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May
19
comment How do you solve $\cos \pi z =0$?
Perhaps the downvotes arise because it's clear that you just copied and pasted the answer from a computer? In fact, your TeX can be produced exactly via the Mathematica command Reduce[Cos[Pi*z] == 0, z] // TeXForm.
May
17
comment Proving basic properties of Hausdorff dimension and measure
@AJY I'm not claiming a complete answer - just the idea and a reference, as you asked for. I suppose the key issue to address your present question is this: if you have disjoint closed sets (and they are bounded), then the they will be positively separated.
May
17
comment Proving basic properties of Hausdorff dimension and measure
@AJY Again, my answer addresses exactly that.
May
17
comment Proving basic properties of Hausdorff dimension and measure
@AJY What? My counter example is stated on the line - i.e. $\mathbb R^1$. The salient point is that countable additivity must be with respect to some $\sigma$-algebra. The answer itself is certainly applicable on $\mathbb R^n$.
May
17
comment Proving basic properties of Hausdorff dimension and measure
@AJY The purpose of an edit is to clarify, correct, or otherwise improve a question - not to change it's content. Your edit deleted one question entirely, thus making Daniel's comment superfluous.
May
12
comment How to compute a negative “Multibrot” set?
@DanielAllenLangdon I'm not too surprised about the disconnectedness. Orbit detection is an expensive operation, particularly near the boundary of the Mandelbrot set. Also, the point at infinity should be counted as an orbit as well. I placed some (not particularly fast) Javascript here that takes these things into account.
May
8
comment Graph modeling using calculus
Must it be a polynomial? This would be very simple using a piecewise defined function.
May
7
comment What is the integral of $e^{a \cdot x+b \cdot y}$ evaluated over the Koch Curve
I would be very surprised if there is any known closed form expression for this integral. For what it's worth (which might not be much), I computed the integral numerically for $a=b=1$ and found a value of $1.8800692$. The inverse symbolic calculator didn't find any particularly promising results.
May
7
comment How to compute a negative “Multibrot” set?
@DanielAllenLangdon I made some updates to my answer, as well as to my Javascript application. Most importantly the Javascript contains code to return the detected period, as well as the detection time. Thus, the image of the bifurcation locus can be colored to indicate the various periods. I would love to know if you use this stuff to improve your own applications. I notice that you've been trying to create a fairly general bifurcation locus generator and I'd be happy to provide advice on that.
May
6
comment What does a 3D periodic solution of a differential equation look like?
@2mkgz I thought so! Oh well. So, I know you've got upvotes available (it's early in the UTC day) but I guess my little response below was not worthy. I can certainly agree to disagree on this silly voting thing but, please, do let me know, if you ever find something I wrote upworthy. It would just make my day. :)
May
6
comment What does a 3D periodic solution of a differential equation look like?
@2mkgz An amazingly incomplete response. I'd actually downvote, if you'd posted an actual answer. :)
May
2
comment How to compute a negative “Multibrot” set?
@DanielAllenLangdon There is a floating point zero that can be produced by simple arithmetic, though, and division by that zero produces a problem. Nonetheless, I was surprised to find it to be a non issue when I coded it in JavaScript. I'm sure the code just misses that scenario due to the improbability of hitting it exactly.
May
1
comment How to compute a negative “Multibrot” set?
Very nice again - thanks!
May
1
comment How to compute a negative “Multibrot” set?
@DanielAllenLangdon Yes, you made it quite clear in your question that you had come to that conclusion after your implementation involving the erroneous bailout. I do hope that you find the mathematical discussion, outline of the algorithm, and Javascript implementation all described in the answer to be of use to you. It's an interesting family and I've certainly enjoyed thinking about it.
May
1
comment How to compute a negative “Multibrot” set?
If you don't mind me asking, how well does your Julia set algorithm work for $p=-2$ and $c=-1/2^{2/3}$?
Apr
30
comment How to compute a negative “Multibrot” set?
Very nice - makes good sense, too. Another approach to generating the Julia set is to simply iterate until you reach the attractive orbit. Since there's just one attractive orbit, you can find it at the outset by iterating from the critical point.
Apr
29
comment How to compute a negative “Multibrot” set?
@Zach466920 Thank you! If $p$ is not an integer, then the function is no longer rational and the analysis becomes much more complicated. If $p$ is not rational, then you'd need to move to the iteration of transcendental functions. There are results along these lines but the situation is really totally different. I would not expect to discover any nice behavior by treating $p$ as a parameter. I've been wrong before, though. :)
Apr
29
comment Is the adjacency matrix of a given graph (OR any graphs isomorphic to a given graph) a Kronecker product, and if so what are the factors?
Welcome to the community. You might also be interested in mathematica.se. In either case, there are useful formatting tools for typeset mathematics and code. I edited your question to make the code more accessible. Have fun!
Apr
23
comment how do i prove that a collection of contractions does not satisfy the open set condition?
To be more blunt, could you please present the IFS that you are working with? The question is interesting and potentially challenging. Given the IFS, someone might very well be interested in the challenge. Without it, certainly no one can answer your question.
Apr
22
comment how do i prove that a collection of contractions does not satisfy the open set condition?
I believe this is a very difficult problem from a purely algorithmic perspective and that there is no known, general algorithm. It might be possible to address your question specifically, if you could present the four functions that comprise the IFS.