Complex Angles and a Review of Some Linear Algebra
These notes started as a simple exploration of the meaning of complex angles. However, they quickly blossomed beyond recognition and required separation into at least three parts. Two of these parts will cover the general computation of angles between subspaces of R^n and the realification/complexification of vector spaces. These are nearly complete, but need some…
Keep readingThe Zeroth Homotopy Group
A discussion of the zeroth homotopy group and its application to the physics groups we care about.
Keep readingNotes on Partitions of Sets
In the course of another project, I realized that I need a better understanding of partitions of sets. In particular, I was curious why one doesn’t speak of ‘morphisms’ or ‘isomorphisms’ of these. I figured that there was some simple explanation, and that these notions were either untenable or trivial. As it happens, partitions of…
Keep readingThe Assumption of Equal apriori Probabilities
As a student, more years ago than I care to count, I was bothered by something we were taught in statistical mechanics. Actually, I was bothered by a lot of what we were taught in stat mech — but I’m going to focus on one particular issue in this post. If you’re a physics student,…
Keep readingSemidirect Products, Group Extensions, Split Exact Sequences, and all that
A detailed discussion of semidirect products, groups extensions, and split exact sequences.
Keep readingWhat happens when you iterate Bayesian Inference with the same data set?
I’ve recently been reviewing Bayesian networks with an eye to learning STAN. One question which occurred to me is the following. Suppose we are interested in the probability distribution 
Be Careful Interpreting Covid-19 Rapid Home Test Results
Now that Covid-19 rapid home tests are widely available, it is important to consider how to interpret their results. In particular, I’m going to address two common misconceptions. To keep things grounded, let’s use some actual data. We’ll assume a false positive rate of 1% and a false negative rate of 35%. These numbers are…
Keep readingFun with Voting in Cambridge, MA
My city of Cambridge, MA is one of a few municipalities which employs ranked choice voting for City Council elections. Unlike most cities, the Mayor is chosen by the City Council and is largely a ceremonial position. Most real power resides with the City Manager, who is appointed for an indefinite term by the City…
Keep readingThe (quasi)-Duality of the Lie Derivative and Exterior Derivative
A short set of notes that arose out of an enigmatic comment I encountered, to the effect that the Lie and exterior derivatives were almost-dual in some sense. I wanted to ferret out what this meant, which turned out to be more involved than anticipated. Along the way, I decided to explore something else I…
Keep readingThe Truth about Stock Prices: 12 Myths
No-fee trading has invited a huge influx of people new to trading. In this article, I discuss the basics of “price formation,” the mechanism by which stock prices are determined. This is a very gentle introduction to market microstructure, and hopefully will be of some small help to new traders who wish to better…
Keep readingTwo-Envelope Problems
In this piece we play with two incredibly counterintuitive puzzles involving a pair of envelopes which contain unknown amounts of money. First we describe a famous fallacy when it is known that one envelope contains double the money in the other. Then we discuss an actual mechanism (probably first discovered by Thomas Cover…
Keep readingDifferential Entropy
A discussion of some of the subtleties of differential entropy. This also contains a review of discrete entropy, various entropy-related information quantities such as mutual information, and a listing of various axiomatic formulations.
Keep readingThe Optics of Camera Lens Stacks (Analysis)
I examine here the behavior of stacked and reversed camera lenses for Macro photography. This paper is mathematical, while in another post I link to code which performs the calculations for us.
Keep readingCardinality
A compilation of useful results involving cardinal numbers (small ones, not huge ones) and arithmetic, along with the cardinalities of certain useful sets. There’s also a small section on bases of infinite-dimensional vector spaces. Proofs and justifications for many of the results are included in an appendix.
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A Place of Sand 