NONDECREASABLE AND WEAKLY NONDECREASABLE DILATATIONS

2016 ◽  
Vol 102 (3) ◽  
pp. 420-434
Author(s):  
GUOWU YAO
Keyword(s):  
Quasiconformal Mapping ◽  
Equivalence Class ◽  
Infinite Number ◽  
Negative Answer ◽  
Quasiconformal Mappings ◽  
Extremal Quasiconformal Mapping

Zhou et al. [‘On weakly non-decreasable quasiconformal mappings’, J. Math. Anal. Appl.386 (2012), 842–847] proved that, in a Teichmüller equivalence class, there exists an extremal quasiconformal mapping with a weakly nondecreasable dilatation. They asked whether a weakly nondecreasable dilatation is a nondecreasable dilatation. The aim of this paper is to give a negative answer to their problem. We also construct a Teichmüller class such that it contains an infinite number of weakly nondecreasable extremal representatives, only one of which is nondecreasable.

scholarly journals Application of quasiconformal mapping methods to the investigation of the explosive processes impact on the environment

2019 ◽  
pp. 17-18
Author(s):  
Kateryna Mykolaiivna Malash ◽  
Andrii Yaroslavovych Bomba
Keyword(s):  
Numerical Methods ◽  
Mathematical Models ◽  
Quasiconformal Mapping ◽  
Quasiconformal Mappings ◽  
Practical Application

The mathematical models used to study explosive processes are given. A class of problems investigating the influence of explosive processes on the environment by the quasiconformal mappings numerical methods are outlined and their practical application are described

scholarly journals On a Theorem of Schwarz Type for Quasiconformal Mappings in Space

1967 ◽  
Vol 29 ◽  
pp. 19-30
Author(s):  
Kazuo Ikoma
Keyword(s):  
Conformal Mapping ◽  
Quasiconformal Mapping ◽  
Quasiconformal Mappings ◽  
Space Domain ◽  
Moebius Transformation ◽  
Modulus Condition

A space ring R is defined as a domain whose complement in the Moebius space consists of two components. The modulus of R can be defined in variously different but essentially equivalent ways (see e.g. Gehring [3] and Krivov [5]), which is denoted by mod R. Following Gehring [2], we refer to a homeomorphism y(x) of a space domain D as a k-quasiconformal mapping, if the modulus conditionis satisfied for all bounded rings R with their closure , where y(R) denotes the image of R by y = y(x). Then, it is evident that the inverse of a k-quasi-conformal mapping is itself k-quasiconformal and that a k1-quasiconformal mapping followed by a k2-quasiconformal one is k1k2-quasiconformal. It is also well known that the restriction of a Moebius transformation to a space domain is equivalent to a 1-quasiconformal mapping of its domain.

scholarly journals ON LENGTH DISTORTIONS WITH RESPECT TO QUADRATIC DIFFERENTIAL METRICS

2014 ◽  
Vol 56 (3) ◽  
pp. 681-689
Author(s):  
ZONGLIANG SUN
Keyword(s):  
Quasiconformal Mapping ◽  
Homotopy Class ◽  
Quadratic Differential ◽  
Riemann Surfaces ◽  
Simple Closed Curve ◽  
Quasiconformal Mappings ◽  
Closed Curve

AbstractIn this paper, we consider the question about length distortions under quasiconformal mappings with respect to quadratic differential metrics. More precisely, let X and Y be closed Riemann surfaces with genus at least 2, and f: X → Y being a K-quasiconformal mapping. Given two quadratic differential metrics |q1| and |q2| with unit areas on X and Y respectively, whether there exists a constant $\mathcal C$ depending only on K such that $\frac{1}{\mathcal C} l_{q_1} (\gamma) \leq l_{q_2} (f(\gamma)) \leq \mathcal C l_{q_1} (\gamma)$ holds for any simple closed curve γ ⊂ X. Here lqi(α) denotes the infimum of the lengths of curves in the homotopy class of α with respect to the metric |qi|, i = 1, 2. We give positive answers to this question, including the aspects that the desired constant ${\mathcal C}$ explicitly depends on q1, q2 and K, and that the constant $\mathcal C$ is universal for all the quantities involved.

POLYGONAL QUASICONFORMAL MAPPINGS AND CHORD-ARC CURVES

2016 ◽  
Vol 95 (1) ◽  
pp. 66-72 ◽  
Author(s):  
SHENGJIN HUO ◽  
SHENGJIAN WU ◽  
HUI GUO
Keyword(s):  
Quasiconformal Mapping ◽  
Quasiconformal Mappings ◽  
Path Connected

In this paper we show that a polygonal quasiconformal mapping always corresponds to a chord-arc curve. Furthermore, we find that the set of curves corresponding to polygonal quasiconformal mappings is path connected in the set of all bounded chord-arc curves.

Composition operators on W 1 X are necessarily induced by quasiconformal mappings

2014 ◽  
Vol 12 (8) ◽  
Author(s):  
Luděk Kleprlík
Keyword(s):  
Quasiconformal Mapping ◽  
Function Space ◽  
Composition Operator ◽  
Composition Operators ◽  
Invariant Function ◽  
Quasiconformal Mappings ◽  
Open Set ◽  
Scaling Property

AbstractLet Ω ⊂ ℝn be an open set and X(Ω) be any rearrangement invariant function space close to L q(Ω), i.e. X has the q-scaling property. We prove that each homeomorphism f which induces the composition operator u ↦ u ℴ f from W 1 X to W 1 X is necessarily a q-quasiconformal mapping. We also give some new results for the sufficiency of this condition for the composition operator.

scholarly journals Stretching and Rotation of Planar Quasiconformal Mappings on a Line

2021 ◽  
Author(s):  
Olli Hirviniemi ◽  
István Prause ◽  
Eero Saksman
Keyword(s):  
Lower Bound ◽  
Hausdorff Dimension ◽  
Quasiconformal Mapping ◽  
Quasiconformal Mappings ◽  
Optimal Estimate ◽  
Holomorphic Motions

AbstractIn this article, we examine stretching and rotation of planar quasiconformal mappings on a line. We show that for almost every point on the line, the set of complex stretching exponents (describing stretching and rotation jointly) is contained in the disk $ \overline {B}(1/(1-k^{4}),k^{2}/(1-k^{4}))$ B ¯ ( 1 / ( 1 − k 4 ) , k 2 / ( 1 − k 4 ) ) . This yields a quadratic improvement over the known optimal estimate for general sets of Hausdorff dimension 1. Our proof is based on holomorphic motions and estimates for dimensions of quasicircles. We also give a lower bound for the dimension of the image of a 1-dimensional subset of a line under a quasiconformal mapping.

scholarly journals Unique extremality, local extremality and extremal non-decreasable dilatations

2007 ◽  
Vol 75 (3) ◽  
pp. 321-329 ◽  
Author(s):  
Guowu Yao
Keyword(s):  
Quasiconformal Mapping ◽  
Unit Circle ◽  
Quasiconformal Mappings ◽  
Boundary Correspondence

Given a quasi-symmetric self-homeomorphism h of the unit circle Sl, let Q(h) be the set of all quasiconformal mappings with the boundary correspondence h. In this paper, it is shown that there exists certain quasi-symmetric homeomorphism h, such that Q(h) satisfies either of the conditions,(1) Q(h) admits a quasiconformal mapping that is both uniquely locally-extremal and uniquely extremal-non-decreasable instead of being uniquely extremal;(2) Q(h) contains infinitely many quasiconformal mappings each of which has an extremal non-decreasable dilatation.An infinitesimal version of this result is also obtained.

scholarly journals Space quasiconformal mappings and Neumann eigenvalues in fractal type domains

2018 ◽  
Vol 25 (2) ◽  
pp. 221-233 ◽  
Author(s):  
Vladimir Gol’dshtein ◽  
Ritva Hurri-Syrjänen ◽  
Alexander Ukhlov
Keyword(s):  
Quasiconformal Mapping ◽  
Laplace Operator ◽  
Composition Operators ◽  
Geometric Theory ◽  
Quasiconformal Mappings ◽  
Neumann Eigenvalues ◽  
Mapping Theory

Abstract We study the variation of Neumann eigenvalues of the p-Laplace operator under quasiconformal perturbations of space domains. This study allows us to obtain the lower estimates of Neumann eigenvalues in fractal type domains. The proposed approach is based on the geometric theory of composition operators in connection with the quasiconformal mapping theory.

scholarly journals On A Property of the Boundary Correspondence under Quasiconformal Mappings

1960 ◽  
Vol 16 ◽  
pp. 185-188 ◽  
Author(s):  
Kazuo Ikoma
Keyword(s):  
Quasiconformal Mapping ◽  
Logarithmic Capacity ◽  
Quasiconformal Mappings ◽  
Maximal Dilatation ◽  
Capacity Zero ◽  
Boundary Correspondence

Let w = f(z) be a quasiconformal mapping, in the sense of Pfluger [5]-Ahlfors [1], with maximal dilatation K, which will be simply referred to a K-QC mapping. As is well known, any K-QC mapping w = f(z) of Im z > 0 onto Im w > 0 can be extended to a homeomorphism from Im z ≧ 0 onto Im w ≧ 0 and hence it transforms any set of logarithmic capacity zero on Im z = 0 into a set with the same property on Im w = 0.

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