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Trial And Error.

The picture is a portrait of a famous Mathematician who was often criticized for not being rigorous enough in his new inventions.

"Anyone who has never made a mistake has never tried anything new." - Albert Einstein


8h
comment Proving that a Sturm-Liouville problem is in the limit-point/-circle case
You were able to verify square integrability with respect to $\frac{1}{x+1}$ on the infinite interval? What type of theorem did you use on the ininite interval? I'm curious. Did you look at a singular point at $\infty$? The reason I'm asking is that everything I was looking at pointed to the limit point case at $\infty$.
12h
comment Solving a particular system of differential equations
Hi, I used $X'(t)$ because it's easier to type than putting the dot above the symbol. The prime notation indicates derivative in general, whereas the dot is usually restricted to a time derivative. When I write $A\cdot B$, I do mean the dot product. You can easily write this out in component form, differentiate and see that you get $\frac{d}{dt}(X\cdot X)=\frac{dX}{dt}\cdot X + X\cdot\frac{dX}{dt}$ where $\frac{dX}{dt}$ is the vector obtained from $X$ by differentiating each of the components. So, most of these operators you can work out fairly easily, though it make take a little thought.
14h
revised Spectral theory - continuous spectrum
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16h
comment Proving that a Sturm-Liouville problem is in the limit-point/-circle case
I'm not saying I know the answers, but I wondered if you had tried the standard substitution to put the equation into the form $-f''+\dots$. Such cases are better cataloged.
16h
comment Solution of the Legendre's ODE using Frobenius Method
Legendre's Equation has regular singular points at $x=\pm 1$, which you can see because $1-x^{2}=0$ at $x=\pm$. But all other finite points are regular points because $1-x^{2}$ does not vanish anywhere else. That means everywhere else you have two linearly-independent McClaurin series solutions. At $x=0$ there are two independent power series solutions that are guaranteed to converge in $|x| < 1$. Existence and uniqueness results are classical when all coefficients are holomorphic and the the highest order coefficient does not vanish in some neighborhood.
23h
answered Spectral theory - continuous spectrum
1d
comment How to raise a matrix to the power of $13$ without boring, repetitive multiplication?
"Is There any way to calculate to avoid boring calculation?" Yes, outsource the job, which is what you did. :)
1d
comment Prove that T = I with Linear Transformations.
@Omnomnomnom, Thanks!
1d
comment Prove that T = I with Linear Transformations.
@Omnomnomnom, you did take a careful look at what I used? I used $T^{2}=I$. That happened at the second = in my equation string.
1d
comment Spectral theory - continuous spectrum
That's all you need. I'll add the example later.
1d
comment What math have I missed as an Engineeering graduate?
Here's someone's classification of top-level subjects: en.wikipedia.org/wiki/…
1d
comment Uniform continuity of the function $x(t)=e^{tA}x$
Yes, that's what I understand. So you have an unbounded group, but a vector x for which U(t)x is uniformly bounded. Seems plausible that might give you uniform continuity for t↦U(t)x in the norm topology, but I don't know.
1d
comment Differential forms and determinants
If you imagine a third vector so that you have a $3\times 3$ matrix, then you have the cofactors associated with the imaginary new vector. So, once you wedge again, you get a determinant. This is definitely related to the permutation definition of determininant, which happens naturally for forms when swapping terms like $dy\wedge dx$ to get $dx\wedge dy$ in order to collect like terms. Any multilinear anti-symmetric scalar form $f(\cdot,\cdot,\cdots,\cdot)$ on $\mathbb{R}^{n}$ is a scalar multiple of the determinant.
1d
comment Proving that a Sturm-Liouville problem is in the limit-point/-circle case
Have you tried letting $y(x)=f((1+x)^{2}/2)$ to get a new equation for $f$?
1d
comment Spectral theory - continuous spectrum
Looks very much like classical Dirac Quantum, doesn't it? :) Are you familiar with basic complex analysis? If so, I'll later add an example of how this works for $V=0$ on $[0,\infty)$, and where you get the spectral density function. These special cases are very instructive. The radial equation for Hydrogen gives you some bound states that vary, and continuous spectrum on $[0,\infty)$ for all of the possible radial equations obtained through separation of variables, and is singular at $r=0$ and $r=\infty$.
1d
revised Spectral theory - continuous spectrum
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