Mathematicians crack code for transforming complex shapes while preserving key properties
Researchers have developed a new mathematical framework that transforms irregular, fractal-like spaces into simpler shapes without losing their essential energy characteristics. The breakthrough could accelerate computational modeling in machine learning, materials science, and data analysis by making complex geometries easier to process while maintaining their mathematical integrity.
Originaltitel: Preserving Besov (fractional Sobolev) energies under sphericalization and flattening
Abstract We introduce a new sphericalization mapping for metric spaces that is applicable in very general situations, including totally disconnected fractal type sets. For an unbounded complete metric space which is uniformly perfect at a base point for large radii and equipped with a doubling measure, we make a more specific construction based on the measure and equip it with a weighted measure. This mapping is then shown to preserve the doubling property of the measure and the Besov (fractional Sobolev) energy. The corresponding results for flattening of bounded complete metric spaces are also obtained. Finally, it is shown that, for the composition of a sphericalization with a flattening, or vice versa, the obtained space is bi-Lipschitz equivalent with the original space and the resulting measure is comparable to the original measure.