The birational geometry thing is interesting, because birational CY 3-folds will have the same Hodge numbers. So I guess if there are birational CY 3-folds one shouldn't expect a unique mirror.
You may as well be doing homological mirror synmetry, since I think there's a recet series of papers that show homological mirror symmetry implies classical mirror synmetry
Applying for stuff made me think about recletters. I hope profs dont base them on things you said at exams. "The candidate does not know the Leibniz rule and displays significant difficulty adding one-digit integers" won't look to good.
first one: "For an arbitrary semisimple Frobenius manifold we construct an integrable hierarchy of Hamiltonian partial differential equations, which we call the Hodge hierarchy. In the particular case of quantum cohomology the tau-function of a solution to the Hodge hierarchy generates the intersection numbers of the Gromov--Witten classes and their descendents along with the characteristic classes of Hodge bundles on the moduli spaces of stable maps.
For the one-dimensional Frobenius manifold the Hodge hierarchy is a deformation of the Korteweg-de Vries hierarchy depending on an infinite number of parameters. We conjecture that this hierarchy is a universal object in the class of scalar Hamiltonian integrable hierarchies possessing tau-functions."
and the other: "We consider the problem of classification of hierarchies of bihamiltonian integrable PDEs that are closely related to the theory of Gromov-Witten invariants and 2D topological field theory, and show how to solve this problem by computing the bihamiltonian cohomologies associated to semisimple bihamiltonian structures of hydrodynamic type."
A friend of mine reviewed a paper once and wrote something along the lines of "the author uses the notation $g_{it}$, which I find needlessly offensive."
One of the integrals I came across these days (during my studies) is $$\int _0^1\int _0^1\text{Li}_2(x y) \text{Li}_2((1-y) x) \ dx \ dy$$
that can be turned into a series, or can be approached by using the integration by parts, but these
ways do not look like as a promising way to go, or I migh...
@JulianRachman Well, the example given is indeed an infinite ordered set
@ADG Well, there is no reason to look for linear dependence of the vectors. Just write up the matrix you did and see what you get. If you do get the identity, then indeed the fourth vector cannot be written as a linear combination of the other three
@JulianRachman But actually, it is even easier to see this. Just take any two wqo which are not isomorphic but have the same cardinality. Clearly they cannot both be isomorphic to the free monoid with the subword order
@ADG Usually I would use that phrase to mean "find additional vectors such that you get a basis". But clearly that requires the original vectors to be linearly independent
But no, $A$ and $A^*$ are also not isomorphic under the domination ordering, as one can see by considering the example given in the answer to the MO question, where $A$ has a unique element which has infinitely many smaller elements, whereas this is not the case for $A^*$
Why is it that when I study intro to algebra, many diverse concrete examples are given immediately, whereas the introductory axiomatic set theory that I have encountered so far gives so few?
If anyone is here I'd like to ask what needed to be googled for solving limit n to inf and defined integral. It's needed to proof some equation for home home assignment. Thanks
@TobiasKildetoft I understand that $\gcd(a, b) = \prod p_i ^{\min(s_i, t_i)}$. However, I'm a bit confused as to how that gives $a^{\min(3,2)}\cdots $.
@user153330 If there would be a little chance to exist the possibility to reverse the time, I'd probably invest my whole energy, power into that, but according to the most important scientists it is not possible. All good stuff is in the past.
@I'manartist i meant to ask: do you really mean 'without pen and paper?' the only way I can imagine that's possible is if it were to vanish trivially, and mathematica doesn't support that.
@RandomVariable does that variable change you told me about works for that integral above? You said that it might work but I didn't ask you if you tried to finish it.