness result. As a first application of this convergence we give, on a sequence of sets, a condition which assures the lower semicontinuity of the Hausdorff measure with respect to this new convergence and we show that this condition is satisfied by any minimizing sequence of Mumford-Shah functional.

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