scholarly journals Quasicircles and width of Jordan curves in CP1

2020 ◽  
Author(s):  
Francesco Bonsante ◽  
Jeffrey Danciger ◽  
Sara Maloni ◽  
Jean‐Marc Schlenker
Keyword(s):  
Jordan Curves

Digital Jordan Curves and Surfaces with Respect to a Closure Operator

2021 ◽  
Vol 179 (1) ◽  
pp. 59-74
Author(s):  
Josef Šlapal
Keyword(s):  
Direct Product ◽  
Jordan Curve ◽  
Closure Operator ◽  
Jordan Curve Theorem ◽  
Closure Operators ◽  
Ternary Relation ◽  
Digital Space ◽  
Jordan Curves ◽  
Curves And Surfaces ◽  
Digital Plane

In this paper, we propose new definitions of digital Jordan curves and digital Jordan surfaces. We start with introducing and studying closure operators on a given set that are associated with n-ary relations (n > 1 an integer) on this set. Discussed are in particular the closure operators associated with certain n-ary relations on the digital line ℤ. Of these relations, we focus on a ternary one equipping the digital plane ℤ2 and the digital space ℤ3 with the closure operator associated with the direct product of two and three, respectively, copies of this ternary relation. The connectedness provided by the closure operator is shown to be suitable for defining digital curves satisfying a digital Jordan curve theorem and digital surfaces satisfying a digital Jordan surface theorem.

On the Hilbert boundary-value problem for Beltrami equations with singularities

2020 ◽  
Vol 17 (4) ◽  
pp. 484-508
Author(s):  
Vladimir Gutlyanskii ◽  
Vladimir Ryazanov ◽  
Eduard Yakubov ◽  
Artyem Yefimushkin
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Beltrami Equation ◽  
General Nature ◽  
Countable Collection ◽  
Smooth Curves ◽  
Beltrami Equations ◽  
Jordan Curves ◽  
Equations Of Mathematical Physics ◽  
Hilbert Boundary Value Problem

We investigate the Hilbert boundary-value problem for Beltrami equations $\overline\partial f=\mu\partial f$ with singularities in generalized quasidisks $D$ whose Jordan boundary $\partial D$ consists of a countable collection of open quasiconformal arcs and, maybe, a countable collection of points. Such generalized quasicircles can be nowhere even locally rectifiable but include, for instance, all piecewise smooth curves, as well as all piecewise Lipschitz Jordan curves. Generally speaking, generalized quasidisks do not satisfy the standard $(A)-$condition in PDE by Ladyzhenskaya-Ural'tseva, in particular, the outer cone touching condition, as well as the quasihyperbolic boundary condition by Gehring-Martio that we assumed in our last paper for the uniformly elliptic Beltrami equations. In essence, here, we admit any countable collection of singularities of the Beltrami equations on the boundary and arbitrary singularities inside the domain $D$ of a general nature. As usual, a point in $\overline D$ is called a singularity of the Beltrami equation, if the dilatation quotient $K_{\mu}:=(1+|\mu|)/(1-|\mu|)$ is not essentially bounded in all its neighborhoods. Presupposing that the coefficients of the problem are arbitrary functions of countable bounded variation and the boundary data are arbitrary measurable with respect to the logarithmic capacity, we prove the existence of regular solutions of the Hilbert boundary-value problem. As a consequence, we derive the existence of nonclassical solutions of the Dirichlet, Neumann, and Poincar\'{e} boundary-value problems for equations of mathematical physics with singularities in anisotropic and inhomogeneous media.

scholarly journals A Note on Rational $$L^p$$ L p Approximation on Jordan Curves

2013 ◽  
Vol 13 (3) ◽  
pp. 425-431 ◽  
Author(s):  
Vilmos Totik
Keyword(s):  
Jordan Curves

scholarly journals Approximation on Wild Jordan Curves

1976 ◽  
Vol s2-13 (2) ◽  
pp. 317-322 ◽  
Author(s):  
Jacob Korevaar ◽  
Herbert Alexander
Keyword(s):  
Jordan Curves

scholarly journals A Characterization of Polynomial Density on Curves via Matrix Algebra

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1231
Author(s):  
Carmen Escribano ◽  
Raquel Gonzalo ◽  
Emilio Torrano
Keyword(s):  
Matrix Algebra ◽  
Optimization Problems ◽  
Positive Semidefinite ◽  
Point Of View ◽  
Alternative Proof ◽  
General Context ◽  
Positive Semidefinite Matrices ◽  
Jordan Curves ◽  
Point Evaluations

In this work, our aim is to obtain conditions to assure polynomial approximation in Hilbert spaces L 2 ( μ ) , with μ a compactly supported measure in the complex plane, in terms of properties of the associated moment matrix with the measure μ . To do it, in the more general context of Hermitian positive semidefinite matrices, we introduce two indexes, γ ( M ) and λ ( M ) , associated with different optimization problems concerning theses matrices. Our main result is a characterization of density of polynomials in the case of measures supported on Jordan curves with non-empty interior using the index γ and other specific index related to it. Moreover, we provide a new point of view of bounded point evaluations associated with a measure in terms of the index γ that will allow us to give an alternative proof of Thomson’s theorem, by using these matrix indexes. We point out that our techniques are based in matrix algebra tools in the framework of Hermitian positive definite matrices and in the computation of certain indexes related to some optimization problems for infinite matrices.

scholarly journals Bilipschitz group actions and homogeneous Jordan curves

1999 ◽  
Vol 43 (4) ◽  
pp. 770-792 ◽  
Author(s):  
David A. Herron ◽  
Volker Mayer
Keyword(s):  
Group Actions ◽  
Jordan Curves

Jordan Curves of Positive Lebesgue Measure

1994 ◽  
pp. 131-143
Author(s):  
Hans Sagan
Keyword(s):  
Lebesgue Measure ◽  
Positive Lebesgue Measure ◽  
Jordan Curves

Jump conditions for a metaharmonic double layer potential on rectifiable closed Jordan curves in R2

2015 ◽  
Vol 115 (2) ◽  
pp. 179-188 ◽  
Author(s):  
Ricardo Abreu Blaya ◽  
Juan Bory Reyes
Keyword(s):  
Double Layer ◽  
Double Layer Potential ◽  
Jump Conditions ◽  
Layer Potential ◽  
Jordan Curves
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