Box integrals on fractal sets

Abstract:

Using fractal self-similarity and functional-expectation relations, the classical theory of box integrals is extended to encompass a new 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 and report various numerical results.