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For Question 1, we want to consider the following figure
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For part a, in the capacity dimension calculation for the set in the above figure, what is epsilon?
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Epsilon, in the capacity dimension calculation, is the side length of each box
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As labeled here, each box is 0.25 wide, so the answer to this is 0.25
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For part b, we want to know what N(epsilon) is
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Recall that N(epsilon), in this calculation, is the number of boxes of size epsilon needed to cover the set
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Notice here, if you count the blue boxes, you need 12 boxes to cover the Julia set, so the answer is 12
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For the remainder of the questions, we値l be considering this plot
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This is a log log plot of 1/epsilon versus X(epsilon)
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For Question 2, we want to consider the power law X(epsilon) scales like 1/epsilon to the nu
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We know that this power law holds in the scaling region of this curve
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Using this information, how would we then approximate nu, given this curve?
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We can approximate nu, given this power law and this plot, using the slope of the line fitted to the curve in region B
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For Question 3, considering the same figure, on this type of plot region B is called a scaling region
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For Question 4, we want to delve into what is causing the three different regions in this figure
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Region B is a scaling region, and the shape of this part of the curve is due to the power law relationship
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This is the actual region we care about in this plot
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Regions A and C are numerical side effects of the algorithm
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Region A is being caused by epsilon being too large
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In this case, the entire set is being covered by a single epsilon ball
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In region C, on the other hand, epsilon is too small
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And in this case each point is being covered by a single epsilon ball
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Regions A and C are numerical artifacts of the algorithm, and are not important to this calculation
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Region B, the so-called scaling region of this power law relationship, is usually what you care about when dealing with this kind of plot