By Feng Gan Zhuang, You Lan Zhu
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Extra resources for 10th Int'l Conference on Numerical Methods in Fluid Dynamics
1. For the Cantor set with positive measure, the value of is significantly different from the dimension. We again observe, for all of the examples, that the cutoff resolution, , is well approximated by the value where the number of isolated points ceases to be zero. Again, this is because all of the underlying sets are perfect. 15(a). 16. 1; this makes sense because both are generated by iterated function systems of similarity transformations. For this example, the and are at ; because there are four self-similar copies at one jumps in and , which gives the theoretically determined third the size, we see ✑ limits of and .
1; this makes sense because both are generated by iterated function systems of similarity transformations. For this example, the and are at ; because there are four self-similar copies at one jumps in and , which gives the theoretically determined third the size, we see ✑ limits of and . This is in close agreement with our numerical estimate of ✕✎ and ✖✎ . 15(b), is also generated by similarities. This time, the lower two have a contraction ratio of and the upper two have a ratio of . 16, this leads to a more complicated staircase pattern in the and graphs.
The sum of the gap lengths is ✚ ; choosing an interval remaining from level ✛ and ✚ to make this length less than one ensures the Cantor set has positive measure. It is easy to recursively generate the end points of the gaps (down to some level) and these points are used as the finite point-set approximation. 15: Cantor sets generated by iterated function systems of four similarity transformations. Both sets have points. (a) Similarities with contraction ratio . (b) The upper two similarities have ratio and the lower two have ratio .
10th Int'l Conference on Numerical Methods in Fluid Dynamics by Feng Gan Zhuang, You Lan Zhu