Search Results

Responding to qwerty's answer-as-comment: I'm going to assume that you are happy with me writing $X \times Y$, and writing ordered pairs $(x,y)$, without me having to expand these into raw set theore …

The full title of the translation appears to be "Almost independence of random variables and the capacity of a secrecy channel" and it appeared in Problems of Information Transmission 32 (1996), no. …

Let $Q$ live in $\mathbb{P}^{2n-1}$, so $Q$ defines a symmetric bilinear form on $\mathbb{C}^{2n}$, and the complex dimension of $Q$ is $2n-2$. Take a flag of isotropic subspaces $0 \subset F_1 \subse …

If a series has a well-defined Cesaro sum, then it has a well-defined Abel sum and they are equal. I think I first learned this from Hardy's Divergent Series; the proof is short enough to give here. …

I can't give you a reference, but I can give you a quick proof. There is nothing special about $Gn(n,2n)$, so I'll prove it for $Gr(k,m)$. Consider the $\binom{m}{k}$ Plucker coordinates. One of the …

I am a little nervous writing this, because it is far from my field. But it was my understanding that there is a much more basic problem in proving the binary Goldbach conjecture by analytic methods. …

At a lower level than Flajolet and Sedgewick, Chapter 9 of Concrete Mathematics (Graham, Knuth and Patashnik) is a good introduction to elementary methods.

At a lower level than Flajolet and Sedgewick, Chapter 5 of generatingfunctionology by Wilf is a good introduction to complex analytic methods. (Yes, my two answers look very similar. As usual in a big …

If you want to know about quantities which (1) have nice generating functions and (2) depend on more than one parameter, the most thorough guide will be found in the papers of Robin Pemantle. to the b …

$\def\CC{\mathbb{C}}$User "anon" points out to me that this is Proposition 8.28 in Milne's notes; see also Example 8.36 for a quasi-finite map $\CC^2 \to \CC^2$ which is not finite. The rest of my ans …

This is false! Let $$A = \begin{bmatrix} 0&0&0&1 \\ &0&1&0 \\ &&0&0 \\ &&&0 \\ \end{bmatrix}.$$ Imposing that $XA=AX$ for upper triangular $X$ gives linear equations on the $10$ entries of $X$. Solvi …

I've been thinking about this, and I want to record a few thoughts. Let $k$ be a field of characteristic $p$, let $X$ be a smooth $n$-dimensional variety, let $D$ be a Cartier divisor and let $U = X \ …

Here is a derivation of $(\ast)$ from the displayed equation $$\sum \frac{z^n}{(1-q)^n [n]_q! (1-t)^n [n]_t!} \sum_{\pi \in S_n} q^{\mathrm{maj}(\pi)} t^{\mathrm{maj}(\pi^{-1})} = \prod_{i,j \geq 0} \ …

Torsten's argument is of course beautiful, but it might be worth recording that there is also a slick combinatorial argument, in case you need to teach this to students without algebraic geometry. (Af …

Yes. The relation you are imposing between $\pi$ and $\sigma$ is called weak order -- specifically, you are saying that $\sigma \leq_W \pi$ where $\leq_W$ is weak order. Every maximal weak chain is al …