MathOverflow

Let $\mathcal{F}$ be a union-closed family with universe (the union of all sets in $\mathcal{F}$) $U(\mathcal{F}) = [m] = \{1, \ldots, m\}$, and $7 \le m \le |\mathcal{F}|/2$. Let $\mathcal{F}_X = \{A \in \mathcal{F} : A \cap \{1,2,3\} = X\}$.
We know that:...

Given a variety of groups $\mathbb{V}$, let $f_\mathbb{V}(n)$ be the smallest $k$ such that some $H\in\mathbb{V}$ with $\vert H\vert=k$ has the property that every group $G\in\mathbb{V}$ with $\vert G\vert\le n$ embeds into $H$.
For example, if $\mathbb{V}$ is the variety of all groups, then by Cayley's theorem we have $f_\mathbb{V}(n)\le n!$. More complicated (= smaller) varieties might drive $f_\mathbb{V}$ up or down - down because we have fewer groups we need to embed, but...

Consider the quartic anharmonic oscillator with Hamiltonian
$$H = \frac{1}{2}p^2 + \frac{1}{2}x^2 + \varepsilon x^4 = 1, \qquad \varepsilon \geq 0,$$
where $\omega = 1$ and $E = 1$ without loss of generality. The turning point $x_+$ satisfies $x_+^2/2 + \varepsilon x_+^4 = 1$, and the exact period is...

Let $U,V\subseteq \mathbb{C}$ be non-empty connected open subsets, let $f:\mathbb{C}\times V\to \mathbb{C}$ be a function and assume that
$f|_{U\times V}$ is holomorphic;
For each $z\in V$, the function $w\mapsto f(w,z)$ is holomorphic on $\mathbb{C}$....