Timeline for Quantum dynamics on varieties and Salmon Prizes
Current License: CC BY-SA 3.0
19 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 22, 2012 at 11:44 | comment | added | John Sidles | @Theo and @Todd, on further literature review it turns out that the Salmon Prize is offered for progress in this class of conjecture, and the question now links to this prize. | |
| Feb 12, 2012 at 23:11 | comment | added | John Sidles | @Theo and @Todd (and others), the amended question has now been posted, and in particular it quotes a passage from Harris' textbook that addresses the "odd circumstance" that questions regarding bilinear algebraic varieties commonly can be answered by elementary methods, yet the analogous questions for generic multilinear varieties are deep. The amended question seeks to clarify that the state-spaces of practical concern to quantum systems engineers nowadays, and hence the mathematical tools of greatest interest, are associated to generic multilinear varieties. Thank you all very much! :) | |
| Feb 12, 2012 at 17:51 | comment | added | John Sidles | @Theo and @Todd (and others), thank you for your considerate remarks! Per Theo's request, I will clarify the question's notation and characterize more precisely the type of reference that is desired. My access to Harris' text via Google Books is (frustratingly!) partial, yet for quantum systems engineers (QSEs) the intent is captured by Harris' passage (page 100) "We should draw a fundamental and important distinction between bi- and tri- or multilinear objects." In practice, QSEs require a mathematical toolset that extends naturally to the multilinear algebraic objects of Harris' remark. | |
| Feb 12, 2012 at 2:21 | comment | added | Theo Johnson-Freyd | that statement is in any undergraduate linear algebra textbook, with all of the conditions you ask for at the top of your question. As a representation theorist, I am very aware that there is a lot to be said about the (algebraic) geometry of solutions to such equations. I assume (without having read Harris) that that's what Harris discusses, and not the (in the 21st century) essentially trivial fact that the existence of $\xi$ satisfying the LHS is controlled by the determinant of $\psi$. | |
| Feb 12, 2012 at 2:15 | comment | added | Theo Johnson-Freyd | @John: Yes, it does seem very odd. Which is why I hope I've been careful to say that it's likely I've misinterpreted your question. Indeed, your question is really asking about your Lion hypotheses, about which I can make no contributions. As written, the SHLT doesn't make any sense --- the LHS of the equation refers to vectors $\xi$ whereas the RHS does not. So I interpreted it as "Given $\psi$ a tensor with two indices ranging from $1$ to $k$, there exists $\xi$ with indices ... such that $\psi = \xi\xi$ iff $\det \psi = 0$." That was the statement I rewrote in my answer. I maintain that | |
| Feb 11, 2012 at 23:11 | comment | added | Michael Bächtold | @John: one does not need the SVD for the converse implication, it is a fundamental property of the determinant: if $\det \psi =0$ then the rows\columns of the matrix must be linearly dependent, and so span at most a k-1 dimensional subspace. So really any book on linear algebra probably proves what you call the SHLT. I think this is what Theo was saying. | |
| Feb 11, 2012 at 20:37 | comment | added | John Sidles |
Michael, your $\Rightarrow$ argument is entirely correct, and the $\Leftarrow$ implication is no harder ... e.g., the SVD of a zero-determinant $\psi$ must have at least one zero principal value, such that the SVD constructs a (non-unique) solution set of $k-1$ pairs $\{\xi_{1r},\xi_{2r}\}$ ... but such constructions are not the answer to the question asked (which was a request for references) and neither do they correspond to the answer that Theo supplied (to be sure, Theo's answer was well-reasoned, but it did not address the question asked).
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| Feb 11, 2012 at 20:03 | comment | added | Todd Trimble | I'm giving a +1 to Theo for a frivolous reason: to bring him closer to a gold reversal badge. (I haven't thought hard whether it completely answers the question, but the last sentence does seem reasonable to me.) | |
| Feb 11, 2012 at 18:38 | comment | added | Michael Bächtold | @John Sidles: if you interpret $\psi$ as a $k\times k$ matrix then the lhs of SHLT can be read as saying that each of the $k$ row vectors of $\psi$ is a linear combination of the $k-1$ vectors $\xi_{1r}$. Which is the same as saying that they must be linearly dependent or $\det \psi=0$. | |
| Feb 11, 2012 at 14:09 | comment | added | John Sidles | @Theo, I have added a technical caveat regarding linearity to the question, in the hope that you might augment your answer to address it... in which event it would be my pleasure to upvote your suggested answer. Thank you! | |
| Feb 11, 2012 at 11:20 | comment | added | John Sidles | @Theo, please let me say that I have great respect for your contributions here on MOFL. However, it seems (to me) that the answer you posted here is the correct answer to a completely different question than was asked. Doesn't it seem odd to you that "a theorem we teach in second-year calculus to every STEM major" gets an entire chapter in Harris' (high-level graduate) text? That is why it would help a great deal if your answer specified explicitly what vector spaces and endomorphisms you were referring to! You might find the question asked is subtler than your initial impression of it. | |
| Feb 11, 2012 at 4:04 | comment | added | Theo Johnson-Freyd | @John: One thing that I think is hard for the mathematicians here to process is that you keep referring to Harris Algebraic Geometry, which is a GTM and not in an easy subject. I might have misunderstood what you mean the second-hand lion theorem to mean, but my understanding of it is a theorem we teach in second-year calculus to every STEM major. (It is somewhat disappointing to hear that they so quickly forget it!) Conversely, typically 100 pages of a GTM is most of a semester of a graduate-level class. For example, I have never read Harris, nor any textbook on algebraic geometry. | |
| Feb 11, 2012 at 3:49 | comment | added | Paul Siegel | The linear algebra course doesn't even have to be that good - I put problems like that on my matrix-algebra-for-engineers exams and the decent students usually get them right. | |
| Feb 10, 2012 at 22:18 | comment | added | John Sidles | The short answer is "no." It is rather like saying, surely STEM professionals have taken a good course on thermodynamics, no? Surely STEM professionals have taken a good course on differential forms, no? Surely STEM professionals have taken a good course on complexity classes, no? An enjoyably hilarious polemic on this subject is Bill Burke's <i>samizdat</i> classic Div, Grad, Curl are Dead (ucolick.org/~burke/home.html). | |
| Feb 10, 2012 at 22:12 | comment | added | Mariano Suárez-Álvarez | Hm. But the displayed statement of Theo is something that anyone who's taken a good course on linear algebra should be able to prove... Surely STEM professionals have taken a linear algebra course, no? | |
| Feb 10, 2012 at 22:07 | comment | added | John Sidles | STEM = "Science, Technology, Engineering, and Mathematics" (en.wikipedia.org/wiki/STEM_fields) | |
| Feb 10, 2012 at 21:57 | comment | added | Mariano Suárez-Álvarez | What is a "STEM professional"? | |
| Feb 10, 2012 at 21:53 | comment | added | John Sidles | @Theo, in the main I agree with you entirely. It's true that (in your phrase) the SHLT is "completely obvious" to all STEM professionals who have worked through the first 99 pages of Joe Harris' Algebraic Geometry. Yet that is a very tiny subset of STEM professionals, and this reflects the paucity of terms like "practical, physics, engineering, dynamics, simulation" (which appear not at all in Harris). What is not obvious (to mathematicians especially) are the transformational applications of Harris' methods. And so I have added a link (to "thermodynamical relations") in this regard. | |
| Feb 10, 2012 at 21:31 | history | answered | Theo Johnson-Freyd | CC BY-SA 3.0 |