MathOverflow

Motivation
Alternative Proof of Fullness of Collection on $D^b(\mathbb P^n)$
Richard Thomas's notes on HPD (page 2) mention an alternative proof (rather than Beilinson's resolution of the diagonal) for the fullness of the strong exceptional collection $\mathcal O(-n+1), \mathcal O(-n+2), \dots, \mathcal O$ on $\mathbb P^{n-1...

The precise conjecture that I want to prove or disprove is the following:
Let $P_n$ denote the path graph with n vertices, $S_n$ the star graph with n vertices, and $T_n$ an arbitrary connected tree with n vertices.
Let $p=(p_1,...,p_n)$, $t=(t_1,...,t_n)$, and $s=(s_1,...,s_n)$ denote vectors of eigenvalues of the graph Laplacians $L(P_n)$, $L(T_n)$, and $L(S_n)$ , respectively....

Let $A = \ell^1(G)$ be the convolution Banach $*$-algebra of a finite (not necessarily abelian) group $G$ with norm $\|\cdot\|_1$,let $\mu \in A$ be a probability measure, and set $\delta = \|\mu * \mu - \mu\|_1 \leq 1/10$. Is it the case that there...

In a paper published in 1985, Shih-Ping Tung observed that an integer $m$ is nonzero if and only if $m=(2x+1)(3y+1)$ for some $x,y\in\mathbb Z$. In fact, we can write a nonzero integer $m$ as$\pm3^a(3y+1)$ with $a\in\mathbb N=\{0,1,2,\ldots\}$ and $y...