Mark McClure
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 2d comment Find analytically the sequence of iterations $x_n$ for Newton's method applied to the function $f(x) = x^2$ with the starting point $x_0 = 1$. @Shalid $\cdots = x_n-x_n/2 = x_n/2$. Now, if you compute a few terms, you oughtta spot a pattern fairly easily. 2d comment Find a Continuous Function with Cantor Set Level Sets @StellaBiderman There is definitely a countable but dense set of points in $[0,2/3]$ for which $\tau^{-1}(c)$ is a Cantor set. I expanded my answer substantially to explain this and presented a couple of examples as well. 2d revised Find a Continuous Function with Cantor Set Level Sets added 3812 characters in body 2d answered What is the condensation set of a fractal? Apr 28 comment How to prove $\sum n/3^n$ converges without ratio test? You can use the integral test, as you can integrate $f(x)=x3^{-x}$ (which is positive and decreasing for sufficiently large $x$) using integration by parts. Apr 27 revised Open source lecture notes and textbooks added 187 characters in body Apr 22 answered Find a Continuous Function with Cantor Set Level Sets Apr 12 revised Is this a valid definition of “self-similar fractal”? added 174 characters in body Apr 12 comment Is this a valid definition of “self-similar fractal”? ... Note also that the question at hand refers specifically to "self-similar fractal*, which I took to mean "self-similar set". Now that is a specific concept and my answer essentially says that "invariant set of an IFS* is exactly correct while "doesn't tile the plane" isn't really relevant. Apr 12 comment Is this a valid definition of “self-similar fractal”? @user1952009 I didn't meant to confuse you! My own feeling is that "fractal geometry" is a subject and is as loosely defined as, say, "linear algebra". We know what we are studying when we do linear algebra - linear transformations, matrices, and many related topics. Similarly, we know what we are studying when we do fractal geometry: Iterated function systems, Hausdorff measure, and many related topics. I think the reason people haven't been able to define "fractal" is because the topics in fractal geometry are equally applicable to simple and complicated sets. Apr 12 revised Is this a valid definition of “self-similar fractal”? added 184 characters in body Apr 12 comment Is this a valid definition of “self-similar fractal”? @user1952009 Yes. Even stronger - the maps must be strict similarity transformations. Note that the maps here are, in fact strict similarity contractions and, therefore, must have unique fixed points, though they might be a little tricky to see. Apr 12 comment Is this a valid definition of “self-similar fractal”? @PyRulez Yes, it is tricky. I hope that the edit helps. Apr 12 revised Is this a valid definition of “self-similar fractal”? added 361 characters in body Apr 12 revised Is this a valid definition of “self-similar fractal”? added 361 characters in body Apr 12 awarded Revival Apr 12 answered What is the Hilbert curve's equation?! Apr 10 answered Is this a valid definition of “self-similar fractal”? Apr 10 comment Prove that the iteration of $\sin(x)$ goes to zero as $n$ goes to $\infty$ Hint: $|\sin(x)| < |x|$ for all non-zero $x\in\mathbb R$. Apr 10 revised Prove that the iteration of $\sin(x)$ goes to zero as $n$ goes to $\infty$ edited tags