August 21, 2017 at 05:53PM

Today I Learned: 1) ...why complex eigenvalues in the Jacobian of a dynamical system at steady state always indicate some kind of spiral flow! In short, it's because every eigenvalue of the Jacobian is associated with an eigenvector that represents the direction in which the *direction* of flow doesn't change. When an eigenvalue has an imaginary component, its associated eigenvector *also* has an imaginary component... which means there's no physical vector on which flow doesn't change direction. The only way for that to happen is if the system flows in a spiral. 2) If you ever see reference to a value "crossing the imaginary axis", it really means a change in sign (positive to negative or vice versa). Specifically, a change in *real-valued* sign -- if you ignore the imaginary part of a value, and the value changes sign, that means it had to cross the imaginary axis. This apparently comes up from time to time in dynamical systems, where the real sign of the eigenvalues of the Jacobian at steady state tells you whether a system is stable (always tends toward a state, like a ball rolling to the bottom of a valley) or unstable (always tends to move away from a state, like a ball rolling off the top of a hill). If the aforementioned eigenvalues are negative, the steady state point is stable; if they're positive, it's unstable; if they're complex-valued, the system spirals; if they're real-valued, the system goes straight in or out. 3) Oook, so, I'm now just over 100 episodes into the podcast "The History of Rome", which is about, uh, the history of Rome. The podcast has so far covered everything from the founding of the city of Rome by Romulus (around 750 BC) to the instatement of Elagabalus as Emperor by the eastern legions (218 AD). I've now listened to the brief histories of twenty-five Roman Emperors, and a pattern has made itself clear: Every time a young man or boy becomes Emperor, they are terrible. Every. Single. Time. They have, so far, been universally incompetent, indulgent, and cruel, and every single time a young emperor is deposed, everyone in Rome seems to be happy to move on. So let this be a lesson -- DON'T PUT YOUNG MEN IN CHARGE OF EMPIRES.