MathOverflow

Conjecture. Let $p > 2$ be an odd prime, and let $a_m$ be given by the discrete sum:$$ a_m = \sum_{j=0}^{m-1} (-1)^{m+{\lfloor (j+1)/p \rfloor}+j} \binom{m-1}{j} \frac{(p(j+1))!}{p^{j+1} (j+1)!}.$$ For all $m \ge 1$, $$v_p(a_m) \ge \left \lfloor...

Lawvere's famous fixed point theorem shows that in any Cartesian-closed category with objects $X,Y$, if there is a weakly point-surjective morphism $f:X\to Y^X$ then every endomorphism $g:Y\to Y$ has a fixed point. This is an extremely general sort of...

I have the following elliptic problem:$$ \Delta u = \operatorname{div}\operatorname{div}S, $$where $S=(S_{i,j})\colon \mathbb{T}^n\to \mathbb{R}^{n\times n} $ is bounded and $\mathbb{T}^n$ is the $n$-dimensional torus. My goal is to prove that $u\in...

Lets make things simple. Suppose we have a compact metric space $(X,d)$ and then some Lipschitz partition of unity exists, say a collection $\mathcal{F}=\{f_n\}$ subordinate to some open cover $\mathcal{U}=\{U_n\}$.
Assume that the maximum Lipschitz constant in $\mathcal{F}$ is $L>0$....