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1d
comment Newton Iteration Function
Very nice analysis. It might be interesting to note that this is unexpected. Typically, we figure that $N(x)=x-f(x)/f'(x)$ has a fixed point at $x_0$ only if $f(x_0)=0$. I guess this example also shows that we can also have a fixed point at $x_0$ if $f'(x_0)=\infty$. Another example would be $\sqrt[3]{x}$.
2d
comment Parametrization by arclength
The question asks for a re-parametrization yielding the same path but with constant speed. Does your answer yield a technique to find that re-parametrization?
2d
comment Parametrization by arclength
Use NIntegrate[...] rather than N[Integrate[...]]. The first goes straight to numerical techniques while the second attempts symbolic evaluation before following back on numerical techniques.
2d
comment Parametrization by arclength
I don't have time to type up an answer at the moment. You might consider asking over on mathematica.se. There is also an example in the Arc Length Parametrization notebook on this page.
2d
comment Parametrization by arclength
Mathematica (and any reasonably good numerical software) can handle this problem very quickly and easily. Numerical computation of the integral from 0 to $2\pi$ took $0.03$ seconds on my machine. It sounds to me like you're trying to compute symbolic results which you then pass to $N$.
2d
comment Parametrization by arclength
To compute the arc length, you find yourself integrating $\sqrt{180 \cos (2 t)+18 \cos (4 t)+259}$ which, as far as I know, as no elementary anti-derivative. The integral can easily be computed numerically, however, and the resulting arc length function can also be inverted numerically. As a result, there is an effective numerical procedure to compute the arc length parametrization.
Apr
14
comment What's the Lebesgue measure of this set?
As already mentioned in the other, more complete, answer.
Apr
14
comment Examples of Fractals From Simple Algorithms
There are many. Can you narrow your focus? How, for example, did you generate your circular image?
Apr
14
comment Is “A New Kind of Science” a new kind of science?
@DavidRicherby My resistance is threefold. First and again, a major point behind my answer is that evaluation of NKS (the book) can and should be done independently of the author and I believe I've done so. Second, I think I've already far exceeded the standards of this site, which allows anonymous users. Finally (and to be quite honest), after my experience with you here, it seems to me that you are the one who is biased. I never did understand your objection to that answer, which seems quite fine to me.
Apr
14
comment Planetary Motion: A comet describe a parabola about the sun
Jack provided a very good answer to your related question here, which you didn't even upvote (I did). You might consider being a little more gracious with the folks who are answering these questions.
Apr
14
comment Is “A New Kind of Science” a new kind of science?
@DavidRicherby Thank you for your clarification. I assure you, I am disinterested by your definition. Have a look at my answer to this question; does it look like the answer of someone pushing a Wolfram agenda to you? Also, when you say my answer "comes across" as balanced, I guess you're implying that it's not. I wonder, would you expect my direct experience with Wolfram to skew me in the direction of a favorable opinion of NKS? Or an unfavorable one? Could be either, I suppose.
Apr
14
comment Drawing 3D level sets/curves
@Josephhooper The level set of a function of $n$ variables lives in $n$-dimensional space - always. For example, the level set of $f(x,y)=x^2+y^2$ is a circle $x^2+y^2=k$ that lives in the plane. The level set of $f(x,y,z)=x^2+y^2+z^2$ is a sphere $x^2+y^2+z^2=k$ that lives in space. You've got $f(x,y,z)=k$ so your level set is a surface that lives in space.
Apr
11
comment Is “A New Kind of Science” a new kind of science?
@DavidRicherby If I wasn't interested, I wouldn't have answered. :) Perhaps, you're implying that I have a conflict of interest? As I am not currently employed at Wolfram Research, I don't believe that I do. I make heavy use of Mathematica in my work but that's hardly a secret and, again, I think that is very separate from my evaluation of NKS.
Apr
11
comment Is “A New Kind of Science” a new kind of science?
@Newb Of course. But then, a major point that I'm trying to get across in my answer is that, ideally, evaluation of the person should not affect evaluation of his work. As such, this seems to be the wrong place to expand on those experiences.
Apr
11
comment Is “A New Kind of Science” a new kind of science?
A reasonable assessment! I would point out, though, that the many of the ideas that he "repackaged" are, in fact, his own. From a personal perspective, I've found his numbering schemes of CAs to be quite useful and his paper Geometry of Binomail Coefficients to be delightful.
Apr
11
comment Is “A New Kind of Science” a new kind of science?
@MikeMiller I actually like the review just because I think it's hysterical, in the same way that harsh reviews can be funny. I don't particularly agree with it, though.
Apr
10
comment Monotone Functions
Hmm... No. A function is monotone increasing if $f(x_1)<f(x_2)$ whenever $x_1<x_2$. A function is monotone decreasing if $f(x_1)>f(x_2)$ whenever $x_1<x_2$. It's not monotone if neither of those happen for all possible choices of $x_1$, $x_2$, and $x_3$. In that case, you must be able to find $x_1<x_2<x_3$ such that $f(x_2)>f(x_1)$ and $f(x_2)>f(x_3)$. (Or maybe the reverse as there might be more than one case to consider.) Now, try to apply the IVT.
Apr
10
comment Monotone Functions
If $f(0)<f(1)$, then there are certainly at least two numbers two numbers between $f(0)$ and $f(1)$. There are infinitely many, in fact. Before applying the intermediate value theorem, though, I'd focus on what it means that a function is not monotone. I guess there's be an $x_1<x_2<x_3$ such that ... ?
Apr
10
comment Monotone Functions
I wonder what would happen if $f$ were not strictly monotone? What possibilities are there?
Apr
8
comment Dimension of a Two-Scale Cantor Set
@CandyMan I just noticed your edit - it looks like you've got it!