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Apr
28
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.
Apr
28
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.
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
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
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
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
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
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
11
comment Fixed point method
Agreed! There's lots of solutions, though.
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
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
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
comment Fixed Point Iteration doesn't converge, how to find its convergence?
@user314580 If you like this answer, you should upvote it. I did. If you evaluate $g'$ at the fixed point, by the way, you get a negative number which explains the oscillations.