|Dror Bar-Natan: Classes: 2003-04: Math 157 - Analysis I:||(88)||
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Required reading. All of Spivak's Chapter 22.
To be handed in. From Spivak Chapter 22: 1 (odd parts), 2 (odd parts), 5, 13.
Recommended for extra practice. From Spivak Chapter 22: 1 (even parts), 2 (even parts), 9, 27, 28, 29.
Just for fun. For some constant number , consider the function . Let be the set of all pairs so that and is a limit of a subsequence of the sequence , , , .... Write a computer program to draw the set in the plane whose axes are and , and if your program and picture are nice, they'll find their place on this class' web site.
It's a hard one, but it's well worth it. The set is way more complex than you would expect, with parts that scream ``structure'' and parts that scream ``mess''. If you've ever heard the word ``chaos'' in a mathematical context before, this is it. And if you've ever seen pictures of the beautiful ``Mandelbrot Set'', our is a close relation.