Last time I presented a class of agent-based models where agents hop around a graph in a stochastic way. Each vertex of the graph is some ‘state’ agents can be in, and each edge is called a ‘transition’. In these models, the probability per time of an agent making a transition and leaving some state can depend on when it arrived at that state. It can also depend on which agents are in other states that are ‘linked’ to that edge—and when those agents arrived. I’ve been trying to generalize this framework to handle processes where agents are born or die—or perhaps more generally, processes where ..read more

It looks like they’ve found protonium in the decay of a heavy particle! Protonium is made of a proton and an antiproton orbiting each other. It lasts a very short time before they annihilate each other. It’s a bit like a hydrogen atom where the electron has been replaced with an antiproton! But it’s much smaller than a hydrogen atom. And unlike a hydrogen atom, which is held together by the electric force, protonium is mainly held together by the strong nuclear force. There are various ways to make protonium. One is to make a bunch of antiprotons and mix them with protons. This was done accid ..read more

  Sometime this year, the star T Corona Borealis will go nova and become much brighter! At least that’s what a lot of astronomers think. So examine the sky between Arcturus and Vega now—and look again if you hear this event has happened. Normally this star is magnitude 10, too dim to see. When it goes nova is should reach magnitude 2 for a week—as bright as the North Star. But why do they think it will go nova this year? How could they possibly know that? It’s done this before. T Corona Borealis is a binary star with a white dwarf and a red giant. The red giant is spewing out gas. The mu ..read more

There’s a lot we don’t know. There’s a lot we can’t know. But can we at least know how much we can’t know? What fraction of mathematical statements are undecidable—that is, can be neither proved nor disproved? There are many ways to make this question precise… but it remains a bit mysterious. The best results I know appear, not in a published paper, but on MathOverflow! In 1998, the Fields-medal winning topologist Michael Freedman conjectured that the fraction of statements that are provable in Peano Arithmetic approaches zero quite rapidly as you go to longer and longer statements:   &n ..read more

The Law of Excluded Middle says that for any statement P, “P or not P” is true. Is this law true? In classical logic it is. But in intuitionistic logic it’s not. So, in intuitionistic logic we can ask what’s the probability that a randomly chosen statement obeys the Law of Excluded Middle. And the answer is “at most 2/3—or else your logic is classical”. This is a very nice new result by Benjamin Bumpus and Zoltan Kocsis: • Benjamin Bumpus, Degree of classicality, Merlin’s Notebook, 27 February 2024. Of course they had to make this more precise before proving it. Just as classical logic is desc ..read more

At first glance it’s amazing that one of the great British composers of the 1400s largely sank from view until his works were rediscovered in 1850. But the reason is not hard to find. When the Puritans took over England, they burned not only witches and heretics, but also books — and music! They hated the complex polyphonic choral music of the Catholics. So, in the history of British music, between the great polyphonists Robert Fayrfax (1465-1521) and John Taverner (1490-1545), there was a kind of gap — a silence — until the Peterhouse Partbooks were rediscovered. These were an extensive co ..read more

Last time I presented a simple, limited class of agent-based models where each agent independently hops around a graph. I wrote: Today the probability for an agent to hop from one vertex of the graph to another by going along some edge will be determined the moment the agent arrives at that vertex. It will depend only on the agent and the various edges leaving that vertex. Later I’ll want this probability to depend on other things too—like whether other agents are at some vertex or other. When we do that, we’ll need to keep updating this probability as the other agents move around. Let me tr ..read more

Andreas Werckmeister (1645–1706) was a musician and expert on the organ. Compared to Kirnberger, his life seems outwardly dull. He got his musical training from his uncles, and from the age of 19 to his death he worked as an organist in three German towns. That’s about all I know. His fame comes from the tremendous impact of his his theoretical writings. Most importantly, in his 1687 book Musikalische Temperatur he described the first ‘well tempered’ tuning systems for keyboards, where every key sounds acceptable but each has its own personality. Johann Sebastian Bach read and was influenced b ..read more

Today I’d like to start explaining an approach to stochastic time evolution for ‘state charts’, a common approach to agent based models. This is ultimately supposed to interact well with Kris Brown’s cool ideas on formulating state charts using category theory. But one step at a time! I’ll start with a very simple framework, too simple for what we need. Later I will make it fancier—unless my work today turns out to be on the wrong track. Today I’ll describe the motion of agents through a graph, where each vertex of the graph represents a possible state. Later I’ll want to generalize this, repl ..read more

Okay, let’s study Kirnberger’s three well-tempered tuning systems! I introduced them last time, but now I’ve developed a new method for drawing tuning systems, which should help us understand them better. As we’ve seen, tuning theory involves two numbers close to 1, called the Pythagorean comma (≈ 1.0136) and the syntonic comma (= 1.0125). While they’re not equal, they’re so close that practical musicians often don’t bother to distinguish them! They call both a comma. So, my new drawing style won’t distinguish the two kinds of comma. Being a mathematician, I would like to say a lot about why w ..read more