Maximal Products of a Given Sum

Back in high school, I came across the following contest problem – “What is the largest product of a set of positive integers totaling 20?”

It’s a fun problem, so don’t rush past the spoiler tags too fast. In this post, we’ll give the solution to this problem and discuss a “continuous” version of this question. Namely, what happens when we’re allowed to include real numbers in our product?

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Revisiting the Product Rule

In differential calculus, the product rule is both simple in form and high in utility. As such, it is typically presented early on in calculus courses, and the proof given is almost always the same.

In this post, we’ll explore the merits of a second proof of the product rule using properties of the logarithm, one that I hope presents a motivated and compelling argument as to why the product rule should look the way it does.

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Formal Groups and Where to Find Them

In 1946, S. Bochner published the paper Formal Lie Groups, in which he noted that several classical theorems (due to Sophus Lie) concerning infinitesimal transformations on Lie groups continue to hold when the (convergent) power series locally representing the group law was replaced by a suitable formal analogue. It was not long before this formalism found far-reaching uses in algebraic number theory and algebraic topology.

Unfortunately, few students see more than two or three explicit (i.e. closed form) group laws before stumbling into the deep end of abstract nonsense. In this article, we’ll see in a rigorous sense why this must be the case, providing along the way a complete classification of polynomial and rational formal group laws (over any reduced ring).

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Games of Limited Information (and Topology)

In this post, we’ll look at one-player games of limited information (sometimes classified as puzzles, not games) through a topological lens, and create for each game a poset of topologies under which topologically indistinguishable points correspond to outcomes that are indiscernible in a limited-information context. Expanding this dictionary, we’ll describe a topology on the outcome space under which the “safe” or “warranted” extension of one’s limited information relates to the continuity of certain maps.

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Notes on the Chebyshev Theorem

The prime number theorem (PNT) was not proven until 1896, but a weaker form (up to constants) was established decades earlier. The earliest proof was due to Chebyshev in 1852, and his work inspired others to take up the mantle and inch towards what they thought would be a proof of the PNT. Here, we show the strength of the Chebyshev method and ask whether it had the power to prove the PNT after all.

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