@ACuriousMind What does this passage mean? Why would the Chern class of a differentiable complex line bundle not be in $\tilde H^2(X,\Bbb Z)$, I thought the whole point was that they are
@0celouvsky It is not saying that the Chern class is not in $\tilde{H}^2$ - it's saying that there is information lost in the passage from $H^2$ to $\tilde{H}^2$, namely the torsion, and that the bundles are classified by $H^2$, not $\tilde{H}^2$, i.e. two different bundles can have the same class in $\tilde{H}^2$, but not in $H^2$.
@0celouvsky the claim is pretty simple: $c_1$ lives in $H^2(X;\Bbb Z)$. The map into real cohomology kills torsion, so you lose info. Am I missing something?
@0celouvsky This discussion is a bit pointless since it depends on what exactly you defined as $c_1$ - but the passage seems to clearly distinguish between the Chern class in $H^2$ and the real Chern class in $\tilde{H}^2$.
@ACuriousMind Earlier I said: When writing the time independent Hamiltonian $$H(t)\psi_n(t) = E_n(t)\psi_n(t).$$ As I understand, we can consider that $\psi_n(t)$ is arbitrary up to a phase factor. That implies that $\psi_n(t)e^{i \phi}$ is considered equivalent to $\psi_n(t)$ in the context of being a stationary state.
@JohnDoe The phase factors aren't functions of anything. Given a vector $\lvert \psi\rangle$, the vector $\mathrm{e}^{\mathrm{i}\phi}$, for any $\phi\in\mathbb{R}$, represents the same state.
@ACuriousMind Yes agreed, ignore the looks for a second, but for a time dependent Hamiltonian, where we consider the stationary states $\psi_{n}(t)$, we assume that these states are unique up to a phase factor which is at most dependent on $n$ and $t$?
@JohnDoe I don't know what you mean by "assuming these states are unique". The point of the phase factor is that vectors that differ by phase denote the same physical state
The phase doesn't depend on anything, you can just choose it arbitrarily
@ACuriousMind Do you maybe know what the condition of no level crossings means when referring the energy eigenvalues and egienfunctions, it usually mentioned with no degeneracy (but it is something else)?
@0celouvsky Right, and that's a totally different claim! That just says that the image $c_1\mapsto (c_1)_{\Bbb C}$ into $H^2(X;\Bbb C)$ is invariant under complex conjugation.
Just another statement about an image of $c_1$, not about $c_1$ itself ;D
@AlwaysConfused In the PSE we like questions to be potentially useful to more than one person. That's one of the reasons we have a downer on homework questions. It isn't obvious to me that a species identification question is useful to more than one person. But, different SE sites have different views on this.
I think I might be the only person interested in non-Hausdorff spacetime on all of PSE
it's fairly niche
Anonymous
@JohnRennie All the species identification question images on SE are probably found by search engines like Google Images. So the keywords in the question and the answers may be useful for people who are trying to identify similar organisms using Google Images, and they will be directed to SE when they click the link. But yeah, at the same time there are way too many species...
@JohnRennie However it is of great value to peoples who actively deals with classification and classification systems (though bit old-fashiond; sorry to say; often receives tremendous peer pressure). It is not like that only one-person meet the species; many people earlier had recorded it.
And identification is no discovery. It is just a recovery of already recorded things. As well there is place of generalization... the structural rules, that's why it is part of science. And the variations in organ-placements do have some 'meaning' to a biologist
Peaches are great if you can get them when they are perfectly ripe. Too often they are hard when you buy them and you have to let them sit around in the hope they'll ripen, which they often fail to do.
@JohnRennie Apparently there's no sound biological difference between a fruit and a vegetable (so my teacher tells me). It just depends on commercial usage. It's no crime to market apples as vegetables ;)
@ACuriousMind Ah well, I guess I made too much of a generalisation >_< But so do youHerr Mod ;) An apple "grows from a pollinated flower" but it's a FALSE FRUIT! HAH!
@Fawad Uhhh...got a more reliable source? "Excreta" normally refers to metabolic waste of an organism, and fruits are certainly not that. What does "excreta" mean to you that fruits belong among them?
> Plants can get rid of excess water by a process like transpiration and guttation. Waste products may be stored in leaves, bark, and fruits. When these dead leaves, bark, and ripe fruits fall off from the tree then waste products in them are get rid off. Waste gets stored in the fruits in the form of solid bodies called Raphides.
I tempted to suggest this is an appropriate comment on the current state of British politics, but then I suspect it is always an appropriate comment on the state of British politics.
@paracetamol yes, I finished lunch some time ago, otherwise I wouldn't be participating in the mindless babbling that currently passes for a chat room :-)
The Federal Security Service of the Russian Federation (FSB; Russian: Федеральная служба безопасности Российской Федерации (ФСБ), tr. Federal'naya sluzhba bezopasnosti Rossiyskoy Federatsii; IPA: [fʲɪdʲɪˈralʲnəjə ˈsluʐbə bʲɪzɐˈpasnəstʲɪ rɐˈsʲijskəj fʲɪdʲɪˈratsɨjɪ]) is the principal security agency of Russia and the main successor agency to the USSR's Committee of State Security (KGB). Its main responsibilities are within the country and include counter-intelligence, internal and border security, counter-terrorism, and surveillance as well as investigating some other types of grave crimes and federal...