- Title
- Expectations on fractal sets
- Creator
- Bailey, David H.; Borwein, Jonathan M.; Crandall, Richard E,; Rose, Michael G.
- Relation
- Office of Computational and Technology Research, Division of Mathematical, Information, and Computational Sciences of the U.S. Department of Energy.DE-AC02-05CH11231
- Relation
- Applied Mathematics and Computation Vol. 220, p. 695-721
- Publisher Link
- http://dx.doi.org/10.1016/j.amc.2013.06.078
- Publisher
- Elsevier
- Resource Type
- journal article
- Date
- 2013
- Description
- Using fractal self-similarity and functional-expectation relations, the classical theory of box integrals – being expectations on unit hypercubes – is extended to a class of fractal “string-generated Cantor sets” (SCSs) embedded in unit hypercubes of arbitrary dimension. Motivated by laboratory studies on the distribution of brain synapses, these SCSs were designed for dimensional freedom – a suitable choice of generating string allows for fine-tuning the fractal dimension of the corresponding set. We also establish closed forms for certain statistical moments on SCSs, develop a precision algorithm for high embedding dimensions, and report various numerical results. The underlying numerical quadrature issues are in themselves quite challenging.
- Subject
- expectations; fractals; self-similarity; numerical quadrature; Monte Carlo methods
- Identifier
- http://hdl.handle.net/1959.13/1295551
- Identifier
- uon:19060
- Identifier
- ISSN:0096-3003
- Language
- eng
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