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 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 Badly approximable points on self-affine sponges and the lower Assouad dimension

2017 ◽  
Vol 39 (3) ◽  
pp. 638-657 ◽  
Author(s):  
TUSHAR DAS ◽  
LIOR FISHMAN ◽  
DAVID SIMMONS ◽  
MARIUSZ URBAŃSKI
Keyword(s):  
Lower Bound ◽  
Hausdorff Dimension ◽  
Diophantine Approximation ◽  
Assouad Dimension ◽  
Badly Approximable Points ◽  
Bounded Below ◽  
Badly Approximable ◽  
Sierpinski Sponges

We highlight a connection between Diophantine approximation and the lower Assouad dimension by using information about the latter to show that the Hausdorff dimension of the set of badly approximable points that lie in certain non-conformal fractals, known as self-affine sponges, is bounded below by the dynamical dimension of these fractals. For self-affine sponges with equal Hausdorff and dynamical dimensions, the set of badly approximable points has full Hausdorff dimension in the sponge. Our results, which are the first to advance beyond the conformal setting, encompass both the case of Sierpiński sponges/carpets (also known as Bedford–McMullen sponges/carpets) and the case of Barański carpets. We use the fact that the lower Assouad dimension of a hyperplane diffuse set constitutes a lower bound for the Hausdorff dimension of the set of badly approximable points in that set.

scholarly journals On the Baire Generic Validity of the t-Multifractal Formalism in Besov and Sobolev Spaces

2019 ◽  
Vol 2019 ◽  
pp. 1-14
Author(s):  
Moez Ben Abid ◽  
Mourad Ben Slimane ◽  
Ines Ben Omrane ◽  
Borhen Halouani
Keyword(s):  
Lower Bound ◽  
Hausdorff Dimension ◽  
Besov Space ◽  
Sobolev Spaces ◽  
Wavelet Coefficients ◽  
Multifractal Formalism ◽  
Spectrum Of A Function ◽  
Set Of Points

The t-multifractal formalism is a formula introduced by Jaffard and Mélot in order to deduce the t-spectrum of a function f from the knowledge of the (p,t)-oscillation exponent of f. The t-spectrum is the Hausdorff dimension of the set of points where f has a given value of pointwise Lt regularity. The (p,t)-oscillation exponent is measured by determining to which oscillation spaces Op,ts (defined in terms of wavelet coefficients) f belongs. In this paper, we first prove embeddings between oscillation and Besov-Sobolev spaces. We deduce a general lower bound for the (p,t)-oscillation exponent. We then show that this lower bound is actually equality generically, in the sense of Baire’s categories, in a given Sobolev or Besov space. We finally investigate the Baire generic validity of the t-multifractal formalism.

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 low-dimensional Ricci limit spaces

2013 ◽  
Vol 209 ◽  
pp. 1-22 ◽  
Author(s):  
Shouhei Honda
Keyword(s):  
Lower Bound ◽  
Hausdorff Dimension ◽  
Ricci Curvature ◽  
Riemannian Manifolds ◽  
Limit Space ◽  
Low Dimensional ◽  
Complete Riemannian Manifolds

AbstractWe call a Gromov–Hausdorff limit of complete Riemannian manifolds with a lower bound of Ricci curvature a Ricci limit space. Furthermore, we prove that any Ricci limit space has integral Hausdorff dimension, provided that its Hausdorff dimension is not greater than 2. We also classify 1-dimensional Ricci limit spaces.

scholarly journals Hausdorff dimension and uniform exponents in dimension two

2018 ◽  
Vol 167 (02) ◽  
pp. 249-284 ◽  
Author(s):  
YANN BUGEAUD ◽  
YITWAH CHEUNG ◽  
NICOLAS CHEVALLIER
Keyword(s):  
Lower Bound ◽  
Hausdorff Dimension ◽  
Packing Dimension ◽  
Two Dimensional ◽  
Joint Work ◽  
Unpublished Work

AbstractIn this paper we prove that the Hausdorff dimension of the set of (nondegenerate) singular two-dimensional vectors with uniform exponentμin (1/2, 1) is equal to 2(1 −μ) forμ⩾$\sqrt2/2$, whereas forμ<$\sqrt2/2$it is greater than 2(1 −μ) and at most equal to (3 − 2μ)(1 − μ)/(1 −μ+μ2). We also establish that this dimension tends to 4/3 (which is the dimension of the set of singular two-dimensional vectors) whenμtends to 1/2. These results improve upon previous estimates of R. Baker, joint work of the first author with M. Laurent, and unpublished work of M. Laurent. Moreover, we prove a lower bound for the packing dimension, which appears to be strictly greater than the Hausdorff dimension for μ ⩾ 0.565. . . .

Mass transference principle for limsup sets generated by rectangles

2015 ◽  
Vol 158 (3) ◽  
pp. 419-437 ◽  
Author(s):  
BAO-WEI WANG ◽  
JUN WU ◽  
JIAN XU
Keyword(s):  
Lower Bound ◽  
Hausdorff Dimension ◽  
Lebesgue Measure ◽  
Hausdorff Measure ◽  
Unit Cube ◽  
Transference Principle ◽  
Theoretical Statement ◽  
Formula Group ◽  
Image Position ◽  
Full Lebesgue Measure

AbstractWe generalise the mass transference principle established by Beresnevich and Velani to limsup sets generated by rectangles. More precisely, let {xn}n⩾1 be a sequence of points in the unit cube [0, 1]d with d ⩾ 1 and {rn}n⩾1 a sequence of positive numbers tending to zero. Under the assumption of full Lebesgue measure theoretical statement of the set \begin{equation*}\big\{x\in [0,1]^d: x\in B(x_n,r_n), \ {{\rm for}\, {\rm infinitely}\, {\rm many}}\ n\in \mathbb{N}\big\},\end{equation*} we determine the lower bound of the Hausdorff dimension and Hausdorff measure of the set \begin{equation*}\big\{x\in [0,1]^d: x\in B^{a}(x_n,r_n), \ {{\rm for}\, {\rm infinitely}\, {\rm many}}\ n\in \mathbb{N}\big\},\end{equation*} where a = (a1, . . ., ad) with 1 ⩽ a1 ⩽ a2 ⩽ . . . ⩽ ad and Ba(x, r) denotes a rectangle with center x and side-length (ra1, ra2,. . .,rad). When a1 = a2 = . . . = ad, the result is included in the setting considered by Beresnevich and Velani.

Diophantine approximation and a lower bound for Hausdorff dimension

Mathematika ◽  
1990 ◽  
Vol 37 (1) ◽  
pp. 59-73 ◽  
Author(s):  
M. M. Dodson ◽  
B. P. Rynne ◽  
J. A. G. Vickers
Keyword(s):  
Lower Bound ◽  
Hausdorff Dimension ◽  
Diophantine Approximation
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