MathOverflow

Let $\| z\|$ denote the distance from $z$ to the closest integer.I am looking for a reference for a lower bound for$$\min_{ B < x < 2B, x \neq \square} \| x^{1/2}\| $$for $B \gg 1$. I assume a lower bound for something like this is available...

Let $M$ be a smooth (Hausdorff, paracompact) and connected $n$-dimensional manifold, $n\in\mathbb{N}$. Given $p\in M$ and $g$ a smooth Riemannian metric on $M$, let $V^g_p\! M\subset T_pM$ be the maximal domain of the exponential map $\exp^g_p$ around...

A (finite) set-theoretic solution to the Yang-Baxter equation is a finite set $X$ with a bijection $r\colon X \times X \to X\times X$ such that $(id \times r)(r\times id)(id \times r)=(r\times id)(id \times r)(r\times id)$.
From $X$ a set-theoretic solution to the Yang-Baxter equation, we obtain an action of the Braid group $B_n$ on $n$ strands on $X^n$, where the $i$'th strand acts by $r$ on the $i$th and $i+1$st factors....

What are some examples of cocomplete categories without equalizers? And what are some examples of cocomplete categories without binary products?
They must exist, but at the moment I don't know any. There are no such categories in CatDat so far (Link), and I would like to change that. Of course, you may also list examples of complete categories without coequalizers (resp., binary coproducts)....