## Can you help me?

November 20, 2015

Some of you may remember the time when I asked for help in getting my little netbook back from the pawn shop. I remain thankful to all of you who contributed to the fund. Unfortunately, I must ask for your help once more.

Currently, I have just started my new job as a remote consultant for a foreign company. The offer given was a bit higher than the salary I was getting at my previous job for a pharmaceutical company, so I took it. Unfortunately, when I resigned, I was informed that it will take two months before all the necessary papers are processed and my (not too sizable) last pay will be released to me. For the new job, due to the bureaucracy involved in delicacy of foreign transfers for remote workers, I don’t expect to see my salary in quite a while.

This puts me in a serious bind. Not having much savings, I don’t know how long I can make the little I have last. My personal accounting tells me that I’m ~ $320.00 short for rent, electricity, phone/Internet service and other necessities. So, I am asking for help from any of you kind readers. If you’re interested in donating some spare change, shoot me an e-mail at this (ROT13-encoded) address: zqd˙ʎʌuzʇ@ʇɐʌǝɥʇʌssqɹǝɥɟuɹʎɔ. Every little bit helps. Thank you very much. I promise, I will always strive to make high-quality contributions to the sites I participate in. ~ Jan ## Yes, I’m still here… December 21, 2014 …and to those guys who showed concern and worry during my long-ish absence, I’m fine. Two important things, though: the hard drive containing a lot of my unpublished research crashed, including a good amount that I have not been able to back up; and, I have been quite busy with a (relatively) new job over the past few months, details of which I cannot discuss here. In short, I’m fine, and I hope you guys will still be here when I’m actually back. Happy Holidays! ~ Jan ## Using the random.org generator in Mathematica May 15, 2012 An answer by the user Emre on the Mathematica StackExchange site introduced me to the random.org random number generation service by Mads Haahr. In the spirit of a previous post of mine, I wanted to be able to use the random.org service through the method plug-in framework for the random number generators of Mathematica. I have thus written a barebones set of routines for the purpose: RandomOrg /: RandomInitializeGenerator[RandomOrg, ___] := RandomOrg["http://www.random.org/"] RandomOrg[___]["GeneratesRealsQ"] := True; RandomOrg[___]["GeneratesIntegersQ"] := True; RandomOrg[___]["SeedGenerator"[seed_]] := RandomOrg["http://www.random.org/"] RandomOrg[url_]["GenerateReals"[n_, {a_, b_}, prec_]] := {a + (b - a) Import[url <> "decimal-fractions/?col=1&format=plain&num=" <> ToString[n] <> "&dec=" <> ToString[Round[prec]] <> "&rnd=new", "List"], RandomOrg[url]} RandomOrg[url_]["GenerateIntegers"[n_, {a_, b_}]] := {Import[url <> "integers/?col=1&base=10&format=plain" <> "&min=" <> ToString[a] <> "&max=" <> ToString[b] <> "&num=" <> ToString[n] <> "&rnd=new", "List"], RandomOrg[url]} Here are a few examples of its use: BlockRandom[SeedRandom[0, Method -> RandomOrg]; RandomReal[{0, 1}, 10, WorkingPrecision -> 20]]  BlockRandom[SeedRandom[0, Method -> RandomOrg]; RandomInteger[{1, 10}, 10]] As a tiny reminder, the site has a daily quota in place that limits the amount of random numbers you can generate using their service in a single day. See their website for more details. ## A short note on Costa’s minimal surface May 14, 2012 The other day, I finally managed to simplify Alfred Gray’s parametric equations for Costa’s minimal surface. I might edit this post later, with details on how to manipulate the Weierstrass elliptic functions that show up in the equations, but enjoy these for now: $\displaystyle x=\frac{\pi u}{2}-\Re\left(\frac{\zeta(u+iv;g_2,0)}{2}-\frac{\pi\,\wp(u+iv;g_2,0)}{\wp^\prime(u+iv;g_2,0)}\right) \\ y=\frac{\pi v}{2}+\Im\left(\frac{\zeta(u+iv;g_2,0)}{2}+\frac{\pi\,\wp(u+iv;g_2,0)}{\wp^\prime(u+iv;g_2,0)}\right) \\ z=\sqrt{\frac{\pi}{8}}\log\left|\frac1{\frac12+\frac{\wp(u+iv;g_2,0)}{\sqrt{g_2}}}-1\right|$ Here, $\wp, \wp^\prime$ and $\zeta$ are respectively the Weierstrass elliptic function, its derivative, and the Weierstrass zeta function, with invariants $g_2=\left(\frac12\mathrm{B}\left(\frac14,\frac14\right)\right)^4=\frac{\Gamma(1/4)^8}{16\pi^2}$ and $0$, and $\mathrm{B}(x,y)$ and $\Gamma(x)$ are the usual beta and gamma functions. The invariants are the ones corresponding to the semi-periods $\omega_1=\frac12$ and $\omega_3=\frac{i}{2}$. The parameter ranges are $0 < u < 1$ and $0 < v < 1$. I was very delighted at my result, that I decided to make a stylized plot of the surface in Mathematica, using Perlin’s simplex noise for the coloring: ## FNV hashing in Mathematica May 4, 2012 I recently needed to do Fowler/Noll/Vo hashing in Mathematica. I figured other people might be have a need for this hash function as well, so here’s my Mathematica implementation: (* FNV hash parameters: bit length, FNV prime, FNV offset *)$FNVData = Append[#, Function[{bi, pr}, Fold[BitXor[Mod[pr #1, 2^bi], #2] &, 0, ToCharacterCode["chongo /\\../\\"]]] @@ #] & /@ Transpose[{2^Range[5, 10], 2^(8 Quotient[2^Range[5, 10] + 5, 12]) + 2^8 + Map[FromDigits[#, 16] &, {"93", "B3", "3B", "63", "57", "8D"}]}];
 FNV[dat_String, bi : _Integer : 32, shift : (0 | 1) : 1] := With[{k = IntegerExponent[bi, 2] - 4}, Fold[BitXor[Mod[$FNVData[[k, 2]] #1, 2^bi], #2] &, shift$FNVData[[k, 3]], ToCharacterCode[dat]]] /; 32 <= bi <= 1024 && BitAnd[bi, bi - 1] == 0 

I made sure to provide for both shift-1 and shift-0 versions in the implementation (though as mentioned on the FNV page, FNV-1 is what should be preferred). The FNV page has a few suggestions to extend FNV hashing to bit lengths that are not a power of two, but I did not bother with them. (Maybe somebody else can try!) It should not be too hard to modify the code to do FNV-1a hashing.

## Quick-and-dirty numerical inversion of the Laplace transform

May 4, 2012

I present here a set of Mathematica routines implementing a quick-and-dirty method for numerically evaluating the inverse Laplace transform. As it is well-known, the problem is a numerically ill-conditioned one, and thus checking the sensibility of your results is very much important here. The value of the following method, however, compared to more elaborate transform inversion methods, is that if one wants a quick estimate for arguments not too far from the imaginary axis, the approximation derived from this method does not take too much effort to evaluate.

The method, due to Herbert Salzer, is a complex-valued $n$-point Gaussian quadrature, where the nodes are the (complex!) roots of (a special case of) the Bessel polynomial. Due to this, if you want to use this method, the function you wish to transform should be able to evaluate complex arguments.

The method below does not give an error estimate; what you can try for verifying the accuracy of your results is to evaluate for an initial number of points (the default is ten points in the routine below), and then increase the number of points taken, and then compare the results you have. (You might also wish to use a sufficiently high value of the option WorkingPrecision.)

Here are the the Mathematica routines:

SalzerRuleData[n_Integer, prec_:MachinePrecision] := Block[{x}, Transpose[Map[{#, (-1)^(n + 1) # ((2 n - 1)/HypergeometricPFQ[{1 - n, n - 1}, {}, #])^2/n} &, x /. NSolve[HypergeometricPFQ[{-n, n}, {}, x], x, prec]]]]

Options[NInverseLaplaceTransform] = {Points -> 10, WorkingPrecision -> MachinePrecision}; NInverseLaplaceTransform[f_, s_, t_, opts:OptionsPattern[]] := Module[{xa, wt}, {xa, wt} = SalzerRuleData[OptionValue[Points], OptionValue[WorkingPrecision]]; Chop[(wt.Function[s, f][1/xa/t])/t]]

As an example, we try the inverse Laplace transform of the function $\frac1{(1+s)^2}$, which is $t\exp(-t)$:
 f[t_] = NInverseLaplaceTransform[(1 + s)^(-2), s, t];
 With[{t = N[2]}, {t Exp[-t], f[t], t Exp[-t] - f[t]}] {0.270671, 0.270671 + 5.68434*10^-14 I, 2.91759*10^-8 - 5.68434*10^-14 I}

(The tiny imaginary part is an artifact of the evaluation; you can use Chop[] if you wish.)

As another example, here is a comparison of the approximate inverse transform of $\frac1{\sqrt{1+s^2}}$ with the true inverse transform, $J_0(t)$ (where $J_0(t)$ is the Bessel function of the first kind):

f[t_] = NInverseLaplaceTransform[1/Sqrt[1 + s^2], s, t, WorkingPrecision -> 25];
 GraphicsRow[{Plot[{f[t], BesselJ[0, t]}, {t, 0, 15}, Frame -> True], Plot[Re[f[t] - BesselJ[0, t]], {t, 0, 15}, Frame -> True, PlotRange -> All]}]

The plot on the left shows that the true function and the approximation are almost indistinguishable within the range plotted. The plot on the right shows the difference between the approximate and the true function.

## Bunnies, teapots, and noise

March 16, 2012

I should be able to have regular Internet access soon. For now, I’d like to share some of the experiments I did with coloring 3D objects with the help of Ken Perlin’s simplex noise in Mathematica. (Relatedly, I’ve found that the implementation of improved Perlin noise supplied in the Mathematica help file has a few subtle flaws, which I was able to fix. That would be for another blog entry, though.)

Here are some of the results of coloring the Utah teapot with simplex noise:

Here are some results of coloring the Stanford bunny with simplex noise:

I never would have thought that Mathematica would ever be able to do graphics as fancy as this. I’ll only note that none of these used the Texture[] construct; both models in fact used simplex noise as solid textures (i.e. an RGBColor[]` associated with each point in the model).

I might write a (series of) more descriptive post(s) on these things much later; for now, these pictures, entirely generated in Mathematica (no post-processing at all!) will have to suffice.