Note: The origins of the Yang-Baxter equation is in physics, not mathematics. Let $V$ be a vector space. Let $R: V \otimes V \to V \otimes V$ be a linear operator. Consider a labelled (for notational convenience) vector space $V_1 \otimes V_2 \otimes V_3$ where $V_1 = V_2 = V_3$. Let $R_{ij} := R...

I understand the solution with calculating the distance the proton would be able to close maximally to the particle, but physically it would be impossible to change the distance from infinity to anything else, right?

I'm learning optics, and I'm struggling to understand how ray diagrams in 2D effectively generalize to real world 3D optical systems. I understand that we are considering systems that are rotationally symmetric around the principal axis. This is fine for rays that lie within a plane that contain...