ap.analysis of pdes – Canonical forms on higher degree Jet bundles similar to the Liouville form

On a smooth manifold of dimension $n$, the application value of the canonical $1$-form, the Liouville form on $T^*(X)$, to the Hamiltonian mechanics is well known; $T^*(X)$ is a degree $1$-Jet bundle. My question is Do canonical forms similar to the Liouville form exist on higher degree Jet bundles?
I ask this because, beyond the invariant sub-principal symbol of a pseudodifferential operator, nothing much seems to be known to handle multiple characteristic problems, especially of the non-involutive
type. I am aware of Ivrii-type Fuchsian operators, already posing great difficulties.

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file transfer – Is it possible to force Windows 10 to use higher TCP window scale?

Microsoft has some detailed information on this:

To set the receive window size to a specific value, add the TcpWindowSize value to the registry subkey specific to your version of Windows. To do so, follow these steps:

Select Start > Run, type Regedit, and then select OK.

Expand the registry subkey specific to your version of Windows:

For Windows 2000, expand the following subkey:

For Windows Server 2003, expand the following subkey:

On the Edit menu, point to New, and then select DWORD Value.

Type TcpWindowSize in the New Value box, and then press Enter

Select Modify on the Edit menu.

Type the desired window size in the Value data box.

kernel – Disable virtual console hotkeys Ctrl+Alt+F7 and higher

On Linux systems Ctrl+Alt+F1 thru F6 open virtual terminals. It seems like there are only 6 of these. Sometimes F7 may also be used for the window manager but at least on my current system (18.04) it is F1 that brings me back to the desktop.

So I have some binds, and due to historical momentum I chose F10 as the key to swap with my caps lock, and so I want to set up various key combination binds with F10, e.g. Shift+F10, Alt+F10, etc. I use these key chords heavily in terminal since it’s kind of a free source of extra keys, F10 is normally out of the way and won’t get used.

So here’s the problem, I can’t really set up a Ctrl+Alt+F10 bind because on Linux it will go to a blank virtual terminal blinking cursor screen. I’d like to know if there is a way to disable this while keeping the 1 thru 6 tty’s available for troubleshooting needs. Barring that it seems like Ctrl+Alt+F10 will be completely off-limits to me.

estimation theory – Spline Interpolation error of higher degree

It is well known that the interpolation error of a cubic spline has at best order $mathcal{O}(h^4)$ which results from polynomials of degree 3.

Can I assume that if one uses polynomials of degree p and the respective function to be interpolated $fin C^p((a,b))$, that the interpolation error of this spline is $mathcal{O}(h^{p+1})$ ?

Is this known in Literature ? (I couldn’t seem to find it.)

higher algebra – Is the rank of free module spectra unique?

$pi_0: Spto Ab$ is a direct sum preserving functor, and it sends $E_1$-ring spectra to rings, and modules over them to modules over them.

In particular you get a functor $pi_0: Mod_Rto Mod_{pi_0(R)}$. If $R^nsimeq R^m$ as $R$-modules, then $pi_0(R)^ncong pi_0(R)^m$ as $pi_0(R)$-modules. So if $pi_0(R)$ is commutative and nonzero (which is a much weaker hypothesis than $R$ having an $E_infty$-structure), this implies $n=m$.

More generally, it suffices that $pi_0(R)$ have the invariant basis property.

Note that conversely, if $pi_0(R)$ does not have the invariant basis property, we can find inverse nonsquare matrices $M,N$ with coefficients in $pi_0(R)$, and you can view them as elements of $pi_0map_R(R^n,R^m)$ ($pi_0map_R(R^m,R^n)$ respectively), and their matrix product corresponds to the composition up to homotopy, so that $R^nsimeq R^m$ as $R$-modules.

So it’s an “if and only if” situation with $pi_0(R)$.

A related claim is the fact that group-completion $K$-theory only sees $pi_0$, namely if $R$ is a ring spectrum, then the group-completion $K$-theory of projective $R$-modules (summands of $R^n$ for some finite $n$, no shifts) is the same as that of $tau_{geq 0}R$, which is the same as that of $pi_0(R)$.

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