- journal articles metadata
European Mathematical Society Publishing House
2024-03-29 02:13:29
3
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=JFG&vol=5&iss=2&update_since=2024-03-29
Journal of Fractal Geometry
J. Fractal Geom.
JFG
2308-1309
2308-1317
Measure and integration
General
10.4171/JFG
http://www.ems-ph.org/doi/10.4171/JFG
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
5
2018
2
Modeling the fractal geometry of Arctic melt ponds using the level sets of random surfaces
Brady
Bowen
University of Utah, Salt Lake City, USA
Courtenay
Strong
University of Utah, Salt Lake City, USA
Kenneth
Golden
University of Utah, Salt Lake City, USA
Fractal geometry, sea ice, Arctic melt ponds, Fourier series, random surfaces, level sets
During the late spring, most of the Arctic Ocean is covered by sea icewith a layer of snow on top. As the snow and sea ice begin to melt, water collects on the surface to form melt ponds. As melting progresses, sparse, disconnected ponds coalesce to form complex, self-similar structures which are connected over large length scales. The boundaries of the ponds undergo a transition in fractal dimension from 1 to about 2 around a critical length scale of 100 square meters, as found previously from area–perimeter data. Melt pond geometry depends strongly on sea ice and snow topography. Here we construct a rather simple model of melt pond boundaries as the intersection of a horizontal plane, representing the water level, with a random surface representing the topography. We show that an autoregressive class of anisotropic random Fourier surfaces provides topographies that yield the observed fractal dimension transition, with the ponds evolving and growing as the plane rises. The results are compared with a partial differential equation model of melt pond evolution that includes much of the physics of the system. Properties of the shift in fractal dimension, such as its amplitude, phase and rate, are shown to depend on the surface anisotropy and autocorrelation length scales in the models. Melting-driven differences between the two models are highlighted.
Geometry
Partial differential equations
Fourier analysis
Geophysics
121
142
10.4171/JFG/58
http://www.ems-ph.org/doi/10.4171/JFG/58
6
4
2018
Sobolev algebra counterexamples
Thierry
Coulhon
Université de Cergy-Pontoise, France
Luke
Rogers
University of Connecticut, Storrs, USA
Sobolev algebra, Bessel potential, analysis on fractals, post-critically finite fractal
In the Euclidean setting the Sobolev spaces $W^{\alpha,p}\cap L^\infty$ are algebras for the pointwise product when $\alpha > 0$ and $p\in(1,\infty)$. This property has recently been extended to a variety of geometric settings. We produce a class of fractal examples where it fails for a wide range of the indices $\alpha,p$.
Functional analysis
Measure and integration
Potential theory
143
164
10.4171/JFG/59
http://www.ems-ph.org/doi/10.4171/JFG/59
6
4
2018
Numerical integration for fractal measures
Jens
Malmquist
University of California, Berkeley, USA
Robert
Strichartz
Cornell University, Ithaca, USA
Numerical integration, fractal measures, p.c.f. self-similar fractals, energy measures, Laplacians, Koksma–Hlawka theorem, Sierpiński gasket
We find estimates for the error in replacing an integral $\int f d\mu$ with respect to a fractal measure $\mu$ with a discrete sum $\sum_{x \in E} w(x) f(x)$ over a given sample set $E$ with weights $w$. Our model is the classical Koksma–Hlawka theorem for integrals over rectangles, where the error is estimated by a product of a discrepancy that depends only on the geometry of the sample set and weights, and variance that depends only on the smoothness of $f$. We deal with p.c.f. self-similar fractals, on which Kigami has constructed notions of energy and Laplacian. We develop generic results where we take the variance to be either the energy of $f$ or the $L^1$ norm of $\laplace f$, and we show how to find the corresponding discrepancies for each variance. We work out the details for a number of interesting examples of sample sets for the Sierpinski gasket, both for the standard self-similar measure and energy measures, and for other fractals.
Measure and integration
Numerical analysis
165
226
10.4171/JFG/60
http://www.ems-ph.org/doi/10.4171/JFG/60
6
4
2018