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.
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).
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.
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.
This post serves as a follow-up article to “Units Groups and the Infinitude of Primes”, and looks at generalizations of those results to a wide class of commutative rings.
This post concerns generalizations of Euclid’s proof of the infinitude of primes to number fields, where it can be viewed as a relationship between the size of the units group and the number of prime ideals in a ring.
The full list of imaginary quadratic fields with unique factorization has been determined, but it remains unknown whether or not infinitely many real quadratic fields are UFDs. Here, we develop a simple criterion for disproving unique factorization based on continued fractions and use this to give upper bounds.
