Jens
Reputation
1448/1000 score
 Jan 30 comment Non Standard Eigenfunction Plots of the Laplacian Over the Unit Square The projector in the case funs[[5]] returns numerically zero, indicating that these states don't belong to the 2D irreducible representation onto which I project. But there are other (1D) representations, each with their own projection operators. You would have to construct them using the rules of group theory and apply them to each eigenstate to isolate the symmetrized components. You'll need a character table for the group and repeat what I did above - it can be automated, but I'll have to leave that to you or someone else for now. Jan 30 comment Non Standard Eigenfunction Plots of the Laplacian Over the Unit Square You can do a lot by applying group theory if there are symmetries. I give an example for the square in my answer. Jan 30 comment Non Standard Eigenfunction Plots of the Laplacian Over the Unit Square The L shape is very different from the square because the latter is invariant under the symmetry group $C_{4v}$ which causes the degeneracies. There is no such symmetry in the L. So the goal of your question isn't clear to me. You're asking about a highly symmetric example, where the degeneracies can be removed by reducing the domain. But that's not a Mathematica problem, and it's also unrelated to what you'd see in the L shape. All I can say then is: degeneracies are something you see in the spectrum and not in the wave functions. Jan 30 comment Non Standard Eigenfunction Plots of the Laplacian Over the Unit Square In more general domains, you won't even see this problem because degeneracies will be very rare. So maybe we aren't really talking about the square here... what shapes are you really interested in? Jan 30 comment Non Standard Eigenfunction Plots of the Laplacian Over the Unit Square The reason is the same as what I said in my comment to your earlier Q: degeneracies can lead to eigenvectors that are arbitrary orthogonal superpositions of the symmetrized states you expected. If your new Q is about achieving this symmetry given the solutions you plotted, then there are several ways to answer it. Is that what you're asking? Jan 30 comment How can I solve a 3D heat transfer partial differential equation? What exactly is the issue you're asking about? What have you tried? Without evidence of some effort, the question is likely to be closed. Jan 30 comment Angle between two vectors in spherical coordinates Also see this Q&A for a pathological case where VectorAngle becomes inaccurate. The three answers there provide different workarounds. Jan 30 comment How to deal with complicated Gaussian integrals in Mathematica? I think it's worth mentioning that the slowness of Moment that would potentially affect this answer appears to have been fixed in Mathematica version 10.3. See also How to efficiently find moments of a multinormal distribution Jan 30 comment How to efficiently find moments of a multinormal distribution? In Mathematica version 10.3 on Mac OS X, I get a much better timing for the moments. Your example yields AbsoluteTiming of 0.004 here. It looks like some serious improvements have happened in the new version. This apparently makes my answer to How to deal with complicated gaussian integrals in Mathematica roughly as fast as yours now... Jan 29 comment Deciphering Coding Shortcuts You can also try to translate short-form expressions by wrapping them in FullForm[Hold[...]]. That displays the long form from which you can go to the help pages more easily. Jan 29 comment How best to write an exponential of differential operators? Nice comparison and combination of different approaches. The fact that polynomials are the objects of interest means that FT is overkill here (except that it's nice ad compact). THere's also this related post that you may find interesting: Defining the Moyal Product in Mathematica. Anyway, it's funny you were at UO and seem to be in Hamburg now. For me that was exactly the other way around. Jan 28 comment Draw Vectors in Spherical Shell and then Animate You may be interested in my answer here as an example of how to do it. Jan 28 comment How best to write an exponential of differential operators? Closely related, possible duplicate: Exponential of a Differential Operator. If that Q&A isn't what you need, could you please point out in the question what is different from the linked one? Jan 27 comment Numerically Solving Helmholtz over the Rectangle - Why does this code only give eigenfunctions of the form $u_{m1}$ You're welcome. The question may be closed, but I think you may be interested in this link for the visualization of the nodal lines. Perhaps you can re-direct your question if the visualization aspect is still something you want to see improved. Jan 27 comment Numerically Solving Helmholtz over the Rectangle - Why does this code only give eigenfunctions of the form $u_{m1}$ Yes, that's correct. Jan 27 comment Numerically Solving Helmholtz over the Rectangle - Why does this code only give eigenfunctions of the form $u_{m1}$ FEM doesn't use assumptions about separability. But this is what the textbook labeling scheme of the eigenfunction is based on. So the only criterion by which FEM is able to sort the solutions is their eigenvalue. Node numbers based on separability provide a different sorting scheme but not in brute-force approaches such as this. I use "energy" above, but you can omit that term. It's an interpretation only. Jan 27 comment Numerically Solving Helmholtz over the Rectangle - Why does this code only give eigenfunctions of the form $u_{m1}$ The eigenfunctions are sorted by their energy eigenvalue, and $u_{1,2}$ is higher than you probably expected. It is found as evIF[[4]]. So there seems to be no problem. Other eigenfunctions may have degeneracies that will lead to superpositions with nodal lines that aren't straight lines. So I think there is no problem with Mathematica's solutions. Jan 27 comment NDEigensystem producing imaginary eigenfrequencies for the vibrations of a cantilever The imaginary eigenvalues turn real if I set Dirichlet conditions on the entire boundary. So I guess it's got something to do with the Neumann conditions. Looks like a bug, but needs further investigation... I was able to suppress the imaginary parts in the first approach (with your Dirichlet condition) by increasing "MaxCellMeasure" -> (1/800). But they don't go away completely... Jan 26 comment HIghlight intersection of two disks It looks to me like the main issue you're after is the same as discussed in the following link: How to embed a filled Region in a Graphics? Jan 26 comment Continuously varying tube radius I did something like that here.