math

Suppose that we want to construct subsets $S_1, \ldots, S_m \subseteq \{1,\ldots,n\}$ with the following properties: 1. $|S_i| \geq k$ for all $i$ 2. $|S_i \cap S_

Here is interesting linear algebra fact: let $A$ be an $n \times n$ matrix and $u$ be a vector such that $u^{\top}A = \lambda u^{\top}$. Then for any matrix $B$, $u^

Consider the following statements: 1. The shape with the largest volume enclosed by a given surface area is the $n$-dimensional sphere. 2. A marginal or sum of log-concave distributions is log-concave. 3.

Here are two strange facts about matrices, which I can prove but not in a satisfying way. 1. If $A$ and $B$ are symmetric matrices satisfying $0 \preceq A \preceq B$, then $A^

Consider a probability distribution ${p(y)}$ on a space ${\mathcal{Y}}$. Suppose we want to construct a set ${\mathcal{P}}$ of probability distributions on ${\mathcal{Y}}$ such that ${p(y)}$ is the maximum-entropy