I have been non-stop struggling with this one problem where I want to show a geodesic stays in a submanifold.
It intuitively makes sense what I want to show, but I have never seen any calculations or anything for a similar problem, and so I am at a loss for how to actually show it.
does it have a geometric characterization? a popular version of this question is where the submanifold in question is the set of fixed points of an involution of the manifold.
I don't think it should be too complicated, though. I can even make it simpler. Like GL(n) as a submanifold of Diff(R^n).
With the L^2 metric.
I think the idea is that you recognize that if $\gamma$ is a geodesic with initial condition $\gamma(0)(x) = \varphi(x) := Ax$ for $A\in GL(n)$ and $\dot\gamma(0)\in T_{\varphi}Diff(\mathbb R^n)$, then you can find a corresponding velocity field $u$ such that $\partial_t u + u\partial_x u = 0$.
But I want to show that $\gamma$ continues to be in GL(n) and so it's not exactly clear how I can use this to show it (at least, to me).
Unless I can cook up what $u$ is for small t.
That equation for $u$ is just the geodesic equation, $\ddot\gamma = 0$, by the way.
Hi, a little group theory problem i got stuck all day and cant seem to be able to solve it. I have a group and i'm supposing that it is left orderable. If I know that $yhy^{-1}=h^{-1}$ and that $h^{-k}<y^{2}<h^k$ and $h^{-k}<y^{-2}<h^{k}$ with $h>1, k>0$ how can I show that $yhy^{-1}>1$? I've tried for a while to work it out with "bare hands calculation" but I think I'm missing something
p-addict, maybe something very close to a polynomial algebra. where a monomial like $x y^5 z^2$ corresponds to taking partial derivatives with respect to $x$ (once), $y$ (five times), and $z$ (two times), hopefully acting on a function space where these partial derivative operators commute.
@leslietownes ah okay, this would make sense - where i am getting this from it says multiplication is composition of operators (i should have mentioned that) so that sounds right
i'll never forget the checker at the grocery store who saw i was buying both vodka and ginger beer and said "moscow mules?" causing me to make them later, when i went in with no plans just randomly buying them together
@leslietownes I was annoyed when I went into the first Amazon Fresh store to see what was there, I bought some oranges, paid with a credit card, but did not fill out any forms or anything and got a receipt emailed to me. They connected my credit card to my email via a web purchase and used that email. I felt somewhat violated about them doing that.
If I don't give you my email address, don't use it.
@leslietownes sorry... i'm not sure i totally understand actually. if we defined it as the algebra generated by the partials, where addition and multiplication is as with polynomials, wouldn't we just get $\mathbb{R}[x_1,\dots,x_n]$? or am i missing something
@leslietownes My wife was weirded out when Ralphs started sending her tampon discounts synched with when she had been buying such things in previous months.
@P-addict e.g. in solving $p(D) f = g$, for $f$ and $g$ in some function space, knowing that $p$ is a polynomial with constant coefficients is only some of the story
you see this even in one variable, where the 'characteristic polynomial' (or characteristic equation) might tell you something about solutions to the homogeneous equation with constant coefficients, but for a non-homogeneous one, look out
i suppose i was confused because the question was asking only about the algebra itself and not its action on a function space, so i thought there was some reason it was asking about the algebra of differential operators as opposed to just asking about the $n$ variable polynomial ring
note that by e.g. clairauts theorem considerations, or whatever that is called, there is already something interesting happening in your space if you can assume that $\partial_x$ and $\partial_y$ commute as operators on it. they don't have to commute in general
what actually is a partial differential equation? is there some encompassing definition which captures the idea of an "equation using partial derivatives" in a very general sense? (probably it is not useful for actually solving PDEs, just wondering if such a general theory of PDEs exists)
there's lots of fairly general theory but i dont think there is anything resembling an all-encompassing definition. almost every paper you see has lots of hypotheses on inputs, outputs, coefficients. all examples of the theory but not sub-examples of something.
unless i'm wrong. someone go check ncatlab and see if a PDE is a co-pushout of a reverse fibration of an infinity-homotopy of actegories.
> More precisely, we choose a category of differentiable spaces and differentiable maps between them, on which there is an endofunctor that takes each space U to a notion of tangent bundle TU, which is assumed to be a vector bundle over U, and takes a map f:U→Y to df:TU→TY.
i guess you can easily axiomatize the easily axiomatizable stuff. i'm not sure how you would fit L^infty boundary data into 'differentiable spaces and differentiable maps'
but if i follow that page, i'm sure eventually i'll find out
what is the largest $n$ for which one can solve within one second a problem using an algorithm that requires $\log_{2}(n)$ operations? Each bit needs $10^{-9}$ seconds. I did it basically as follows:
$$1 (bit) \to 10^{-9}$$ $$\log_{2}(n) \to t$$
so, we have $t = 10^{-9} \times \log_{2}(n)$, so since algorithm has to run within 1 second, so we have $t=1$, so we have $10^{9} = \log_{2}(n)$??
@leslietownes as it turns out the discussion that followed my question is a spew of unnecessary information and the solution to the problem is much easier.
