Timeline for What is an "integrable hierarchy"? (to a mathematician)
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 23, 2017 at 6:29 | vote | accept | Saal Hardali | ||
| Nov 20, 2017 at 21:34 | comment | added | Tom Copeland | The polynomial in Ex. 3.14 is also the power sum rep of the complete homogeneous symmetric polynomial $h_3$. See the Newton Identities at Wikipedia. The power sums in terms of the $h_n $ are the Faber partition polynomials in the indeterminates $h_n$ mod signs. | |
| Nov 13, 2017 at 20:57 | comment | added | Tom Copeland | Kazarian and Lando "Combinatorial solutions to integrable hierarchies" might do the job. | |
| Nov 13, 2017 at 18:21 | comment | added | Carlo Beenakker | sorry, I don't know of such a pedestrian intro; perhaps this merits a separate question? | |
| Nov 13, 2017 at 2:43 | comment | added | Tom Copeland | Could you give a ref (freely available pdf) that explains clearly to pedestrians the computations inherent in Example 3.14? | |
| Nov 12, 2017 at 21:48 | comment | added | Tom Copeland | As I thought, so the first o.g.f. is related to the refined Lah polynomials, oeis.org/A130561, and the Example 3.14, to the refined Stirling polynomials of the first kind, a.k.a., the cycle index polynomials for the symmetric groups, oeis.org/A036039. | |
| Nov 12, 2017 at 20:49 | comment | added | Carlo Beenakker | @TomCopeland --- This is a mistake, arising from two different definitions of Schur polynomials; they are denoted $S_n$ and $s_n$ in this paper, page 12. Shapiro defines $S_n$ on page 32, but then in example 3.14 he should have used $s_n$. If you start from the polynomial $S_n$ you should replace $p_k$ by $p_k/k$ to get $s_n$. | |
| Nov 12, 2017 at 18:18 | comment | added | Tom Copeland | The definition of the elementary Schur polynomials using the o.g.f. on pg. 32 in Shapiro's introductory paper and the specific polynomial in Example 3.14 don't agree. Correct interpretation? | |
| Nov 11, 2017 at 16:27 | history | edited | Carlo Beenakker | CC BY-SA 3.0 |
added 98 characters in body
|
| Nov 11, 2017 at 16:25 | comment | added | Carlo Beenakker | This would be an integrable system for a finite number of degrees of freedom, but the concept of an integrable hierarchy is typically used in the infinite-dimensional case. | |
| Nov 11, 2017 at 16:16 | comment | added | Saal Hardali | Sounds like you're describing an integrable system. | |
| Nov 11, 2017 at 16:08 | history | edited | Carlo Beenakker | CC BY-SA 3.0 |
added 169 characters in body
|
| Nov 11, 2017 at 16:03 | history | edited | Carlo Beenakker | CC BY-SA 3.0 |
added 169 characters in body
|
| Nov 11, 2017 at 15:38 | history | answered | Carlo Beenakker | CC BY-SA 3.0 |