All Questions

I am curious if anything is known about the following. Let $\Gamma$ be a finite abelian group, and let $\chi$ be its character table, normalized so that it is a unitary matrix. E.g., if $\Gamma$ is $\...

Suppose that sequences $v_1,\ldots,v_m\in \{-1,+1\}^n$ and $u_1,\ldots,u_m\in \{-1,+1\}^n$ are isometric in $\mathbb{R}^n$, i.e. $u_j=Qv_j$ for some orthogonal matrix $Q$. What is the largest $m$ such ...

There are already detailed results on the solutions of linear equations over fields, but I'd like to inquire about any good conclusions regarding the solutions of linear equations over the integer ...

I'm looking for combinatorial proofs of the convolutional identity COP below and its specializations I) and II). (Edit 6/2/2023: A combinatorial proof is sketched in a blog post by Mike Spivey of a ...

$\DeclareMathOperator{\GL}{\operatorname{GL}}$Let $G$ be an elementary abelian $p$-group of rank $2$. Let $\alpha, \beta :G\rightarrow \GL_{n}(\Bbb Z / p\Bbb Z)$ be two injective homomorphisms. The ...

I consider two sequences of numbers: $A=\{a_1,...,a_{m-1},n\}$ and $B=\{n-a_{m-1},...,n-a_1,n\}$, where $a_1 < a_2 < ... < a_{m-1} < n$ and $\gcd(A) = \gcd(B) = 1$. I investigate the ...