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Feb
4
revised Where does the series of real Mandelbrot lobes end?
edited tags
Feb
4
comment Where does the series of real Mandelbrot lobes end?
Are you asking about the end of the period doubling cascade? If so, I think the period 16 lobe is centered approximately at -1.3969453597045611. Once you know a few of them, you can use Feigenbaum's first constant to get a decent approximation to the next which can be used as a seed to get a higher precision approximation with Newton's method. This allows you to get a pretty good estimate of the limit. Also, why do you think there's a Misiurewicz point point at the end of all that?
Feb
1
revised Why we cannot apply pole placement for the following system?
Deleted complex dynamics
Jan
30
comment Show that $E_\mu$ has no periodic points that are not fixed points
@sequence I recommend that you write down the definition of "increasing function", i.e. $f$ is increasing if $x_1<x_2$ implies ... Next, suppose that $x_1$ and $x_2$ form an orbit of period two. You should find that you are close to a contradiction yielding the result for period two orbits. Working with an orbit of greater length is just a bit more work.
Jan
30
comment Show that $E_\mu$ has no periodic points that are not fixed points
Does the fact that it's an increasing function help?
Jan
30
comment Repeating decimal notation of 1/53 on WolframAlpha vs notation on Wikipedia
It's quite common in numerical analysis to normalize the numbers so that the first digit is non-zero.
Jan
30
comment What is so special about the Schwarz Inequality?
I recommend you have a look at The Cauchy-Schwarz Master Class. There you will find that "The typical chapter in this course is built around the solution of a small set of challenge problems. Sometimes a challenge problem is drawn from one of the world’s famous mathematical competitions, but more often a problem is chosen because it illustrates a mathematical technique of wide applicability." And that's the point - so many applications have their root in this one elegant inequality.
Jan
28
comment Are the solutions to $1+1/2^s+1/3^s=0$ known?
What exactly are you trying to do? I can reproduce figures 4 and 5 from Borwein's paper fairly easily, if that's the kind of thing you want.
Jan
27
revised Curves Bounding Complex Function
edited body
Jan
27
answered Curves Bounding Complex Function
Jan
26
comment For which angles we know the $\sin$ value algebraically (exact)?
That makes sense - thanks!
Jan
26
comment For which angles we know the $\sin$ value algebraically (exact)?
+1 - I used this no problem for $\pi/7$ and $\pi/9$. But, I don't follow your comment that every rational multiple of $\pi$ has trig functions expressible in $n$th roots for suitable $n$. We could certainly express them in terms of roots of a polynomial system obtained by expanding out the $\text{cis}(2pi/n)^n$. If we can solve that in terms of roots, then you're correct. I obtained an irreducible 10th degree polynomial for $\pi/11$, however.
Jan
26
comment Are the vertices of a Voronoi diagram obtained from a Sierpinski attractor also a kind of attractor?
That does look pretty cool!
Jan
20
awarded  Enlightened
Jan
20
awarded  Nice Answer
Jan
18
answered How does Mathematica calculate sin(Pi/5)?
Jan
18
comment Simple, stable $n$-body orbits in the plane with some fixed bodies allowed
@EliRose Awesome! I notice on your StackOverflow info page that you've got some Python tags. It seems that SciPy's odeint command works swell for this.
Jan
18
comment How to draw a Mandelbrot Set with the connecting filaments visible?
@JerryGuern I think your mistaken. Those swirls do, in fact envelope filaments. The fact that colors highlight filaments is exactly the reason that Hubbard suggested coloring these pictures in the first place.
Jan
18
comment How to draw a Mandelbrot Set with the connecting filaments visible?
I don't see what you're talking about. There appear to be zillions of filaments in that first image to me.
Jan
17
comment Simple, stable $n$-body orbits in the plane with some fixed bodies allowed
@EliRose I think that the main issue is that the forces and, therefore, the accelerations scale in a completely different way.