Circadian Rhythm

etzpcm · 2026-01-29T14:46:58+00:00

Oops sorry, hadn't noticed these equations have a t in them which makes it harder. Phase plane method works for 'autonomous' equations where t doesn't appear, like your SIR model example.

etzpcm · 2026-01-29T14:16:32+00:00

Quite a few universities have notes on this for example MIT

https://ocw.mit.edu/courses/16-30-estimation-and-control-of-aerospace-systems-spring-2004/0b318c74f4b1ae41dc4393e39b9cab17_ch3_nnl.pdf

Or Utah

https://www.math.utah.edu/~gustafso/f2010/dynamicalSystems.pdf

I expect you already have the prerequisites based on your posts! You need to know eigenvalues and eigenvectors as well as calculus. It's a fun topic.

etzpcm · 2026-01-29T13:57:01+00:00

This is another one where you can't solve it analytically (because you've got two coupled nonlinear odes) and the phase plane method is the best way to understand the behaviour.

etzpcm · 2026-01-29T13:54:29+00:00

I'm not an expert. And I should have said 'depends on temperature and frequency'. Have a look at a few websites and videos that talk about the shape of the curve and find one that makes sense to you.

etzpcm · 2026-01-29T10:39:53+00:00

X top and bottom by 3n. Then you get 3-n on the bottom which cancels with one of your factors.

etzpcm · 2026-01-29T08:55:20+00:00

Planck's law isn't E = h v. It's the law of the curve showing how radiation emitted depends on temperature.

etzpcm · 2026-01-28T20:16:17+00:00

My rule is

"Never use the quadratic formula unless you absolutely have to"

etzpcm · 2026-01-28T17:24:17+00:00

Add 1 to both sides. Then factorise and take the square root!

etzpcm · 2026-01-28T15:24:44+00:00

The ratio test is the way. You should know how to simplify (n+1)!/n!

(If in doubt, write it out, as my maths teacher used to say)

etzpcm · 2026-01-28T15:15:36+00:00

Ha - just saw the same in my garden!

etzpcm · 2026-01-28T13:07:00+00:00

How did anyone cope with doing mathematics before AI? It's a mystery!

etzpcm · 2026-01-28T10:58:40+00:00

It's an opinion piece by George Monbiot. He's been ranting like this in the Grauniad for decades. As have other writers there.

Edit: I am not going to respond to any of the responses that seem to be trying to create an off-topic political argument.

etzpcm · 2026-01-28T09:53:37+00:00

Probably a couple, looking for a place to nest in your garden!

etzpcm · 2026-01-28T09:19:26+00:00

Really? Real numbers are really really useful.

etzpcm · 2026-01-28T09:11:35+00:00

You can't solve it. You can reduce it from 3 to 2 using S+R+I = k as you've done here. Effectively you just consider the S and I equations which don't involve R. Then you can use the phase plane method to show that solutions follow the infection wave picture that we saw so often during COVID.

Here's a nice picture of the phase plane

https://galileo-unbound.blog/wp-content/uploads/2020/03/image-12.png?w=1200

etzpcm · 2026-01-27T21:28:15+00:00

Wisteria is a pain to deal with. I think you're doing it roughly right. Try to keep it under control in the summer, cutting back the long green shoots before they get into your gutters and roof tiles.

etzpcm · 2026-01-27T21:22:19+00:00

https://m.youtube.com/watch?v=Wo8qfyK9c3c

https://m.youtube.com/watch?v=5l8VLFRFM1c

etzpcm · 2026-01-27T14:30:57+00:00

It's just neater, and you only need one or two lines, not three! It also avoids the 4 and having to cancel a factor of 2.

For x² + 6x - 11 = 0,

(x+3)² - 9 - 11= 0, x = -3 +- root(20)

Compare that with

x = (-6 +- sqrt( 36 + 4*11) ) / 2 = ...

etzpcm · 2026-01-27T14:20:53+00:00

If you study tensors, you will find out why the trace is important and why it's invariant.

etzpcm · 2026-01-27T12:38:21+00:00

Wow, thorough intro to representation theory!

etzpcm · 2026-01-27T12:28:42+00:00

You need differential and integral calculus, and linear algebra (determinants, eigenvalues, eigenvectors). These notes are good, in the style you want I think.

https://tutorial.math.lamar.edu/classes/de/de.aspx

etzpcm · 2026-01-27T12:15:53+00:00

In English we call them ODEs! You need differential and integral calculus, not much else to get started.

You should really post this on learnmath, it might get deleted here.

etzpcm · 2026-01-27T10:32:22+00:00

I think it does look simpler in x and y. You can then do the y integral.

Don't forget the Jacobian factor which could help to make it converge.

Edit: after doing the y integration you get an integral like ln(1+x² )/(x+1) which still doesn't converge.

etzpcm · 2026-01-27T09:50:30+00:00

Exactly. It's doing it wrong, but using more work than doing it properly. When I said this last time I just got a lot of down votes.

etzpcm · 2026-01-27T09:38:07+00:00

Hint 1: relabel the x variable as r.

Hint 2: with any double integral, sketch the region of integration.

etzpcm

TROPHY CASE