What are the scenes & monsters on the covers?

What are the scenes being portrayed on the covers of the Player’s Handbook (PHB), Dungeon Master’s Guide (DMG) and Monster Manual (MM) and what monsters do the scenes contain?

(Click for larger images.)
Player's Handbook Cover
Dungeon Master's Guide Cover
Monster Manual Cover

usability – Appropriate use of and child-proof alternatives to screwed-down battery covers

On lots of electronic devices with replaceable batteries, the battery cover is secured by a screw.

I understand that it may be necessary to protect against child access to the battery bay, but using a screw necessitates the use of a screwdriver to open the battery cover, which can be inconvenient. It annoys me to no end that I need to open my toolbox, grab my screwdriver, and unscrew the darn thing to replace the batteries.

When is using a screw appropriate for securing a battery cover? Under what circumstances is a simple latch closure inadequate for a battery cover? Are there safety regulations that cover this?


It appears that there are legal requirements that specify that toys intended for young children must have secured battery covers. However, this question covers more than just toys, but pretty much any electronic device that has replaceable batteries. Aside from safety and regulatory requirements, when is it better to use a screw instead of a latch closure to secure the battery cover?

Also, what alternatives are there to a screwed-down battery cover that can prevent young children from accessing batteries while being more convenient than a screw closure?

Any such battery compartment must be “inaccessible” such that “it is not physically exposed by reason of a sealed casing and does not become exposed through reasonably foreseeable use and abuse of the product” (source). This may be very difficult to attain in a toolless manner, but any ideas are welcome.

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?

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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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