Like every Sunday, the young Riemann went to Snickers (coffee shop) to read about new developments in stochastic processes, a casual pursuit of his. Having spread his papers all over the empty wooden table in a well lit corner of the ship, he couldn't help but hear that Leonie was there too. It shouldn't have been that much of a surprise, given the math department was straight around the corner, but Riemann just didn't take her for a coffee gal. She and her mater were getting louder and soon swear word fueled anxiety was governing the room. To be sure, nobody could overhear the discussion she was heaving with her mates, even when in a classroom. Today they were fighting about the right definition of a graph. As always, Leonie took the stance that you should define all object without any reference to sets. Or any notion of sets. Riemann was interested in the discussion a such, but still bothered by the fact that his sanctuary - the only place where he could hit on girls while at the same time bringing his Greeks to paper - was overtaken by the intellectual barbarians he had to endure the whole week. When he found he was starting the same page for the fourth time, he decided to stand up and go over. Five minutes after the decision he actually stood up. One minute later, he took the first step in Lenoie's direction. He was a bit puzzled what his mouth would come up with, how he'd stop these loud folks from ruining his calm but caffeinated math experience.
Like every Sunday...
sickest fuck
>It shouldn't have been that much of a surprise, given the math department was straight around the corner, but Riemann just didn't take her for a coffee gal. She and her mater were getting louder and soon swear word fueled anxiety was governing the room. To be sure, nobody could overhear the discussion she was heaving with her mates, even when in a classroom
sounds pretty comfy. "ching chong ping pong durka durka mohammad jihad" are the only sounds that govern the halls of academia in my university.
i'd like to see a young riemann try and concentrate over that noise
I feel like doing some shit writing.
Any wishes where I should take the story?
I also have a question.
Given an infinite series [math] (a_k) [/math] and the (truncated) function
[math] f_m(x) = x + \sum_{n=2}^m a_k x^k [/math]
what's the expansion of
[math] f_m(g_m^{-1}) [/math]
where [math] g_m [/math] is the inverse of the limit of [math] f_\infty [/math] truncated at m itself.
There's an explicit formula for the coefficients of the inverse function of an analytical function, so this problem can also be tackled analytically. It's work, however.
So e.g. consider the expansions of
e^x - 1
and its inverse
log(1+x)
to m'th order and concatenate those.
It will by x plus som small polynomial expression with coefficients depending on m and which goes to zero for m to infinite (as in that case, those functions are perfect inverses of one another).
There's a formula for this in terms of the coefficients of the exponential function.
>inverse of the limit of f∞ truncated at m itself.
this isn't always invertible, right? are we just assuming an inverse exists over all of $\mathbb{R}$?
The series should make for an analytic function everywhere and then, if you're interested in tackling it, you got
if that's the case, then given some y, isn't the single root of d(f_inf(x) - y)^2 / dx your inverse x?
Not sure what you're saying.
We're given a sequence (a_k) that makes for an analytical function f. It determines an inverse g with coefficients of series expansion around x=0 are given in terms of those of f (the inverse is given by the Lagrange inversion theorem). So we now have actually two series.
Now choose any m and consider both functions to be expanded to that power m. Those are two polynomials. Concatenate those. The result will be x + some polynomial with smallest coefficient m+1 and highest coefficient m^2. (See pic).
The question is what this polynomial is, as a function of (a_k) and m.
What we already know about it is that for m to infinity, it's coefficients will all go to 0. This is because of m to infinity, we get the concatenation of the analytical function and its inverse.