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I'm the little trashcan that could!


15h
comment How can we plot the complex roots of an equation?
This is neat, and I think it answers the OP's question better than my answer.
15h
revised How can we plot the complex roots of an equation?
deleted 45 characters in body
15h
comment How can we plot the complex roots of an equation?
@RahulNarain: Yup, you're right, good catch. I fixed it by adding an Evaluate to the definition of expression.
1d
comment Optimization in MMA for time resolved spectroscopy
Yup, that's the article I sent to you 8 months ago LOL :) But yeah, that's sort of what I was thinking. Dunno if it will be useful or not, but I can certainly help you determine whether it could be applicable for your project. However, I don't think you'll be able to do a 104-parameter fit (maybe 20 maximum?), unless your data is very good.
1d
comment Optimization in MMA for time resolved spectroscopy
If your data is good enough to do this, then this will afford you many, many orders of magnitude computational speedup, since it essentially converts a volumetrically-enormous 104-dimensional problem into 104 separate 1-D problems.
1d
comment Optimization in MMA for time resolved spectroscopy
In other words, because the 2D spectrum has the form $$\psi(\lambda,t)=\sum_{j=1}^{N_\text{comp}}c_j(t)\epsilon_j(\lambda),$$ the 2D dataset will admit a rank-1 decomposition $$D=\sum_{j=1}^N\sigma_j\mathbf{u}_j\otimes\mathbf{v}_j$$ which is precisely the singular value decomposition of the dataset. By selecting the $N_\text{comp}$ largest principle components, you can directly determine the $c_j(t)$ and $\epsilon_j(\lambda)$ and bypass the fitting process altogether. Then you can just deconvolve the IRF from the $c_j$, and do $N_\text{comp}$ 1-dimensional simple-exponential fits.
1d
comment Optimization in MMA for time resolved spectroscopy
This should scale well and has the advantage of being model independent and thus avoids fitting altogether, so you don't have to worry about searching though a 104-dimensional parameter space, and should execute in a couple milliseconds. However, that requires the full 2D ($\lambda$ vs time) dataset. Could you post that, if you happen to have it?
1d
comment Optimization in MMA for time resolved spectroscopy
Ok, so one more question: you say that you have multiple datasets (which correspond to multiple wavelengths as detected via time-resolved spectroscopy), but the dataset you linked is only a single wavelength. Do you have more datasets at more wavelengths, so that you can get a 2-dimensional picture of the decay associated spectra over multiple wavelengths? If so, then due to the separability of the DAS model, the 2D data should be decomposable into a sum of rank-1 matrices.
1d
comment Optimization in MMA for time resolved spectroscopy
@yashar: Oh ok, so this is sort of like your question about decay associated spectra.
1d
comment Optimization in MMA for time resolved spectroscopy
One question: you say "The common ones are lifetimes of the decay of detected signals and those which are not shared are the amplitudes", which implies that $\tau_1=\tau_2=\tau_3:=\tau_\text{eff}$. However, that seems sort of nonsensical from a fitting perspective, since your model can then be simplified to $m(t)=(a_1+a_2+a_3)\exp(-t/\tau_\text{eff})=a_\text{eff}\exp(-t/\tau_\text{eff})‌​$, which means that your model has two redundant degrees of freedom. Am I missing something?
1d
awarded  Nice Answer
Oct
26
revised Trouble with Fourier transform of Exp[-Sqrt[x]]
added 33 characters in body
Oct
26
comment Trouble with Fourier transform of Exp[-Sqrt[x]]
@bullwinkle: They're spherical Bessel functions $j_n(z)$, not ordinary Bessel functions $J_n(z)$ (see the article I hyperlinked in my answer). In this case $j_0(z)=\text{sinc}(z)=\sin(z)/z$. I used 4 \[Pi] Integrate[(E^-r r Sin[k r])/k, {r, 0, \[Infinity]}, Assumptions -> k > 0].
Oct
26
comment Poets of the 19th century
@Jens: 'Ode to a large shotgun' immediately springs to mind.
Oct
26
comment Trouble with Fourier transform of Exp[-Sqrt[x]]
@bullwinkle: One minor note: the Fourier transform definition in the linked article is nonunitary, whereas Mathematica's definition of FourierTransform is unitary, so you should include an extra factor of $(2\pi)^{-3/2}$ (I think).
Oct
26
answered Trouble with Fourier transform of Exp[-Sqrt[x]]
Oct
25
comment Trouble with Fourier transform of Exp[-Sqrt[x]]
I'd like to point out that a scaling of variables allows you to reduce this to the Fourier-Bessel Transform of a decaying exponential, which can then be converted into the reciprocal space profile by a scaling. I'll post this as a partial answer in a while.
Oct
25
comment Trouble with Fourier transform of Exp[-Sqrt[x]]
I also have had some problems (using Mathematica 10.0.1) on this and related examples. I'll post either an answer to this, or a separate question when I have time to look further.
Oct
23
comment Why does integration of a radical times HeavisideTheta give a conditional expression?
I tried it on Mathematica 10.0.1 and I correctly get (2 y^(3/2))/3 for the first three integrals (and as you probably know already, the fourth one is correct), so this seems like it could be version-specific. Can anyone else confirm this?
Oct
23
comment How can we plot the complex roots of an equation?
@Mr.Wizard: a link to the code to make the second picture is here.