## 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:

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.

## Youtube theatre mode covered by search bar

I have a laptop that I like to watch youtube on while doing other things on my large monitor, but I noticed that when using the theater mode, a significant area of the video is covered by the search bar. On the big monitor it works fine, but I want to use the small one so that I have the big screen free. Is there any way to fix this? Ideally I’d prefer not to use fullscreen because I personally despise it.

## gt.geometric topology – Is there an orientable prime manifold covered by a non-prime manifold?

A manifold is called prime if whenever it is homeomorphic to a connected sum, one of the two factors is homeomorphic to a sphere.

Is there an example of a finite covering $$pi : N to M$$ of closed orientable manifolds where $$M$$ is prime and $$N$$ is not?

There are no examples in dimensions two or three. If one is willing to forgo the orientability requirement, then there are examples in dimension three. In this paper, Row constructs infinitely many topologically distinct, irreducible (and hence prime), closed 3-manifolds with the property that none of their orientable covering spaces are prime.

There are examples where $$N$$ is prime and $$M$$ is not, such as the double covering $$pi : S^1times S^2 to mathbb{RP}^3#mathbb{RP}^3$$.

## integration testing – Hypothetically if every scenario were covered by an end-to-end tests, would unit tests still have any value?

Note: I’m asking about the strategy behind unit / integration / end-to-end tests, not about classifying tests as one or the other.

More so in the past than present, it was expensive to write and run end-to-end tests for every possible scenario. Now though, for example with the increased use of test fixtures / emulators, or even lower latency / higher query limits to APIs, it’s more feasible. But of course there’s always gonna be human error. We can never be sure we thought of every scenario.

Still I’m asking anyway, hypothetically, given an oracle that tells us every possible scenario, or to put it another way, discounting scenarios other than exactly the ones with end-to-end tests, how might unit tests still be valuable?

What I’m wondering is if, unit-integration tests are kind of an “alternate approach” to thinking about tests, and that’s the advantage of writing them even when aiming for “100%” end-to-end coverage: because they might catch a scenario missed by human error. But eliminate human error in thinking up scenarios, and what do they do?

Some ideas I can come up with (but I’m hoping for even stronger answers!)

• They encourage a useful coding methodology, e.g. TDD.
• On a breaking dependency (or dependent API) change, they reveal precisely where.

## plotting – Line is covered by PlotMarkers in ListLinePlot

Suppose I plot the data and fitting line with black point and red line

``````imgPlotMarkers00 =
ListLinePlot[{Range[60] + RandomReal[{-1, 1}/2, 60], Range[60]},
PlotMarkers -> {Automatic, None}, PlotStyle -> {Black, Red},
Joined -> {False, True}]
``````

However, I found in the output figure

the black points cover up the red line. How could I move the red line to the top layer of figure? Is their a option for this?

This is a simple example. In real case I have more than one data-fit line pairs to plot and their legends to placed.

## It is unclear, because the rules as written (RAW) do not address this.

In this situation, a Floating tensioner disc it floats on a surface, and the surface suddenly becomes a hole. Unfortunately, the rules do not provide guidance on what exactly happens next, as the disc is not expected to be above a hole more than 10 feet deep.

The mechanics of the spell suggests that it was not intended for this scenario, as it normally "floats 3 feet above the ground" and "can move across uneven terrain, up or down stairs, slopes, and the like, but cannot cross a change of elevation 10 feet or more. "

The spell description does not specify the speed of how fast the disc moves, either horizontally or vertically. The disc itself is a "horizontal force plane", which may or may not mean that the plane is fixed in place.

So there are several possible outcomes, depending on the interpretation:

1. The disk is not stationary and would therefore fall off like any other object affected by gravity. It falls until it floats 3 feet on a surface, or exceeds the maximum distance of the caster.

2. The disc is affected by gravity, but "cannot cross an elevation change of 10 feet or more." If the hole is more than 10 feet deep, the disc remains in place. Otherwise, the disc falls off, as described above.

3. The disc is a magical force and therefore gravity does not apply, so it remains floating in place.

Finally, the DM would need to decide what happens.

## gn. general topology: continuous functions covered in connected \$ T_2 \$ spaces

Yes $$X$$ is a topological space, we leave $$C (X)$$ be the collection of continuous functions $$f: X a X$$. We say that $$f, g in C (X)$$ get together Yes there are $$x in X$$ with $$f (x) = g (x)$$. We say that $$D subseteq C (X)$$ is a cover for $$C (X)$$ yes for every $$f in C (X)$$ There is $$g in D$$ such that $$f$$ Y $$g$$ get together.

by $$C ( mathbb {R})$$A boring example of a cover is the collection of all the constant functions. A more interesting example is the following accounting cover: for $$k in mathbb {Z}$$ leave $$f_k$$ be defined by $$x maps to x + k$$ for all $$x in mathbb {R}$$. Leave $$c_0$$ be the constant $$0$$ function. Then $${f_k: k in mathbb {Z} } cup {c_0 }$$ is a cover for $$C ( mathbb {R})$$.

This motivates the following question: if $$kappa$$ he is an infinite cardinal and $$X$$ is a connected $$T_2$$-space with $$| X | = 2 ^ { kappa}$$, does $$C (X)$$ have a cardinality cover $$kappa$$?

## triangles: find the distance covered by a corner of the plate in a revolution around the fixed plate.

An equilateral triangular plate on the "a" side is rolling without sliding on the periphery of another identical fixed equilateral triangular plate as shown.

Find the distance covered by a corner of the plate in a revolution around the fixed plate.

I could have an approximate idea of ​​the path, but I cannot express it mathematically since I cannot find a point that is at a fixed distance during the revolution.

Can anyone suggest a way? Any help would be appreciated.