dg.differential geometry – Non orientable, closed manifold covered by two contractible charts

No. Recall that the Lusternik-Schnirelmann category of a space $X$, denoted $operatorname{cat}(X)$, is the minumum $k$ such that $X$ may be covered by open sets $U_0,U_1,ldots, U_k$ such that each inclusion is null-homotopic. The standard lower bound for LS-category is the cup-length of reduced cohomology: if $R$ is a commutative ring, and $x_1,ldots, x_kin tilde{H}^*(X;R)$ are cohomology classes whose cup product $x_1cdot cdots cdot x_kin tilde{H}^*(X;R)$ is non-zero, then $operatorname{cat}(X)leq k$. The proof of this is a nice exercise in the naturality of relative cup products.

Now suppose $N$ is a closed non-orientable $n$-manifold which is covered by two contractible charts $U_0$ and $U_1$. It follows that $operatorname{cat}(N)=1$ (it can’t be $0$, because closed manifolds are never contractible), and all cup products in $tilde{H}^*(N;mathbb{Z}/2)$ are trivial.

Since $N$ is non-orientable, its first Stiefel-Whitney class $w_1(N)in H^1(N;mathbb{Z}/2)$ is non-zero. Poincaré duality gives a non-singular pairing $H^1(N;mathbb{Z}/2)times H^{n-1}(N;mathbb{Z}/2)to H^n(N;mathbb{Z}/2)$, in particular $w_1(N)cdot yneq 0$ for some $yin H^{n-1}(N;mathbb{Z}/2)$. Contradiction.

dnd 5e – Is a Manual of Golems’ spell slot restriction covered by multiclassed spell slots?

There are many cases in which multiclassing spell slots can create unclear areas in the rules, but I don’t think this is one. The only requirement from the manual of golems is that you have at least two slots.

For multiclassing spellcasters slots are determined by your combined spellcaster level and do not belong to any one class.

Spell Slots. You determine your available spell slots by adding together all your levels in (spellcasting classes). Use this total to determine your spell slots by consulting the Multiclass Spellcaster table.

Looking at the table referenced, you see that a wizard 5/cleric 5 does indeed have two available 5th level slots. There is no differentiation in the rules that one of those slots is a wizard slot and one is a cleric slot. You can use those 5th level slots to cast any spell you know from either class.

Seeing that we do not need to differentiate where those slots came from (and the rules don’t differentiate them anyways), they would meet the manual of golems restriction with no caveats.

One confusing point might be that multiclassed spellcasters do indeed use their individual classes for one aspect in particular: determining what spells they know and/or can prepare.

Spells Known and Prepared. You determine what spells you know and can prepare for each class individually, as if you were a single-classed member of that class.

However, this is only the rule for determining what spells you know not how many.

So, while it is true that the cleric/wizard does not actually know/cannot prepare any 5th level spells (only upcast from lower levels), that should not matter in this case.

The manual of golems does not require a character to use or expend any 5th level spells or even spell slots to use the item. It only requires that they have them. And, as shown above, they definitely have the required number of slots.

The spell slot restriction on the manual of golems seems to be a restriction put in place to ensure that only powerful spellcasters can use it. Allowing multiclassed spellcasters to use it with combined slots not only follows the rules but also still abides by the presumed intent of the spell. Even though their spellcasting prowess is split between more than one discipline, they are still a very powerful spellcaster.

From a balance/rules perspective, multiclassing will never allow a character to be able to use this item before a single-classed full spellcaster would. In fact, many combinations will actually slow down spell progression meaning the multiclassed character would have to be a higher level in order to use it. This seems very straightforward, well within the rules, and fair.

System calls to GPL covered by linking exception?

I’d like to release a project of mine under MIT. It makes system calls to a dependency that is GPL-3. Is this covered by the linking exception?

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physics – How to calculate distance covered by the beam? (friction force problem)

Question:
A uniform beam of weight W and length L is initially in position AB. As the cable is pulled over the pulley C, the beam first slides on the floor and is then raised, with its end A still sliding. If $mu$ be the coefficient of friction between the beam and the floor, calculate the distance a that the beam will slide before it will begin to rise. Ans. $L(1-mu)$

My try:
I drew the F.B.D. like this,

Now, as the beam slides, so,
Force for which the beam slides F = $Psin 45^0-mu N$
and from equation of equilibrium,
$$Sigma F_y=0\
implies Pcos 45^0+N=W\
implies N=W-Pcos 45^0$$

$therefore F = Psin 45^0-mu (W-Pcos 45^0)\
implies frac Wgtimes a’=Psin 45^0-mu (W-Pcos 45^0)\
implies a’ = frac gW(Psin 45^0-mu (W-Pcos 45^0))$

But, I can’t understand how I can relate the distance a with the acceleration a’, as there is no final velocity given. Please anyone suggest any approach.

algebraic geometry – A K-space covered by finitely many open subspaces which are affine varieties is irreducible

I’m following the wonderful class notes (“Introduction to algebraic geometry”)(1)

In the section discussing gluing of affine varieties, there is the following exercise:

Let $(X,mathcal{O}_X)$ be a k-space that can be covered by finitely many open subspaces $U_i$ that are affine varieties, such that all intersections $U_i cap U_j$ are non-empty. Show that X is irreducible and is therefore a pre-variety

I assumed to the contrary that $X=Ycup Z $ for some $Y,Zsubsetneq X$ closed sets. Then, for any i,
$U_i=(Ycap U_i) cup (Zcap U_i)$, and that is an closed cover of an affine variety, and so, either $U_isubset Y$ or $U_i subset Z$, and here I got stuck.
I think that if I’ll show that for any i, WLOG, $U_isubset Y$ then it will be a contradiction, but I don’t see how, and if, I could reach that coclusions.

I will appreciate any leads.

Additionally, and extra materials about the process of gluing affine varieties will be highly appreciated! I can’t seem to grasp it at all.
(1): https://www.math.ru.nl/personal/bmoonen/AlgGeom/alggeom.pdf

How can I know whether some parts of a path are not covered by cellular network?

Given a path on some navigation program such as Google Maps or Waze in the United States, how can I know ahead of time (= before driving) whether some parts of the path are not covered by cellular network (i.e., not possible to make phone calls or access Internet)?

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linear algebra – For a compact set $S$ covered with a finite number of neighborhoods, why is the following inequality true?

Let $S$ be a compact set which is covered by a finite number of neighborhoods $N(a_i,r_i)$ with $i={1,2,cdots,k}$. Let $x,ynotin {S}cap{N(a_i,r_i)}$, how to show that $|x-y|gequnderset{i}{min}r_i$ ?

Any thoughts or suggestions are greatly appreciated.

google chrome – Duolingo home screen is covered by a grey modal background, disabling the page from responding to clicks

Here’s a workaround in Chrome, Firefox, and Safari.

In short, we’re going to find the code that’s creating that grey layer and delete it. You don’t need to be tech savvy.

  1. Right-click on one of the margins on the page (as close to the edge as possible), then go to “Inspect” (Chrome) or “Inspect Element” (Firefox).

    In Safari, first you’ll have to go to Preferences > Advanced and enable “Show Develop menu in menu bar”. For other browsers, just Google how to “Inspect.” It’s a pretty universal concept these days.

  2. This should open up a scary-looking panel. Here, we need to find the code, specifically what’s called a “<div> block,” causing the problem. Your panel may look very different than mine, which looks like this:

    enter image description here

    You may have to “navigate” a bit to find the line I’ve highlighted. You can click the tiny arrows pointing right and down to do this. Here’s what you’re looking for.

    <body>
        <noscript>...</noscript>
        <noscript>...</noscript>
        <div id="root" onClick>
            <div>
                <div>...</div>
                <div class="_1yGfG _160QG"> <-- Might be different codes for you.
                  ::before <-- Almost there! Keep going...
                    <div>
                        <div class="_16E8f _18rH6 _39TEz _3wo9p"></div> <-- This is the bad one.
    

    I’m betting right now it’s _16E8f for everyone, but I’m not sure. One hint you’ve found the right one is when you hover your mouse over that line, the whole page should highlight.

  3. Select the line (click on it) and hit Backspace or Delete. The grey layer should have disappeared.

Note to Duolingo developers:

I think it’s happening because of a tracking pixel failing to load in a modal. Here’s the console error I’m seeing:

Refused to execute script from 'https://t.myvisualiq.net/impression_pixel?r=3422676887&et=i&ago=212&ao=546&aca=24380054&si=1781800&ci=135195264&pi=276804369&ad=472248560&advt=4468394&chnl=-7&vndr=115&sz=6586&u=pt=i' because its MIME type ('image/gif') is not executable.

An easier solution than trying to debug this problem might be to just allowing modals to close when a user clicks anywhere in the grey region, which is the expected behavior these days anyway.