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Apr
10
revised Prove that the iteration of $\sin(x)$ goes to zero as $n$ goes to $\infty$
edited tags
Apr
8
revised Find if a fixed-point iteration converges for a certain root
added 298 characters in body
Apr
8
answered Find if a fixed-point iteration converges for a certain root
Apr
8
comment Fractal fundamentals
The Mandelbrot set does not display self-similarity. On the contrary, you can actually tell where you are in the Mandelbrot set by observing the local structure. This behavior is described nicely in this paper by Bob Devaney. On page 3, he calls the Mandelbrot set "the antithesis of a fractal" because of this.
Apr
8
comment Fractal fundamentals
Your set is the attractor of an iterated function system, close to one that generates a 9 fold polygasket. Have a look at this poly-gasket visualization. Set the number of sides to 9 and the scaling ratio to 0.25. You will generate an image very close to yours.
Apr
1
comment Ways to determine $\pi$
@HansLundmark None of the questions that you link address the key issue here - namely, how do you compute some particular digit of $\pi$ without computing all the previous digits of $\pi$. There are several questions (like this one) obtained by searching the site for spigot algorithm for pi that do address that issue.
Mar
22
revised Projection of Antoine's necklace
deleted 164 characters in body
Mar
14
comment One dimensional integration that Mathematica cannot do
Mathematica generated several answers for me, depending on exactly what Assumptions I used. This is why it's typically a good idea to include your code when asking a question about a a computer language.
Mar
14
comment How do I plot the following 3d parametric surface in Mathematica?
There is a StackExchange site for Mathematica, though this question would likely be closed rather quickly. If you do ask there, be sure to have a look at Wolfram's documentation on ParametricPlot3D first and, if you're still having problems, be sure to include the code for your attempt and any error message that are produced.
Mar
11
comment Fixed point method
Agreed! There's lots of solutions, though.
Mar
11
revised Fixed point method
deleted 12 characters in body; edited tags
Mar
11
comment Fixed point method
I think the real issue is that he's not only considering one branch of the arcsine.
Mar
11
comment Fixed point method
You need to consider the other possible branches of the arcsine function - try iterating $f(x) = (\sin ^{-1}\left(\cos (\pi x)/9\right)+\pi)/\pi$, for example.
Mar
11
comment Deriving convergence region of iterative formula
Yes, that agrees with my image. I guess that $|F(z)|>|z|$ for points inside the interior curve but they still don't exit the region bounded by the exterior curve.
Mar
11
revised Deriving convergence region of iterative formula
edited tags
Mar
11
comment Compute Hausdorff dimension of cantor set.
Have a look at section 4.1 of Falconer's Fractal Geometry - in particular, his lemma 4.2, "The Mass Distribution Principle". This is a basic tool for obtaining lower bounds for Hausdorff measure and he presents several examples of its use.
Mar
11
awarded  Necromancer
Mar
11
comment Deriving convergence region of iterative formula
I believe that curve is one component of $|f(z)| = |z|$. Initial points outside that curve map to larger numbers, which is why you get the divergence.
Mar
11
comment Wolfram Alpha wrong answers on $(-8)^{1/3}$ and more?
You can enter cbrt(x) for the cube root function in Wolfram|Alpha, which (as implemented in that particular piece of software) is different from $x^{1/3}$.
Mar
11
revised cubic root of negative numbers
edited tags