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Brad Klee
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Hi Daniel, Oh no! There was a copy-paste error in the first expansion, it was supposed to be: Series[Exp[-Pi Hypergeometric2F1[1/2, 1/2, 1, 1 - x]/Hypergeometric2F1[1/2, 1/2, 1, x]], {x, 0, 5}] and you can test the following by series...
Hi Ed, Yes, I like this one-to-five cross tiling, and years ago I spent some time calculating a set of channels that could probably be encoded to force the substitution hierarchy. Here's a printout from one of my notebooks: ![Colored Cross...
How about This? Graphics3D[Transpose[{{Red, Green, Blue}, MapIndexed[ Function[{dat, ind}, MapIndexed[Sphere[{#2[[1]], 5 ind[[1]], 200 #1}, 1] &, dat]], {RF, DL, GBM}]}], Boxed -> False] ![data][1] ...
In [A Rational Approach to Pi][1] Beukers asserts on the right column of page 377 that the integral, $$I_n=\int_{0}^{\pi/4}d\phi \Big( 4 \sin(\phi) \big(\sin(\phi)-\cos(\phi)\big) \Big)^n,$$ Produces a rational approximation to $\pi$, which...
A few months ago, the thread [Circle Morphing into a triangle][1] came up. This thread falls short of discussing curvature flows, including the famous curve shortening flow. This is an important topic, leading to the more general Ricci flow, [as...
After more calculation, we can generalize the [quadratic case][1] to the next higher cubic case, as [first discussed here][2]. Code for verifying the induction hypothesis and all base cases follows: t1 = AbsoluteTiming[ ...
Hey thanks for the plaudit. I realize that these elliptic functions involve some complicated calculus, but think that the effort is worthwhile. Though elliptic curves are easy-enough to understand, especially in Edward's normal form, some of the...
I think you should be careful with this plot, and it looks like you are headed in a wrong direction. The function you give isn't $2\pi$ periodic, as can easily be seen by explicit computation of function values V[r_, \[CurlyPhi]_] := Exp[-(r...
Hi Dan, It would help if you distribute an example STL file. Your question may be answered using regions. See for example: [Integrate Regions][1] You may also need **Eigensystem** or **Eigenvectors**. Hope it helps. Brad Edit: ...
Hi, I agree with others that the sign does not matter here. The roots are the only values of interest. It's worthwhile -- though somewhat circular -- to calculate the characteristic polynomial as follows: Poly = Times @@ ((x - #) & /@...