I was playing around with a model of ellipses evolving over time, and it occurred to me that one could interpret a positive vector as the eigenvalues of a positive definite matrix. We may then consider the dynamics of the eigenvalues.

Here we continue our introduction to linear dynamical systems. In Part 1, we discussed state and dynamics. In Part 2, we will discuss noise.

Challenge: Design a system where execution involves solving multiple optimization problems (not as part of a larger, more difficult problem!)

Consider the robust geometric program

Mexico City, 5am, 1935. A man enters a dirty bar. “A gin and tonic.”

May the Lord bless his land with the precious dew from heaven above and with the deep waters that lie below; with the best the sun brings forth and the finest the moon can yield; with the choicest gifts of the ancient mountains and the fruitfulness of the everlasting hills; with the best gifts of the earth and its fullness and the favor of him who dwelt in the burning bush.

“Hi everyone, I’m new here. Do you know where Unit 813 is? That’s the dwelling I’ve been assigned.”

There are some who say too much has been sacrificed in the name of human equality. Others say too little. I will point out to both sides that all humans are currently, and have always been, totally equal. Allow me to explain.

Daenerys Wins

Cultivation

Dear Leonidas,

Following your sage counsel has paid off, as usual. I’ve got all Athens excited about moving to Mars! Would you or any Spartans like to come?

Thanks again,
Pericles

It is often said that the exchange is the atomic unit of the market. You give the grocery store money in exchange for food. You give the bank money to hold in exchange for interest. You give your employer labor in exchange for income.

People of Athens, a great challenge beckons before us. To restore a shattered Earth. Ravaged by climate change. Impoverished by COVID-19. Threatened by rising authoritarianism. Restoring our world, will be a painstaking, unenviable, super-boring task.

Dear Leonidas,

It is with some embarrassment that I write to you. The situation in Athens is out of control. Everyone is trapped in their homes because of plague. A demagogue has taken over the Assembly. I have always envied the steady character of the Spartan state, though I cannot accept your conformism. I need your counsel. How can Athens be restored?

Respectfully yours,
Pericles

What is tired at dawn and rested at dusk?

Harry asked Hagrid: “Why do you love animals so much?”
Hagrid: “Humanity.”

Wood

Harry Potter and the Field of Dharma

This post is inspired by Mike Duncan’s Revolutions podcast.

Eric Wang has written a very astute article over at his blog. I recommend you read it and think it over before continuing. Welcome back! I hope you enjoy my (stream of consciousness) response.

Today we depart from our usual subject of linear dynamical systems. I wrote the linked short story as a response to an article by Ross Douthat on the state of the Catholic Church. I never got a response from him, but perhaps you’ll find it interesting.

One reason I believe that log-linear dynamical systems have so much potential as a modeling tool is that they measure the relative change in quantities rather than the absolute change. For example, suppose we’re tracking the heat of a system \(h_t\). The absolute change from time \(t\) to \(t+1\) is given by \(h_{t+1} - h_t\). The relative change is given by \(h_{t+1}/h_t\). I also call the relative change the geometric change, since it is based on ratios.

I realize that most people have never learned about linear dynamical systems. It’s a pity, because it’s a very beautiful and useful area of applied math. In the next few posts I’m going to give a crash course on linear dynamical systems, which I hope will convey some sense of why they’re important. For more information check out my advisor’s class. I’ll start here with the most fundamental of the fundamentals.

In my PhD, I studied convex optimization, which is a sub-field of numerical optimization. The essence of convex optimization is restricting what’s allowed in an optimization model, in return for getting an optimization problem that can be solved quickly and reliably, with generic frameworks like CVXPY.