A Neighborhood of Infinity

Introduction I want to return to something I've mentioned a couple of times in the past - the fact that applying certain type constructors performs a tensor product.

Introduction One game I like to play is decomposing code and algorithms as compositions of simpler pieces even when they already seem as simple as they could be.

Biased posteriors the hard way I was previously interested to see how die rolls in an RPG appear when conditioned on you having survived an unlikely situation.

1 We only hear about the survivors In the classic Star Trek episode Errand of Mercy, Spock computes the chance of success: CAPTAIN JAMES T.

A more stable version of this article can be found on github.

This is something I put on github years ago but I probably should have put it here.

Google have stopped supporting the Chart API so all of the mathematics notation below is missing.

Some pointers to things not in this blog One reason I haven't blogged much recently is that my tolerance for blogger.com has reached its limit and I've been too lazy to build my own platform supporting mathematics and code.

Why does water fall out of an inverted cup?

Important Note Google have stopped supporting the Chart API so all of the mathematics notation below is missing.

Introduction I've been reading a little about concentration inequalities recently.

Introduction Popular science writing about quantum mechanics leaves many people full of questions about the status of photons.

Introduction TL;DR The behaviour of a certain kind of delay component has a formal similarity to Löb's theorem which gives a way to embed part of provability logic into electronic circuits.

Introduction Sometimes you want to differentiate the expected value of something.

Introduction A popular question in mathematics is this: given a function \(f\), what is its "square root" \(g\) in the sense that \(g(g(x)) = f(x)\).

Introduction Gabriel Gonzalez has written quite a bit about the practical applications of free monads.

Attempted segue Since I first wrote about profunctors there has been quite a bit of activity in the area so I think it's about time I revisited them.

Introduction Google have stopped supporting the Chart API so all of the mathematics notation below is missing.

Introduction Programming languages and libraries for numerical work tend not to place a lot of emphasis on the types of their data.

> {-# LANGUAGE RankNTypes, MultiParamTypeClasses, TypeOperators #-} Introduction After I spoke at BayHac 2014 about free monads I was asked about cofree comonads.

Introduction There are two broad approaches to problem solving that I see frequently in mathematics and computing.

Introduction As Dan Doel points out here, the gadget Free that turns a functor into a monad is itself a kind of monad, though not the usual kind of monad we find in Haskell.

Introduction A while back I talked about the idea of reinversion of control using the continuation monad to wrest control back from an interface that only wants to call you, but doesn't want you to call them back.

Introduction Suppose we are given a function of boolean arguments that returns a boolean result.

Introduction Recall the standard cartoon sketch of the proof of Gödel's first incompleteness theorem.