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Ultrafilters in topology
Remark: To forestall empty set difficulties, whenever I talk about arbitrary sets in this post they will be non-empty. We continue our exploration of ultrafilters from the previous post. Recall that a...
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Lower bounds on off-diagonal Ramsey numbers
The goal of this post is to prove the following elementary lower bound for off-diagonal Ramsey numbers (where is fixed and we are interested in the asymptotic behavior as gets large): The proof does...
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Noncommutative probability
The traditional mathematical axiomatization of probability, due to Kolmogorov, begins with a probability space and constructs random variables as certain functions . But start doing any probability and...
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Finite noncommutative probability, the Born rule, and wave function collapse
The previous post on noncommutative probability was too long to leave much room for examples of random algebras. In this post we will describe all finite-dimensional random algebras with faithful...
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Moments, Hankel determinants, orthogonal polynomials, Motzkin paths, and...
Previously we described all finite-dimensional random algebras with faithful states. In this post we will describe states on the infinite-dimensional -algebra . Along the way we will run into and...
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Noncommutative probability and group theory
There are, roughly speaking, two kinds of algebras that can be functorially constructed from a group . The kind which is covariantly functorial is some variation on the group algebra , which is the...
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Fixed points of random permutations
The following two results are straightforward and reasonably well-known exercises in combinatorics: The number of permutations on elements with no fixed points (derangements) is approximately . The...
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Short cycles in random permutations
Previously we showed that the distribution of fixed points of a random permutation of elements behaves asymptotically (in the limit as ) like a Poisson random variable with parameter . As it turns out,...
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Small factors in random polynomials over a finite field
Previously I mentioned very briefly Granville’s The Anatomy of Integers and Permutations, which explores an analogy between prime factorizations of integers and cycle decompositions of permutations....
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Maximum entropy from Bayes’ theorem
The principle of maximum entropy asserts that when trying to determine an unknown probability distribution (for example, the distribution of possible results that occur when you toss a possibly unfair...
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