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Jul
21
answered Construction of Rauzy Fractals with substitutions without a fixed point
Jul
20
comment Example of polynomial in dynamics
You can rule out the mulitbrot family, since they all have a single critical point, namely the origin.
Jul
20
comment Example of polynomial in dynamics
Very nice! It might be worth pointing out that it's easy to show that $1$ is in the Julia set using the characterization that the Julia set is the closure of the repelling fixed points. First, $-1/2$ is in the Julia set since it's a repulsive fixed point (i.e. a point of period $1$). Then, $1$ is in the Julia set since it's the inverse image of $-1/2$ and the Julia set is backward invariant.
Jul
15
comment Why do we say “radius” of convergence?
@ElliotG Your question in the title is 'Why do we say “radius” of convergence?' This response is an important part of that answer.
Jul
12
reviewed Approve Topological Semi conjugacy between Henon map and Logistic Map
Jul
10
comment Estimating the value of an improper integral numerically
@TanMath When you say that you "can choose what values of x I want to evaluate f(x) at", that certainly sounds like a routine. If you have only samples, then you can probably interpolate first. Have a look at scipy.interpolate.
Jul
10
comment Estimating the value of an improper integral numerically
If you're using python, why don't you use the quad command defined in scipy's integrate package? You don't need a mathematical formula, just a python function (possibly, defined as a routine) that returns a real number. Though, this only works over bounded intervals, you can likely use joriki's excellent advice, which you should upvote.
Jul
4
comment Apollonian gasket
I still don't see how your $C_4$ is well defined. If I understand your question correctly, though, then a somewhat simpler but still recursive formula for the curvatures $\kappa_n=1/r_n$ might be: $\kappa_n = 2 (\kappa_1 + \kappa_2 + \kappa_{n-1}) - \kappa_{n-2}$.
Jul
4
revised Apollonian gasket
added 37 characters in body
Jul
4
comment Apollonian gasket
What's the $n^{\text{th}}$ circle? Keep in mind that, given three mutually tangent circles, we expect two more circles mutually tangent to all three. Not just one. So your $C_4$ is not uniquely determined.
Jun
27
comment Why does the boundary of the Mandelbrot set contain a cardioid?
One comment on the linked comment: I retract my statement that the boundary of the Mandelbrot set contains a line segment.
Jun
27
comment Why does the boundary of the Mandelbrot set contain a cardioid?
Yes, this is essentially how I would have responded. The fact that this cardioid lies on the boundary of the Mandelbrot set can be proved using rational external rays. It turns out that every point of the form $e^{2\pi i t}(2-e^{2\pi i t})$ for rational $t$ is the base of a bulb off of the main cardioid and that some external ray lands exactly at that point.
Jun
27
comment Can a fractal be a manifold? if so: will its boundary (if exists) be strictly one dimension lower?
@A.P. It can, in fact, be proved that every complex parameter on the cardioid $c=e^{i t} \left(2-e^{i t}\right)/4$ is on the boundary of the Mandelbrot set. I can't find a convenient pointer to that fact at the moment and proving it would take us a bit too far astray from this question. Perhaps you could raise the question on the site? I'd answer it this evening, if no one else has by then.
Jun
27
comment Can a fractal be a manifold? if so: will its boundary (if exists) be strictly one dimension lower?
@iadvd I guess you mean a smooth manifold, which is certainly a common assumption. Otherwise, the graph of a nowhere differentiable function can be both a manifold and a fractal. The boundary of the Mandelbrot set contains a line segment, a circle, and a cardioid. So I guess a fractal can contain a smooth manifold.
Jun
26
comment Is :$\frac{\Bbb d}{\Bbb d x}$ a chaotic operator in infinite-dimensional Hilbert space?
@user51189 The question is on hold still because it still doesn't have enough reopen votes; I was just the fourth to vote to reopen so it needs one more. On the other hand, I think that Noah's comment contains a fine pointer.
Jun
19
revised What is the difference between $f(f^{-1}(U))$ and $f^{-1}(f(U))$?
deleted 3 characters in body
Jun
19
comment How to solve this integral and have a result in term of arccos?
@SimonS I believe he used Mathematica, actually. In fact, the exact TeX in the answer is produced by the Mathematica command TeXForm[Integrate[Sqrt[R/r - 1], r]]. I've told him before that this is not acceptable and he should at least acknowledge the source. I have no idea why he refuses.
Jun
15
comment How many vertices are in the Koch Snowflake?
@LinusS. We have now carefully defined a vertex as a vertex of one of the polygonal approximations. These correspond to the (finite) addresses. If you allow the branches to have infinite length, you can then determine other points on the Koch curve. This is analogous to the fact that the terminating decimal expansions determine only countably many real numbers. You need non-terminating expansions get the remaining reals.
Jun
14
answered How many vertices are in the Koch Snowflake?
Jun
12
comment How many vertices are in the Koch Snowflake?
Supposing that by "vertex" you mean one of the vertices of the standard, polygonal approximations to the Koch curve, then perhaps the answer to this question might help? That indicates a technique to place the vertices in 1-1 correspondence with finite strings of symbols so, if you can enumerate those, then you've enumerated the vertices.