@MikeMiller Actually, I computed $w(X = \Bbb{RP}^2 \times \Bbb{RP}^2)$. Say $w$ of the first factor is $1 + a + a^2$ and the second is $1 + b + b^2$. Then $w_1(X) = a + b$, $w_2(X) = a^2 + ab + b^2$, $w_3(X) = a^2b + ab^2$ and $w_4(X) = a^2b^2$. I don't see a top dimensional $w_1^iw_2^jw_3^kw_4^\ell$ which is distinct from the ones of $\Bbb{CP}^2$, curiously.
Suppose $(M,g)$ is a Riemannian manifold with affine connection $\nabla$. In order to prove that for every path $\gamma:I\to M$ and for every $V,W\in\mathcal T(\gamma)$, if
$$\partial_t\langle V,W\rangle=\langle D_tV,W\rangle+\langle V,D_tW\rangle$$
then connection is compatible with metric, for ...
@DanielFischer In this answer, why does repeated integration by parts only give the first term of an asymptotic expansion for the integral $\int_{0}^{\pi/4} \cos(xt^{2}) \tan^{2}(t) \, dt $ as $x \to \infty$? The accepted answer says that it's because the second term blows up at the $t=0$ limit. But by choosing $1- \frac{\cos(xt^{2})}{2x}$ for the antiderivative of $t \sin(xt^{2})$, the second boundary term vanishes at the $t=0$ limit.
Is the issue the size of the remainder after integrating by parts twice?
@RandomVariable I haven't analysed it completely, but I think the problem is that you then get an integral $\int_0^{\pi/4} \bigl(1 - \cos (xt^2)\bigr)F(t)\,dt$ with an $F$ that blows up quadratically at $0$. If you split the integral at $x^{-c}$, the first part is basically $Ax^{2-3c}$, and the rest is dominated by $x^{2c}$. With $c < 2/5$ you have something that blows up as $x\to\infty$, so you don't get $O(x^{-2})$ from the whole thing. More careful analysis may get the $O(x^{-3/2})$.
hello all, can anyone answer an operations research question for me?
I know the policy is not to "ask to ask" but still
I'm stuck trying to learn sensitivity analysis. How does one derive what the value of the dual price is for a given resource if I've obtained an optimal solution via simplex
@pingOfDoom, better to start by searching on the main site when it's a field that's not very common. I am not aware of anyone who frequents chat who knows OR.
Start with an element of E0 and the first prime (2).
Exp(exp(x)).
( 2 iterations of exp )
Let y1 (x) be the integral of that.
Clearly y1 is an element of E1 and not of E0.
Now take the second prime (3)
Y1(y1(y1(x)))
( 3 iterations of y1 )
Let y2 (x) be the integral of that.
Clearly y2 is an ...
I'm sorry to interrupt guys, but I have a little question which is too "small" to be posted on the site. The question is, what does $[K:F]$ stands for?, for instance a theorem says "If $\alpha$ is an algebraic number of degree $n$ then $[\mathbb Q[\alpha]:\mathbb Q]=n$"
I?m sorry if this is not the place to ask such a question
@TedShifrin Howdy! Yea I've only been by a few times at odd hours, never while you were here. Been trying to take your advice of course, get more work done by spending less time in math chat.
For real analysis, Abbott's Understanding Analysis is relatively readable, but still it'll be very tough if you haven't done proof mathematics seriously.
I taught a year-long math major course out of it, but I worked hard to put in more proofs and derivations of things. One of the things that most fascinated me was about signal processing (about band-limited exact reconstruction) ... also Kelvin angle for the angle of the wake behind a boat in a lake. Such amazing stuff.
Have a quick complex analysis question for you all. Say $f(x) = \cos{x}/(x^2 + 1)$, the nth coefficient in the Laurent (Taylor) expansion around 0 is given by the residue of $\cos{x}/(x^2 + 1) x^{-n-1}$ at 0. Can we go to the 'other side of the path
You can still define residues with an essential singularity. The problem is that $\infty$ is not an isolated singularity, so it's not an essential singularity.
Just write out the Taylor series of $\frac{\cos z}{z^2+1}$ at $0$. That's easy to do ... unless you have to get arbitrary $n$. What exactly are you trying to do?
Oh I see, how can we tell it's not isolated in this case? If I remember the definition is if $f$ is holomorphic in $\lvert z \rvert > R$ then the singularity at $\infty$ is isolated
Oh, duh, I'm sorry. I was thinking $\cos$ was in the denominator ... I'm a dope.
BUTTTTT ... remember that what makes sense is residue of a meromorphic 1-form $f(z)dz$, so at infinity you have to put in the correct transformation. You need the residue of $-\frac{f(1/z)}{z^2}$ at $0$.
But why in the world would you want to do residues at 3 points instead of 1 ??
I mean, this is a powerful thing to do sometimes ... but ...
Anyhow, it's not that bad to actually write down what the terms of $\sum (-1)^j \frac{z^{2j}}{(2j)!} \sum (-1)^k z^{2k}$ are ...
Yea, it's a bit odd. What I'ma ctually trying to do is understand very precisely what we mean when we write an asymptotic expansion ofa function, like $f(k) k \to \infty \sim \frac{a_1}{k} + \frac{a_2}{k^2}$
(And Yea I did write that down, and compute them, but maybe I made a mistake somewhere because I didnt get the same coefficients as the taylor expansion around 0)
And I realize that in the case of integer power of $k$, for example, what we're doing is a Laurent expansion
It's just a way of shifting indices from what I wrote down to get a $z^{-1}$ term. It's doing absolutely nothing different from the Taylor series expansion.
And all I'm trying to check right now is that if you have a function like $(\cos{k})/(k^2 + 1)$ that has an asymptotic expansion with cosines and not jsut powers of $k$ that you don't end up getting spurious terms by look at residues
@Ali Today I taught and am climbing. After this I'm helping a friend shop and watching Mr. Robot. Yesterday I read a paper about the cohomology of Hilbert manifolds which was pretty good, but not really a priority.
Here's the question: suppose you're given $f(k)$ and a claimed asymptotic expansion for $f(k)$ when k is large. How exactly do you go about checking that the expansion you were given is correct? If the terms int he expansion are integer powers of $k$ then it's obvious, you're just doing Laurent expansion, you can check by computing residues. But what if the expansion has other things in it, oscillating terms like this $\cos{k}$.
Yea in this problem, it's actually known how the function decays in this case, and I'm just trying to see how we can compute the coefficients of the decay, given the function $f$
Haha, yea that's what I feared. If its just $1/k^2$ or $1/k^4$ that' easy becuase I know I can extend $f$ to the complex plane and relate these coefficients to the residues at the poles somewhere in the complex plane, but this oscillaitng stuff I'm not sure if something similar is possible
I want ot relate it to residues because in general solving for the full scattering amplitude is hard. But maybe solving for the residue about some pole isn't quite so hard.
@Ali It's hard to give an inspiring answer to that sort of question anymore since most days I'm not reading or learning new things, I'm trying to prove dumb lemmas or playing with toy computations.
@Kevin: Bottom line is that you can compute a residue at infinity when there's an essential singularity, but the residue theorem only applies when you have a meromorphic 1-form, not when you have essential singularities.
The 1-form $f(z)\,dz$ does not make sense at $\infty$ with your $f(z)$, so there's no way to do what you were talking about at the very beginning.
Okay, excellent. That's what I was thinking after my simple example didn't work out and and I double-checked all the calculations. The essential singularity spoils things.
That's alright, I'm used to it. I'm forever trying to do some ridiculous thing that no mathematics person would do, and it always takes a bit to explain either because I'm failing to grasp something or because it's utterly opaque why in God's name I'd be trying to do this anyway
Anyway, is California working out for you @TedShifrin?
That's good. You picked a pretty impeccable time to relocate given the political hullabaloo. We've had signs and protests and billboards and all kinds of nonsense.
Start with an element of E0 and the first prime (2).
Exp(exp(x)).
( 2 iterations of exp )
Let y1 (x) be the integral of that.
Clearly y1 is an element of E1 and not of E0.
Now take the second prime (3)
Y1(y1(y1(x)))
( 3 iterations of y1 )
Let y2 (x) be the integral of that.
Clearly y2 is an ...
