MathOverflow

On a board, a sequence of n + 1 real numbers is written, with n ≥ 1,formed by the number 1 followed by a geometric progression of n terms whose firstterm is equal to 1 and whose common ratio is equal to √2, that is, the numbers written on the board are...

A strong subtree $T \subseteq 2^{<\omega}$ is a perfect tree of infinite height such that for all $s,t \in T$ for which $|s| = |t|$, $s$ splits in $T$ iff $t$ splits in $T$. The Ramsey theory of strong subtrees was studied by Milliken in A Partition...

Find the smallest number $n$ such that almost all natural numbers can be represented as the sum $$a_1^{a_{p(1)}}+a_2^{a_{p(2)}}+\dots+a_n^{a_{p(n)}}$$ where $a_1,\dots,a_n$ are pairwise distinct natural numbers, $p$ is a permutation of the set $\{1,...

Let $P,Q$ be two real orthogonal projections on $\mathbb R^n$, and assume that they are permutation similar. More specifically, assume that each of them is permutation similar to a block diagonal matrix of the form
$$\operatorname{diag}\!\left(\frac1m J_m,\frac1m J_m,\dots,\frac1m J_m\right).$$...