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 Oct10 comment Prove that the sum of three consecutive squares, minus two is a multiple of 3 That is a clever shift compared to John's answer. Dec3 comment Independent components of trace? I don't think there is an explicit explanation, other than it is a number, and it corresponds to the 1D representation of $SO(3)$. One of the nice consequences of it being a representation, is representations don't mix with the others under the group action. This simplifies some analyses, and can allow you to block diagonalize the matrix. Now, it is not independent of the diagonal elements, but the use of the trace is effectively a change of variables. Dec3 comment Independent components of trace? @Hunter I've added an additional explanation. Let me know if it helps. Sep12 comment Pure maths vs applied @jkn as a counter-point to that, my experience with math students though was interesting. In all of the advanced calc courses, I was always way ahead of what they were doing/understood because of the physics courses, and simultaneously, the physics courses made more sense because of the math. Being ahead of the math students did not apply to real analysis as nobody was ahead in that class. Overall, though, they complemented each other very well. Sep12 comment Eigenvalues and the Characteristic Equation Additionally, if some $\lambda_i = 0$, then $A$ is not invertible, and the eigenspace corresponding to $\lambda_i$ is the null space of $A$. Aug5 comment Prove that matrix $A = U^{-1}HU$ is Hermitian I believe you meant to post this on Mathematics. This site is for the software program Mathematica, not mathematics. Jun1 comment Generating the partners in a multi-dimensional irreducible representation. Since this has received no traction here, I've asked over at mathoverflow. Jan30 comment Pythagorean Theorem Proof Without Words (request for words) Brilliant. Nice answer. Nov23 comment Compute $\mathbf v \mathbf A^{-1}\mathbf v^\top$ in a numerically stable way I found cholesky in the parallel directory: decompositions. Nov17 comment Possible bug in Mathematica? Also, Integrate doesn't do anything with an integer assumption (see the comments), so you have to watch out for them yourself. Nov17 comment Possible bug in Mathematica? Sorry, mixing messages there. $\int^\pi_{-\pi} \cos(qx) = 2$ when $q = 0$, but $0$ otherwise. But, you get the wrong answer if you integrate first and then set $q=0$. So, I was wondering if we're running into something similar with regards to $r$ in your integrals, but I didn't write it out as fully as I should have. Nov17 comment Possible bug in Mathematica? I have no idea why it ends being the exact negative. A guess is that like $\int^\pi_{-\pi} \cos(qx)$ with $q\in\mathbb{Z}$, there are multiple solutions depending on $r$, and by not fixing its value, Mathematica blithely chooses the incorrect one. Nov17 comment Possible bug in Mathematica? Some quick notes. Symbolically integrating both pieces, I get (-i r^2 + r^(3/2) (i FresnelC[Sqrt[r]] + FresnelS[Sqrt[2]]))/Pi for $\int f$ and its negative for the second integral. However, integrating them both with a definite value of $r$, I get different results. For $r=5$, the sum is $1.19-5.00i$, and for $r=6$, the sum is $3.30-9.00i$, rounded, of course. I wonder if this is similar to the problem of integrating $\cos(q x)$ with $q \in \mathbb{Z}$. Nov16 comment How to solve integral in Mathematica? Can you do me a favor and give some details as to the mathematics problem you're trying to solve? If we can understand what it is you're trying to accomplish, we may be able to provide better answers. Nov16 comment How to make Runge-Kutta for solving nonlinear ODE system in Mathematica @George, I just downloaded your notebook, and in it you're using the form \[DifferentialD]f/\[DifferentialD]t which is incorrect as DifferentialD has no meaning by itself. Instead, you're looking for \[PartialD]. Nov15 comment How to make Runge-Kutta for solving nonlinear ODE system in Mathematica @Georde, also the NDSolve plug-ins tutorial gives explicit detail of setting up an RK4 integration. Admittedly, not all of it is germane to your question, but it does lay out the algorithm for you. Nov15 comment Group under matrix multiplication As an added point, since your inverse did not have the same form as the other elements of the group, you should have been immediately suspicious of it, especially since you have already proven closure. Nov9 comment What an Hermitian power of a normal matrix say about the original matrix? @dan, your argument is circular. $(QDQ^{-1})^{*} = QDQ^{-1}$ relies on $A$ being Hermitian. If $A$ is normal, then it may have complex eigenvalues, so $D^{*} \neq D$. Nov9 comment Calculate periodicity in 1-dimensional array with noise Here's a tutorial on correlations in Mathematica. It's late here, so I don't have time to write up a complete answer. Nov8 comment How to solve integral in Mathematica? @George, my apologies for not getting back to you sooner. I have a couple of questions. Why should $H$ be diagonal? I don't see any reason, a priori, why it should be. So, I'd check to see that c[n,m], h, and g are giving you what you expect. If not, let me know what you expect them to be, and we'll see if we can get them there.