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Last Notes

2026-06-27 18:54:19 CEST

While France roasted and England sweltered, Scotland had some fine sunny days - a rarety. Yesterday, however, a cold fog rolled into Edinburgh.
https://media.mathstodon.xyz/media_attachments/files/116/823/106/115/243/372/original/fbe656cb4fbd326d.jpg

2025-11-21 21:59:35 CET

- reply

- Wow, that's great - thanks! So somehow
\graph [layered layout, level distance=20mm]
created the spacings of the entries within each row? I'm wondering why the space between 10000 and 01000 is so much bigger than the other spaces in that row - that's the main thing I'd want to change, along with the symmetrically large space between 01111 and 10111.
I imagine some TikZ expert could figure out the intelligent way to change the labels. I might do it a dumb way myself.

2025-11-16 12:59:47 CET

Beavers are really important.
After the huge wildfires in Oregon in 2022, a biologist went out to survey the damage. Not only were the forests blackened, thriving trout populations in the streams were gone, choked to death by ash. “I was in total shock. It just looked like devastation.”
Then he stumbled upon something even more surprising: roughly five acres of pristine greenery in an otherwise burned-out area! At the center were 8 active beaver dams.
But this was more than a refuge from the fire. While fish had disappeared upstream of these dams, the downstream water was crystal clear — and trout were thriving as though the fire had never happened! The beaver dams were acting as a water treatment plant.
[Paraphrased from this article: https://www.scientificamerican.com/article/beaver-dams-help-wildfire-ravaged-ecosystems-recover-long-after-flames-subside/]
https://media.mathstodon.xyz/media_attachments/files/115/559/251/351/143/483/original/c72a567db14206c7.webp

2025-05-31 20:04:17 CEST

- reply

- since I'm a mathematical physicist I know that when Hilbert was asked
"How is mathematics at Göttingen, now that it is free from the Jewish influence?”
he replied, “There is no mathematics in Göttingen, anymore”.
And I know very well the history of the Manhattan Project, full of smart physicists who fled the Nazis and built a bomb to make sure the Nazis didn't do it first.

2024-12-26 23:01:34 CET

Hardcore math post:
After my spectacularly popular post on Alfvén waves, there's nothing to do but write a much less popular post on the math behind them. People say magnetic field lines in a plasma can act like stretched rubber bands: if they're bent, they try to straighten out. There's a Wikipedia article on this idea, which is called 'magnetic tension':
https://en.wikipedia.org/wiki/Magnetic_tension#Mathematical_statement
and I'd like to understand it.
They start with the equations of 'ideal magnetohydrodynamics' where you have a plasma with velocity vector field 𝐯, pressure p, density ρ and electric current vector field 𝐉 in a magnetic field 𝐁. It's called 'ideal' because we're assuming the plasma conducts electricity so well that the electric field is zero.
One of these equations, called the 'Cauchy momentum equation', says the plasma accelerates due to two forces:
the gradient of pressure, or more precisely −∇p, and
the force caused by the magnetic field, which is 𝐉 × 𝐁 as usual - this is called the Lorentz force.
They then do some fun vector calculus manipulations. Assuming |𝐁| is constant everywhere, they find a term in the force whose magnitude is the curvature κ of the magnetic field line, which points to the center of curvature of the magnetic field - where you approximate the magnetic field locally as going around a circle of radius 1/κ. See the picture to understand what the hell I just said.
I think I follow the calculation, but this is a force on the plasma, right? Not the magnetic field line per se! So why do they interpret it as the field line wanting to straighten out? If we knew 𝐯 is proportional to 𝐁 that would make sense, but they don't say anything about that.
https://media.mathstodon.xyz/media_attachments/files/113/721/349/151/384/313/original/e3ad8806d7d78792.png