scholarly journals Intersection of Continua and Rectifiable Curves

2013 ◽  
Vol 57 (2) ◽  
pp. 339-345 ◽  
Author(s):  
Richárd Balka ◽  
Viktor Harangi
Keyword(s):  
Hausdorff Dimension ◽  
Rectifiable Curve ◽  
Rectifiable Curves

AbstractWe prove that for any non-degenerate continuum K ⊆ ℝd there exists a rectifiable curve such that its intersection with K has Hausdorff dimension 1. This answers a question of Kirchheim.

scholarly journals A note on rectifiable curves

1956 ◽  
Vol 40 ◽  
pp. 12-14
Author(s):  
W. F. Newns
Keyword(s):  
Bounded Variation ◽  
Plane Curve ◽  
Arc Length ◽  
Absolutely Continuous ◽  
Rectifiable Curve ◽  
Continuous Complex ◽  
Image Position ◽  
Rectifiable Curves ◽  
Complex Valued ◽  
Equality Holding

Let f be a continuous complex-valued function of a real parameter whose real and imaginary parts are of bounded variation in the range (a, b) of the parameter, so that the range of f is a rectifiable plane curve. The main results connecting the arc-length s with the parametrization are as follows:Theorem 1 (Tonelli). For any rectifiable curve,equality holding for all α, β (a ≤ α < β ≤ b) if and only if f is absolutely continuous in (a, b).

SINGULAR CURVES AND THE PLATEAU PROBLEM

1991 ◽  
Vol 02 (01) ◽  
pp. 1-16 ◽  
Author(s):  
JOEL HASS
Keyword(s):  
Plateau Problem ◽  
Rectifiable Curve ◽  
Rectifiable Curves ◽  
Least Area

Douglas and Rado showed that an embedded rectifiable curve bounds a disk of least area. It is shown that this result holds for all rectifiable curves, including singular ones.

scholarly journals Minimal F2-flows

1985 ◽  
Vol 39 (2) ◽  
pp. 187-193
Author(s):  
Daniel Berend
Keyword(s):  
Hausdorff Dimension ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Affine Transformations ◽  
Dimensional Torus ◽  
Minimal Sets ◽  
Necessary And Sufficient

AbstractLet σ be an ergodic endomorphism of the r–dimensional torus and Π a semigroup generated by two affine transformations lying above σ. We show that the flow defined by Π admits minimal sets of positive Hausdorff dimension and we give necessary and sufficient conditions for this flow to be minimal.

Length distortion and the Hausdorff dimension of limit sets

2000 ◽  
Vol 122 (3) ◽  
pp. 465-482 ◽  
Author(s):  
Martin Bridgeman ◽  
Edward C. Taylor
Keyword(s):  
Hausdorff Dimension ◽  
Limit Sets ◽  
Length Distortion

scholarly journals A Generalization of the Hausdorff Dimension Theorem for Deterministic Fractals

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1546
Author(s):  
Mohsen Soltanifar
Keyword(s):  
Hausdorff Dimension ◽  
Lebesgue Measure ◽  
Virtual World ◽  
Fractal Dimensions ◽  
Definition Of ◽  
The Given

How many fractals exist in nature or the virtual world? In this paper, we partially answer the second question using Mandelbrot’s fundamental definition of fractals and their quantities of the Hausdorff dimension and Lebesgue measure. We prove the existence of aleph-two of virtual fractals with a Hausdorff dimension of a bi-variate function of them and the given Lebesgue measure. The question remains unanswered for other fractal dimensions.

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