Mathematics

well i agree with that part saying $e^x\approx1+x$

@Thorgott there's a ton of algebraic analogues for that right

in what sense

I was thinking of this math.stackexchange.com/questions/2639485/…

bottom answer, very end

it's one of many great stackexchange answers that don't get much attention

I actually thanked that guy in a comment haha, he took the time to write two answers

My calculation above isn't particularly hard, it's just variance of the sum of two independent RV, and one very modest application of the itô isometry, but damn it looks ugly

substituting that $e^x\approx1+x$ makes it surprisingly concise tho

12:39 AM

ah, I guess this is saying that the germs defined by $e^x$ and $1+x$ at $0$ are zero modulo $(x^2)$

which is just the algebraic way of saying that they're equal up to second order at $0$ ig

yeh that's what I meant with algebraic analogue

yeah, I imagine this generalizes to stuff involving jets and sheaves or sth

I'd like to draw a level curve of a Gaussian surface, i.e. $f(x,y)=Ce^{-(x+y)^2/D}$. I'd like the peak of the Gaussian function to lie in the first quadrant, and be elongated (egg shaped, but converging to a ridge) and symmetric around $y=x$. Any tips on how to find such a Gaussian function? I've tried Desmos, but to no avail.

northerner

12:56 AM

A bit pandantic but when people speak of translations (i.e. vertical or horizontal) does that mean original and translated function are related, or could they just happen to differ by a constant? For example if I had f(x)=x+1 and an unrelated function g(x)=x+2 would it be correct to say g(x) is f(x) translated +1 units vertically? I guess I'm thinking in context of where the functions have a meaning, like a physical system.

TL;DR is it correct to say g(x)=x+2 is a vertical translation of f(x)=x+1 or do you need more context?

@Thorgott "Nilpotents in a ring correspond to tangent vectors. Or more generally, to jets, which are the higher order Taylor approximations."

apparently, it does

I don't understand how to make sense of that first part. What do tangent vectors have to do with nilpotents.

@Thorgott You mean the difference?

@northerner There is no vertical unless you talk about the graphs in the cartesian plane. Once you say graphs, then it's clear.

The algebraic description of tangent vectors at $p$ I know is as being naturally dual to $\mathfrak{m}_p/\mathfrak{m}_p^2$ ($\mathfrak{m}_p$ being the ideal of germs vanishing at $p$) (similar idea: you "linearize" by modding out "higher order terms"), but no nilpotents in sight.

@Ted right

@Thor: Yes, $\bar x$ is nilpotent in $k[x]/(x^2)$.

@schn Sounds like you want $(x-y)^2$ where you have $(x+y)^2$.

1:11 AM

Right, I guess an analogy is to think of nilpotents as infinitesimal elements of various orders. That's how $\mathbb{R}[\varepsilon]/(\varepsilon^2)$ is the algebraists poor model of differential calculus up to first order and I believe jets are a more sophisticated version of a similar idea.

Yes, that's right.

but how do I enter tangent vectors into these pictures

A mapping from the coordinate ring of a variety to $k[\epsilon]/(\epsilon^2)$ gives a tangent vector to the variety. Mumble Spec somewhere.

northerner

@TedShifrin ok so translations only apply when you're talking about graphs?

You need a geometric set to translate, in my book, @northerner.

1:13 AM

@TedShifrin Indeed. I’m attaching an image of what I’d like. I’d like the peak to be centered in the first quadrant where $0 \leq x,y \leq 1$.

ah, that sounds reasonable

is there an analogy for this on manifolds

Sure, Thor. Just use smooth functions and encode derivations.

Oh, so you want the max at a single point on the diagonal, @schn. Probably need something like $f(x-a)f(y-a)$ where $f$ is the single-variable Gaussian.

@TedShifrin Looks good. How would one elongate it, like make it more egg shaped?

wolframalpha.com/input/…

Now the level curves are very circular.

1:34 AM

Think about a linear map that distorts the plane as you want and compose functions.

ah, a derivation of $C_p^{\infty}(M)$ is the same thing as a ring hom $C_p^{\infty}(M)\rightarrow\mathbb{R}[\varepsilon]/(\varepsilon^2)$, which is evaluation at $p$ on the first coordinate

and ig we can think of that as mapping a germ to a possible infinitesimal first-order approximation

You truncate the germ, right?

Damn, sometimes, the internet does forget

Some 6 years ago, I took the combinatorics course by robert sedgewick on coursera and devised some ludicrous solution to a difference equation. All the data from that course are lost though, they delete all the old data after some time

I guess it is (on an arbitrary manifold, we don't quite have a Taylor expansion, but it should probably work out in coordinates?), but of course we can "trunctuate in different directions" (those being the tangent vectors)

1:50 AM

You need coordinates ultimately, Thor, but remember the trick that you can write any smooth function as $f(x)=f(0)+\sum x_ig_i(x)$.

@TedShifrin “Distorts the plane...” How? : ) Some change of coordinates?