User:IntegralPython/sandbox: Difference between revisions

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==Open set condition==

In [[fractal geometry]], the '''open set condition''' ('''OSC''') is a commonly imposed condition on self-similar fractals. In some sense, the condition imposes restrictions on the overlap in a fractal construction.<ref>{{cite journal |last1=Bandt |first1=Christoph |last2= Viet Hung |first2= Nguyen |last3 = Rao |first3 = Hui | title=On the Open Set Condition for Self-Similar Fractals | journal=Proceedings of the American Mathematical Society | volume=134 | year=2006 | pages=1369–74 | issue=5 | url=http://www.jstor.org/stable/4097989| url-access=limited}}</ref> Specifically, given an iterative[[iterated function system]] of [[contraction mapping| contractive mappings]] fψ_''i'', the open set condition requires that there exists a nonempty, open set S satisfying two conditions:

#<math> \bigcup_{i=1}^m\psi_i (V) \subseteq V, </math>

# Each <math>\psi_i (V)</math> is pairwise disjoint.

Introduced in 1946 by P.A.P Moran,<ref>Moran, P.A.P. (1946) Additive Functions of Intervals and Hausdorff Measure. Proceedings-Cambridge Philosophical Society, 42, 15-23.

Thehttps://doi.org/10.1017/S0305004100022684</ref> the open set condition is used to compute the dimensions of certain self-similar fractals, notably the Sierpinski Gasket.

https://www.jstor.org/stable/4097989?read-now=1&refreqid=excelsior%3A19b81930eca74e0a264d556ab56211ae&seq=1#page_scan_tab_contents

===Computing Hausdorff measure===

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:<math> \left(\frac{1}{2}\right)^s+\left(\frac{1}{2}\right)^s+\left(\frac{1}{2}\right)^s = 3 \left(\frac{1}{2}\right)^s =1. </math>

Taking [[natural logarithm]]s of both sides of the above equation, we can solve for ''s'', that is: ''s'' = ln(3)/ln(2). The Sierpinski gasket is self-similar and satisfies the OSC. In general a set ''E'' which is a fixed point of a mapping

: <math> A \mapsto \psi(A) = \bigcup_{i=1}^m \psi_i(A) </math>

is self-similar if and only if the intersections

:<math> H^s\left(\psi_i(E) \cap \psi_j(E)\right) =0, </math>

where ''s'' is the Hausdorff dimension of ''E'' and ''H^s'' denotes [[Hausdorff measure]]. This is clear in the case of the Sierpinski gasket (the intersections are just points), but is also true more generally. Indeed, under the same conditions as above, the unique fixed point of ψ is self-similar.

==Hand-eye calibration problem==