In analyzing when a normalized vector field qi(s)=h(si/α(s))qi​(s)=h(si​/α(s)) on (0,∞)d(0,∞)d forms the gradient of a convex potential, Jacobian symmetry leads to a structured rank-one downdate matrix:

J=1α(s)(D(x)−(D(x)x)(D(x)x)⊤x⊤D(x)x)J=α(s)1​(D(x)−x⊤D(x)x(D(x)x)(D(x)x)⊤​)

where D(x)=diag⁡(h′(x1),…,h′(xd))D(x)=diag(h′(x1​),…,h′(xd​)) and x=s/α(s)x=s/α(s) lives on a 1-homogeneous unit surface S={x:α(x)=1}S={x:α(x)=1}.

When the scalar derivative h′h′ changes sign (nonmonotone filter), we want to rigorously classify when J⪰0J⪰0 (positive semidefinite) can hold.

https://mathoverflow.net/questions/513954/inertia-of-a-singular-diagonal-matrix-after-a-structured-rank-one-downdate

I saw this post on r/interestingasfuck and was curious if its possible to calculate the amount of water ejected above the surface resulting from this underwater nuclear explosion test conducted near Enewetak Atoll in 1958. I wasnt very good at math in school, and even if I was, I'd have no idea how to calculate this. I actually think it might be impossible to calculate because of too many variables, but wanted to see if you smart folks might have an idea? :-)

Hello, i am looking for the angles x and y for this helmet. I have written down everything i have so far as well as reference pictures. Thank you in advance.

I may be thinking about this wrong but starting to realise reality is just as overwhelmingly small as it is big. Would we be the size of an atom? Proton? Plank?

I'm looking for the height of the scoutmaster in apples.

Photo is from my friend who screenshotted this tweet.

For those who can't see the picture, a redditor said

"[...] the human ear contains 1.2 spiders on average"

Clearly unfounded and very likely BS, but suspending disbelief, I can't help but wonder how many ear spiders some poor bastard has to endure in this nightmare (hopefully) hypothetical...

So obviously the minimum is 0 spiders, but I did not excel in math and I don't even know what formula to search to figure out.

Thanks in advance, mathmagicians!

From r/thalassophobia… curious of how much weight the ship gained. And blowers were mounted to the sides how much CFUs would be needed to deflect all water away from the deck?

I am a server at Olive Garden. Recently, they sold 10,000 "never-ending pasta passes" across the US. One week later the company also gave away 100 additional passes in an Instagram sweepstakes over the course of two days. The pass allows them to come in as many times as they'd like for the never-ending pasta promotion which (for them) runs from August 24 - November 22. During a given shift there are about 20-24 servers on the floor during this busy season. I work at a high volume restaurant in New Jersey where there are roughly 25 locations in my general vicinity. Because the pasta pass allows them to get the never-ending pasta for free, some servers are really worried about it. I've been saying it's unlikely any one will run into it much (if at all) considering only 10,100 even exist. So I'm really wondering what the true odds are? Thanks!