May 1st
Today I learned a heuristic for the fact there are no global holomorphic $1$-forms on (say) the Riemann sphere. (continue reading...)
May 2021
May 2nd
Today I learned the proof that Berstein polynomials uniformly approximate continuous functions on $[0,1],$ from the Wikipedia page . (continue reading...)
May 3rd
Today I learned (finally) the proof that all mere propositions are sets. (continue reading...)
May 4th
Today I learned another application of Cheboratev's theorem: irreducible polynomials have on average one root modulo primes. (continue reading...)
May 5th
Today I learned the proof of Penney's nontransitive coin-flipping game. (continue reading...)
May 6th
Today I learned the definition of covariance, from the Wikipedia page . (continue reading...)
May 7th
Today I learned Tate's proof of the Riemann-Roch theorem for curves over finite fields.\todo{} (continue reading...)
May 8th
Today I learned the power iteration method to compute the largest eigenvalue of a matrix, from (say) the Wikipedia page . (continue reading...)
May 9th
Today I learned some properties of the type $S^1.$ (continue reading...)
May 10th
Today I learned another another application of Chebotarev because these are fun. (continue reading...)
May 11th
Today I learned a little about maps between totally ordered sets. (continue reading...)
May 12th
Today I learned an alternative presentation of the Dirichlet form of the Chebotarev application from a few days ago, from here . (continue reading...)
May 13th
Today I learned a little combinatorial reciprocity, from Combinatorial Reciprocity Theorems . (continue reading...)
May 14th
Today I learned a little more about doing algebra over posets, from Combinatorial Reciprocity Theorems . (continue reading...)
May 15th
Today I learned about incidence algebras taking advantage of the digraph structure of posets, from Combinatorial Reciprocity Theorems as usual. (continue reading...)
May 16th
Today I learned how to create an order-preserving bijection from $\QQ$ into $\QQ\setminus\{0\},$ which is the back-and-forth trick. (continue reading...)
May 17th
Today I learned the finish of the proof of the poset combinatorial reciprocity theorem, from Combinatorial Reciprocity Theorems . (continue reading...)
May 18th
Today I learned about the chromatic polynomial of a graph, from Combinatorial Reciprocity Theorems . (continue reading...)
May 19th
Today I learned about (undirected) graph orientations. (continue reading...)
May 20th
Today I learned some examples of the chromatic polynomial. (continue reading...)
May 21st
Today I learned the proof of a zero-free region of $\zeta(s),$ modulo some lemmas.\todo{} (continue reading...)
May 22nd
Today I learned a few things about infinite descent. (continue reading...)
May 23rd
Today I learned some examples of poset M\"obius inversion.\todo{}% Boolean lattice% number theory using Z under divides% I think the proofs are just by brute force showing they satisfy the recurrence + uniqueness (continue reading...)
May 24th
Today I learned the proof of the graph coloring combinatorial reciprocity theorem.\todo{}% prove the deletion-contraction functional equation: f_G = f_(G/e) + f_(G\e)% this is done by restricting colorings with c(v)=c(w) to an acyclic orientation in G/e% the others go to G\e; I'm pretty sure there's no overcounting% finish by induction on #E(G) (continue reading...)
May 25th
Today I learned the proof of some of the lemmas for the zero-free region.\todo{} (continue reading...)
May 26th
Today I learned the proof that there are infinitely many primes start with the string $1234,$ from here Ultimately, this is a size issue: the distance between $1234\cdot10^n$ and $1235\cdot10^n$ is potentially too large to avoid $\pi(n)\sim n/\log n.$ (continue reading...)
May 27th
Today I learned that the intersection of all prime ideals in a ring is set the of nilpotent elements, from Undergraduate Commutative Algebra. (continue reading...)
May 28th
Today I learned some background and analogues of the near-miss identity $e^\pi-\pi\approx20,$ from here . (continue reading...)
May 29th
Today I learned a heuristic reason why $\pi^2\approx10,$ from Noam Elkies . (continue reading...)
May 30th
Today I learned about Euler's continued fraction formula, from the Wikipedia page. (continue reading...)
May 31st
Today I learned a proof that $\ZZ[x]$ is a unique factorization domain, using Gauss's lemma. (continue reading...)