hahaha - a blast from the past, but spot on!

Thank you Joel! Your kind words mean a lot. I wish you the same!

the only thing you care about is the ORIENTATION of the angle: the arrow that points in F(s0) goes "down" from the real axis. Instead, the angle the arrow that points to F(s1) is positive 'cause it follows the positive orientation of angles, hence it's positive even if the arrow itself is in the 3rd quadrant

@@Arty_x_g Ah - of course! That’s what I was missing! Thank you very much for taking the time to respond - kindest, Ralph

It's always nice seeing how that kind of insight and intuition can generalise - or not. Definitely a fun part of learning. Thanks for watching!

@@richard_pates here’s a fun question, can you generalize the Hamilton Cayley theorem in general using SVD?

@@joshuaiosevich3727 interesting thought! I'm a bit unsure where I'd start - the connection between the characteristic polynomial and eigenvalues rather than singular values might be tricky to get around. But I've been wrong plenty of times before, and I know there are all sorts of generalisations of the cayley hamilton theorem into other more exotic algebraic situations, so there could be something!

Thank you! I made this with manim. For all the collision detection I used the manim-physics package which makes use of pymunk I think

@@richard_pates could you kindly share the code.

Good question! The answer is yes, x* does not have to be the origin. I'll make a few extra comments though: 1. In some sense it is no real loss of generality to assume that x*=0. This is because we can always change our coordinate system through y=x-x*, and then rewrite all our dynamics in the new state variable y. In these coordinates the equilibrium point will be y*=0 2. If we have more than one equilibrium in omega, the theorem is still valid. However you will not be able to show the stronger stability condition of asymptotic stability. This is because at both the equilibrium points f(x)=0, and so dot{V}=0 at both the points, and so we cannot have dot{V}<0 everywhere in omega. This means that when you want to show asymptotic stability it is important that the region omega is chosen so that it only has one equilibrium point in it

@@richard_pates thanks for explanation. I have another question respect region Omega. It could get any form ?? For example be an open region or has an annulus form?

very good, and very difficult question! This is actually PhD level or maybe even above. The types of question are very similar. We can ask, for example, what subset of the state-space can we get to from a particular point. We would call this the reachable set from that point. However actually finding this set can be very difficult! If you are very interested, the following link might be a good place to get started. Or it may be enough to convince you that you are happy to work with linear models, even if it is just an approximation! inria.hal.science/hal-02421207/document

and if the 2-coloring method is powerful, there's a refinement of the 3 coloring method by doing this coloring instead of the other: ...123123... ...312312... ...231231... ...123123... any solvable figure should pass both colorings, refining the 3 coloring method as there are figures that pass one of the ways of coloring with 3 colors but doesn't pass the other way. I wonder if even this method is less powerful than the 2-coloring.

Is he alright?