Nonvanishing univalent functions

Duren, Peter; Schober, Glenn

doi:10.1007/bf01214860

Cited by 116 publications

(166 citation statements)

References 20 publications

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“…Hence from (2.10) we find 12) exactly as obtained in §2.2 by different means. Such results are readily extended to the case of flow driven by many sources or sinks [40], or even through slits [42].…”

Section: The Schwarz Functionsupporting

confidence: 62%

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Two-dimensional Stokes and Hele-Shaw flows with free surfaces

Cummings¹,

Howison²,

King

1999

Eur. J. Appl. Math.

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We discuss the application of complex variable methods to Hele-Shaw flows and twodimensional Stokes flows, both with free boundaries. We outline the theory for the former, in the case where surface tension effects at the moving boundary are ignored. We review the application of complex variable methods to Stokes flows both with and without surface tension, and we explore the parallels between the two problems. We give a detailed discussion of conserved quantities for Stokes flows, and relate them to the Schwarz function of the moving boundary and to the Baiocchi transform of the Airy stress function. We compare the results with the corresponding results for Hele-Shaw flows, the principal consequence being that for Hele-Shaw flows the singularities of the Schwarz function are controlled in the physical plane, while for Stokes flow they are controlled in an auxiliary mapping plane. We illustrate the results with the explicit solutions to specific initial value problems. The results shed light on the construction of solutions to Stokes flows with more than one driving singularity, and on the closely related issue of momentum conservation, which is important in Stokes flows, although it does not arise in Hele-Shaw flows. We also discuss blow-up of zero-surface-tension Stokes flows, and consider a class of weak solutions, valid beyond blow-up, which are obtained as the zero-surface-tension limit of flows with positive surface tension.

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Section: The Schwarz Functionsupporting

confidence: 62%

“…where K( · ) denotes the Koebe map of univalent function theory [12]. It is defined by K(ζ) = ζ/(1+ζ) 2 , and maps |ζ| < 1 onto the whole complex plane, minus the semi-infinite line segment (−∞, −1/4].…”

Section: The Momentsmentioning

confidence: 99%

Two-dimensional Stokes and Hele-Shaw flows with free surfaces

Cummings¹,

Howison²,

King

1999

Eur. J. Appl. Math.

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“…We review some elementary facts about conformal (aka univalent) mappings, as may be found in [Dur83], for example. Let D ⊂ C be some domain.…”

Section: Conformal Mapsmentioning

confidence: 99%

Scaling limits of loop-erased random walks and uniform spanning trees

2000

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Abstract. The uniform spanning tree (UST) and the loop-erased random walk (LERW) are strongly related probabilistic processes. We consider the limits of these models on a fine grid in the plane, as the mesh goes to zero. Although the existence of scaling limits is still unproven, subsequential scaling limits can be defined in various ways, and do exist. We establish some basic a.s. properties of these subsequential scaling limits in the plane. It is proved that any LERW subsequential scaling limit is a simple path, and that the trunk of any UST subsequential scaling limit is a topological tree, which is dense in the plane.The scaling limits of these processes are conjectured to be conformally invariant in dimension 2. We make a precise statement of the conformal invariance conjecture for the LERW, and show that this conjecture implies an explicit construction of the scaling limit, as follows. Consider the Löwner differential equationwith boundary values f (z, 0) = z, in the range z ∈ U = {w ∈ C : |w| < 1}, t 0. We choose ζ(t) := B(−2t), where B(t) is Brownian motion on ∂U starting at a random-uniform point in ∂U. Assuming the conformal invariance of the LERW scaling limit in the plane, we prove that the scaling limit of LERW from 0 to ∂U has the same law as that of the path f (ζ(t), t) (where f (z, t) is extended continuously to ∂U × (−∞, 0]). We believe that a variation of this process gives the scaling limit of the boundary of macroscopic critical percolation clusters.1991 Mathematics Subject Classification. Primary 60D05. Secondary 60J15.

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“…introduces two essential geometric quantities: R 0 and a 0 , respectively, the uniquely defined (outer) conformal radius (Pólya & Szegö 1951) and the conformal centre (Pommerenke 1975;Duren 1983). The (external) conformal mapping provides a unique but not explicit definition of R 0 and a 0 , which can be interpreted as the characteristic size and location of the obstacle.…”

Section: (A) External Conformal Mappingmentioning

confidence: 99%

Hydrodynamic object recognition using pressure sensing

2010

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Hydrodynamic sensing is instrumental to fish and some amphibians. It also represents, for underwater vehicles, an alternative way of sensing the fluid environment when visual and acoustic sensing are limited. To assess the effectiveness of hydrodynamic sensing and gain insight into its capabilities and limitations, we investigated the forward and inverse problem of detection and identification, using the hydrodynamic pressure in the neighbourhood, of a stationary obstacle described using a general shape representation. Based on conformal mapping and a general normalization procedure, our obstacle representation accounts for all specific features of progressive perceptual hydrodynamic imaging reported experimentally. Size, location and shape are encoded separately. The shape representation rests upon an asymptotic series which embodies the progressive character of hydrodynamic imaging through pressure sensing. A dynamic filtering method is used to invert noisy nonlinear pressure signals for the shape parameters. The results highlight the dependence of the sensitivity of hydrodynamic sensing not only on the relative distance to the disturbance but also its bearing.

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Nonvanishing univalent functions

Cited by 116 publications

References 20 publications

Two-dimensional Stokes and Hele-Shaw flows with free surfaces

Two-dimensional Stokes and Hele-Shaw flows with free surfaces

Scaling limits of loop-erased random walks and uniform spanning trees

Hydrodynamic object recognition using pressure sensing

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