Reputation
15,006
Top tag
Next privilege 20,000 Rep.
Access 'trusted user' tools
Badges
20 45
Newest
 Enlightened
Impact
~308k people reached

13h
comment Are there sets of zero measure and full Hausdorff dimension?
@JulienGodawatta It is a better fit here, anyway.
2d
comment How to find the root of a polynomial function closest to the initial guess?
+1! Though, I fixed a couple typos so that it would run. Why the import of abs from math? There's a built in already.
2d
comment Roots of iterations of polynomials
@Watson - No problem! To answer your question, the Julia set is invariant under application of the function, so the Julia sets of $f$ and $f^n$ are identical. The Julia set of $f^n+id$ should be close, but not identical.
2d
comment How do find the numerical average of $x^x$ from $(-4,-2)$?
Have you seen this answer on concerning the $x^x$ spindle? If you choose the principal branch, then you can use ordinary integration to get an average value of $0.00704628 - 0.0121195i$.
2d
comment How to find Misiurewicz Points without solving huge polynomials? (Mandelbrot Set)
The parabolic and Misiurewicz points can be parametrized by the rational numbers between 0 and 1 using the idea of an external ray. The Wikipedia page provides a serviceable introduction, this paper provides a modern account, and Milnor's book is excellent as well. External rays are not hard to compute and you can use one to get close to a Misiurewicz point. I could provide more details over on Mathematica.se, if you like.
2d
comment Roots of iterations of polynomials
@lhf Good point! I used that to help me improve my answer a bit.
Feb
4
comment How many values does $\sqrt{\sqrt{i}}$ have?
@DavidC.Ullrich Thank you!
Feb
4
comment How many values does $\sqrt{\sqrt{i}}$ have?
@DavidC.Ullrich The question is tagged wolfram-alpha and expresses clear confusion over the response provided by WolframAlpha. Whether that's the main subject matter of the question or not, it's clear that an answer providing insight into the behavior provided by that specific software is, at least, germane to the discussion. I've edited my answer to indicate that my intention is to address that specific aspect. Incidentally, I personally wrote the code that generates much of the Alpha output, so I think it's safe to say that I know what I'm talking about, at least, in this context.
Feb
4
comment How many values does $\sqrt{\sqrt{i}}$ have?
@DavidC.Ullrich No, I am not claiming that Mathematica is the universal authority. I am answering a question about the behavior of a particular software tool, however, so I think that a proper understanding of that behavior requires an understanding of the conventions employed by that software.
Feb
4
comment How many values does $\sqrt{\sqrt{i}}$ have?
@DavidC.Ullrich As I said in my response to this same comment on my answer, the question concerns WolframAlpha's response, which is built on top of Mathematica's Sqrt function which is well known to return the principal square root.
Feb
4
comment How many values does $\sqrt{\sqrt{i}}$ have?
@DavidC.Ullrich Exactly - and that's what Mathematica's Sqrt function has returned since 1988.
Feb
4
comment How many values does $\sqrt{\sqrt{i}}$ have?
I must disagree with this. The square root function is perfectly well defined as a function. Have a look at my answer if you'd like to understand why Alpha responds the way that it does.
Feb
4
comment Where does the series of real Mandelbrot lobes end?
The estimate you have, -1.42625, does in fact yield a nearly super-attractive orbit of period 16, but it's not part of the period doubling cascade. Rather, it's the center of a baby-brot well to the left of the period doubling cascade. I agree that Misiurewicz points are certainly outside the period doubling cascade but you should be able to find them arbitrarily close by. You might find information on the $c$-value you seek under the name Feigenbaum attractor.
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?
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
26
comment For which angles we know the $\sin$ value algebraically (exact)?
That makes sense - thanks!