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1
Content available remote Persistence landscapes of affine fractals
Catanzaro Michael J., Przybylski Lee, Weber Eric S.
Demonstratio Mathematica
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2022
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Vol. 55, nr 1
163--192
EN
We develop a method for calculating the persistence landscapes of affine fractals using the parameters of the corresponding transformations. Given an iterated function system of affine transformations that satisfies a certain compatibility condition, we prove that there exists an affine transformation acting on the space of persistence landscapes, which intertwines the action of the iterated function system. This latter affine transformation is a strict contraction and its unique fixed point is the persistence landscape of the affine fractal. We present several examples of the theory as well as confirm the main results through simulations.
2
Content available remote Most Cantor sets in RN are in general position with respect to all projections
Frolkina Olga
Bulletin of the Polish Academy of Sciences. Mathematics
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2022
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Vol. 70, no. 1
53--62
EN
We prove the theorem stated in the title. This answers a question of John Cobb (1994). We also consider the case of the Hilbert space ℓ2.
3
Content available Dimension of the intersection of certain Cantor sets in the plane
Pedersen Steen, Staw Vincent T.
Opuscula Mathematica
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2021
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Vol. 41, no. 2
227--244
EN
In this paper we consider a retained digits Cantor set T based on digit expansions with Gaussian integer base. Let F be the set all x such that the intersection of T with its translate by x is non-empty and let Fβ be the subset of F consisting of all x such that the dimension of the intersection of T with its translate by x is β times the dimension of T. We find conditions on the retained digits sets under which Fβ is dense in F for all 0 ≤ β ≤ 1. The main novelty in this paper is that multiplication the Gaussian integer base corresponds to an irrational (in fact transcendental) rotation in the complex plane.
4
Content available remote Sampling and interpolation of cumulative distribution functions of Cantor sets in [0, 1]
Byars Allison, Camrud Evan, Harding Steven N., McCarty Sarah, Sullivan Keith, Weber Eric S.
Demonstratio Mathematica
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2021
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Vol. 54, nr 1
85--109
EN
Cantor sets are constructed from iteratively removing sections of intervals. This process yields a cumulative distribution function (CDF), constructed from the invariant Borel probability measure associated with their iterated function systems. Under appropriate assumptions, we identify sampling schemes of such CDFs, meaning that the underlying Cantor set can be reconstructed from sufficiently many samples of its CDF. To this end, we prove that two Cantor sets have almost-nowhere intersection with respect to their corresponding invariant measures.
5
Content available On intersections of Cantor sets: Hausdorff measure
Pedersen S., Phillips J. D.
Opuscula Mathematica
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2013
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Vol. 33, no. 3
575--598
EN
We establish formulas for bounds on the Haudorff measure of the intersection of certain Cantor sets with their translates. As a consequence we obtain a formula for the Hausdorff dimensions of these intersections.
6
The Cantor set generated by a simple evolutionary model
Karcz-Dulęba I.
Zeszyty Naukowe Uniwersytetu Jagiellońskiego. Prace Informatyczne
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2000
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z. 10
29-36
EN
Distributions of traits in a population provide important information about evolution of the population itself. In this paper an analysis of traits distributions in a phenotypic evolution is presented. A very simple model of evolution is under consideration - infinite populations evolve in one dimension space of a bimodal fitness function. The analysis of dynamic behavior of a population yields an interesting result concerning generation of the subsequent distributions. It appears that every normal distribution generates two offspring normal distributions. The evolutionary process initialized with a single normal distribution grows up to 2t normal distributions after t generations. The evolution of normal distributions is described equivalently by evolution of their parameters: means and variances. The evolution of distributions' means resembles fractals generated by an Iterated Function System (IFS). Equations describing the location of distrbutions' means in the next generation define contractive affine transformations. The defined iterative system maps the interval [0,1] into the Cantor set after infinite number of iterations.
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