MathOverflow

This came up while trying to calculate the variety associated with a certain cohomology ring.
Let $k$ be an algebraically closed field of characteristic 2. Consider the following ring which we endow with a non-standard grading. $$R = \frac{k[x,y,z]}{(y^2+xz)} \qquad\text{where} \qquad |x| = 1,\; |y| = 2,\; |z| = 3$$Set $X = \operatorname{Proj...

Edit (June 2026). Since posting this question, the aggregate equidistribution statement of Q1 below has been formally verified in Lean 4 + Mathlib end-to-end. The complete proof — spectral gap, tensor factorisation, Fourier inversion, exponential discrepancy...

In graph theory, a cactus is a connected graph in which any two simple cycles have at most one vertex in common. Equivalently, it is a connected graph in which every edge belongs to at most one simple cycle.
The following is well known about the upper border for cactus graph class....

Let $d$ and $N$ be two large comparable integers, for example assume$$N,d \to \infty, \quad d/N \to \gamma \in (0,\infty).$$Let $w_1,\dotsc,w_N$ be iid from $N(0,(1/d)I_d)$ and let $f:\mathbb R \to \mathbb R$ be such that $|f(x)| \le \exp(cx^2/2)$, for...