## Smooth covers rescaling the symplectic form

Let $$(M, omega)$$ be a connected closed symplectic manifold of dimension $$2n$$.

Assume there exist smooth covering maps $$phi_k:Mto M$$ such that $$phi_k^* omega=sqrt(n){k}omega$$ for all $$kgeq 1$$. Is $$n=1$$ then?

## dg.differential geometry – Covers of a 3-manifold pull back a cohomology class to any algebraic multiple

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## at.algebraic topology – Smooth covers pulling back a cohomology class to any algebraic multiple

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## Holomorphic covers pulling back the volume form to any integer multiple

Let $$M$$ be a closed connected complex manifold with $$mathrm{dim}:M=n$$. Can there exist holomorphic covering maps $$phi_k:Mto M$$ for all integers $$kgeq 1$$ such that $$phi_k^*:H^n(M, mathbb{Z})to H^n(M, mathbb{Z})$$ is multiplication by $$k$$?

## ag.algebraic geometry – Open covers by ind-affine ind-schemes

Many apologies if this is totally standard! I couldn’t find it in the literature.

Background definitions:

• A presheaf $$X: textbf{Aff}^text{op} to textbf{Set}$$ is an ind-scheme if it is a filtered colimit of schemes, where we regard a scheme as a presheaf via the Yoneda embedding (and then restrict to affine schemes), and the colimit is taken in the category $$text{Psh}(textbf{Aff})$$ of presheaves (i.e. the colimit is taken “pointwise”). One can impose the condition that all of the transition maps are closed immersions, and then call the result a strict ind-scheme.

• Let’s say that $$X$$ is an ind-affine ind-scheme if it is a filtered colimit of affine schemes. (Side question: Is this the right terminology? It seems there are several distinct notions with this name in the literature.)

Context:

• It is well known that a presheaf $$X in text{Psh}(textbf{Aff})$$ is represented by a scheme if and only if it satisfies the following two conditions: 1) $$X$$ is a Zariski sheaf and, 2) There is a covering of $$X$$ by open affine subfunctors.

• Every ind-scheme is a Zariski sheaf (in fact satisfies the sheaf condition for the fpqc topology).

• The affine Grassmannian (which is an ind-scheme) admits an open cover by ind-affine ind-schemes. See, e.g. the extremely helpful paper by Timo Richarz “Basics on Affine Grassmannians”.

Question:

Is there a similar characterization for (strict or otherwise) ind-schemes? I.e. is something like the following true: a presheaf $$X$$ is an ind-scheme if and only if 1) $$X$$ is a Zariski sheaf and, 2) $$X$$ admits a covering by open subfunctors which are ind-affine ind-schemes? Is it at least sufficient? If not, is there a name for the presheaves that satisfy this property?

## I will create eye caching book covers for \$10

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## referral request – What do Sylow 2 subgroups for Schur look like that covers groups of finite simple groups?

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## How to remove Nikon strap rings with plastic covers?

I have done this on my D800, I guess they are exactly the same.

So first remove the plastic clips, just push them HARD onto the mounting post on the camera, they should just pop out.

Then remove the triangle rings, they are A LOT, so you may need a flat-head screwdriver to open the end, just twist them around like a keychain and they pop out.

I NEVER wear a strap, and these are very annoying jerking all the time …

## video – How to embed covers in MP4 without errors?

I want to add artwork to my MP4 file. How do I do this successfully / without errors?
I have tried these methods with numerous MP4 files, none worked.
If you want to replicate the bugs / errors, here are the sample files that I used for this question.

I have tried the following:

1. ffmpeg -i sample.mp4 -i sample.png -map 0 -map 1 -c copy -disposition:v:0 attached_pic sample_w_artwork.mp4

as stated by Lukas

with this error:

(mp4 @ 0000019ee4852280) Could not find tag for codec h264 in stream #0, codec not currently supported in container
Could not write header for output file #0 (incorrect codec parameters ?): Invalid argument


ffmpeg 4.2.2 (Windows compilation from Zeranoe)

1. atomicparsley sample.mp4 --artwork sample.png --overWrite

unsuccessfully; AtomicParsley

I get this message when I first run:

Started writing to temp file.
Progress: =============================================>100%|
Finished writing to temp file.


Running next time (umpteenth time) gives this:

Updating metadata...   completed.

1. Add artwork via GUI Tag Editor using both available versions: i686-w64 and x86_64-w64

unsuccessfully; create a sample.mp4.bak file, even though the software says the operation was successful

I've also tried using MP4art as suggested by a comment on another issue in this regard. It didn't include a link, so I searched and found another problem about it. A comment also recommended MP4art, with a link being dead.

I think I have used some other methods as well, but I can't think of them at the moment. It will update if I remember correctly.

Any help is really appreciated.