MathOverflow

In Cheng’s 1975 paper Eigenvalue comparison theorems and its geometric applications, he proved that for an $n$-dimensional closed Riemannian manifold with nonnegative Ricci curvature,$$\lambda_1(M) \le \frac{n^2\pi^2}{\operatorname{diam}(M)^2} \qquad...

Let $L$ denote the Rogers dilogarithm,$$L(x) = \operatorname{Li}_2(x) + \tfrac12\log(x)\log(1-x), \qquad 0<x<1,$$normalized so that $L(1)=\pi^2/6$.
Claim. Suppose$$m\,L(e_1) + L(e_2) \in \mathbf{Q}\cdot\frac{\pi^2}{6}$$is a non-trivial two-term identity with integer $m>2$, arising from such a Rogers array system, where $e_1,e_2$ are algebraic and $K=\mathbf{Q}(e_1,e_2)$. Then:...

Let $G$ be a finite group. Consider the set $\{[H]: H\in N(G)\}$, where $N(G)$ is the set of all normal subgroup of $G$. Here $[H]=\{K\in N(G): G/H \cong G/K\}$. Define an order $\le$ on the set as follows: $[ H]\le [K] \iff $ there is an injective homomorphism...

[Now crossposted at math.stackexchange]
Let $\mathcal{B} = \{B_1,\ldots,B_m\}$, $\mathcal{C} = \{C_1,\ldots,C_m\}$, $\mathcal{D} = \{D_1,\ldots,D_m\}$ be three families of sets, with pairwise intersection between families not necessarily empty, and $A$ be another set such that $A \not\in ...