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Results tagged with integrable-systems
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user 11260
3
votes
Why the Riccati equation $\frac{\mathrm{d} y}{\mathrm{d} x} =ax^{m}+by^{2}$ has an elementar...
For $m\neq -2$ the solution contains the Bessel function $J_{\nu}\left(\frac{2 \sqrt{a b}\, x^{1+m/2}}{m+2}\right)$ with index $\nu=\pm(m+2)^{-1}$, $\nu=-1\pm(m+2)^{-1}$, and $\nu=1\pm(m+2)^{-1}$. The …
- 188k
5
votes
Accepted
Is the logistic map $x_{n+1}=r x_n (1-x_n)$ exactly solvable for any $r$ other than $-2,2,4$?
Explicit solutions for arbitrary $r$ exist in various forms:
Logistic map: an
analytical solution (1995) represents the solution as a power of
a transfer matrix.
An explicit solution
for the logisti …
- 188k
2
votes
What is the expectation/variance of the GOE (Airy-1) point process on a partition of the rea...
According to arXiv:0909.2677 (page 9 and 14), the expectation values for GOE and GUE are the same, while the variance in the GOE is twice that in the GUE.
I also note that, according to J. Hägg's Ph. …
- 188k
16
votes
Accepted
What is an "integrable hierarchy"? (to a mathematician)
An integrable hierarchy is another name for a system of commuting Hamiltonian flows. The word "hierarchy" is used because a countably infinite number of commuting flows is obtained recursively.
[For t …
- 188k
32
votes
What is the amplituhedron?
I would think that this presentation by Jaroslav Trnka, given here in Utrecht, goes at least some way towards a mathematical definition of the amplituhedron.
To skip the physical motivation, start at …
- 188k