is it viable to see wether a limit exists as x approaches a certain value c (not infinity) to plug in (c+$\epsilon$) and (c-$\epsilon$) and see wether those are the same
@KevinDriscoll That's pretty much dead on. It's the eigenspectrum of a particular 4-by-4 Hamiltonian as a function of a control parameter.
(If it were a dispersion relation, that control parameter would be (pseudo)-momentum. In this case, it's actually time i.e. this is the (adiabatic) spectrum of H(t).)
well it's also easy to see it's impossible for $a = b$. that would imply that $2a^{n} = c^{n}$, which means that $a = \frac{c}{2^{\frac{1}{n}}}$, which can only be an integer for $n = 1$.
You drew the same picture I did. You went from the origin to the point of contact of the two spheres to the center of the outer sphere to the outermost vertex of the cube, and you said the sum of the three lengths was the diagonal of the small cube.
I should have discussed this in my multivariable course. Computing the volume of the n-ball is an exercise, but I didn't think to put this paradoxical stuff in as a postscript.
But that leads to "weird things" in QFT, so you do analytic continuation to 'imaginary time' and therefore have four spatial dimensions (Euclidean instead of Minkowski).
And one way of doing that is so-called dimensional regularization. Which basically amounts to saying "You want n=4? Okay, let's take $n=4-\epsilon$ and analytically continue to $\epsilon=0$."
Let $Y$ be a Poisson $\lambda$ random variable, and define $X=I_{[Y>0]}$. Compute $E(Y\,|\,X)$ as a function of $X$ and find $E(|Y-X|)$.
So far, I found the mass function of $X$, $$f_X(x)=\begin{cases}
P(X=0)&=e^{-\lambda}\\
P(X=1)&= 1-e^{-\lambda}
\end{cases}$$
I don't see how to f...
This paper ( arxiv.org/abs/hep-th/0401052 ) dated 2004 claims it is proven there is no odd perfect numbers, but in 2013 (at least) they are still looking. What happened?
I am currently working on the following exercise:
Let $G$ be a group of order $pq$, where $p$ and $q$ are primes and $p > q$. Prove that $G = N \rtimes H$ for some subgroups $N$ and $H$ of orders $p$ and $q$, respectively.
Now, I just proved before this that for $G$ where $|G|=pq$ for $p$, ...
@MikeMiller I am somewhat surprised that one got through even arXiv's minimal screening, as I thought one of the things they screened for were papers in blatantly wrong categories.
I mean, this is the sort of thing one would expect to find on vixra
@Bhargav If it digs x distance into the elastic wall, then F = kx is the force exerted on the car by Hooke's law. Suppose initial velocity of the car is v right before it hits the wall. Then 0^2 = v^2 - 2 (F/m) x.
There isn't anything more to it. Anyway, no, this is the mathematics chat. You'd get more help in the physics chat (h-bar)
Actually scrap that; the force exerted on the car is not constant.
Indeed, in 0^2 = v^2 - 2(F/m)x if I replace F by (0 + F)/2 that's exactly what you get. The correction comes from the fact that you average the force exerted on the car to model it with a constant acceleration.
@TobiasKildetoft I think if $f$ is a modular form on $\Bbb H^2$ then $f(z)dz$ is a PSL_2(Z)-invariant differential form. So quotienting gives you a differential form on a modular surface downstairs.
I've seen the well ordering principle proved from the induction one before @DHMO but I'm not sure if the proof can be carried out in PA and I don't have time to think about it now, sorry
(Also @DHMO sorry for going off on the proof, I had only read the most recent messages and I thought you were just looking for a proof by induction)
I usually only see Peano when a class is trying to give an intro to proofs, and want to do some axiomatic buildup but don't feel like going into /too/ much detail on ZFC
And I guess in the incompleteness theorems you're trying to formulate stuff in any system that can contain Peano, so you have to appeal there
But beyond that, you usually couple it with some notion of subsets
Well anyway, good luck, I will need to go to bed because classes and stuff
@user379685 it seems to me that there will be a lot of substitutions involved I would try to rewrite the numerator as sqrt(u^2-1) or something. Sorry this is the best I can think of. May be try posting it as a question