Notes 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 sets turned out to be of limited utility in the particular project that had inspired their exploration. However, the subject itself turned out to be richer than I had imagined. Partly for this reason and partly from a desire to be comprehensive and meticulous (though…
Read MoreThe 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, you almost certainly have encountered something called the “assumption of equal apriori probabilities”. This says that every microstate in the microcanonical ensemble has an equal probability. Putting aside whether such an assumption is philosophically or physically justified, we can ask whether the statement itself has…
Read MoreSemidirect Products, Group Extensions, Split Exact Sequences, and all that
A detailed discussion of semidirect products, groups extensions, and split exact sequences.
Read MoreThe (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 never had properly understood: the nature of integration from a topological perspective. This led to an exploration of the equivalence of de Rham and singular cohomology.
Read MoreDifferential 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.
Read MoreCardinality
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 