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Definitions: An $n\times n$ Hadamard matrix (HM for short) is a matrix whose entries are either $1$ or $−1$ and whose rows are mutually orthogonal. If $A$ is a symmetric matrix, then $A = A^T$ and if $...

Let $s$ and $d$ be non-negative integers with $0\leq s<d$ and let $v,u\in \mathbb{R}^d$ be vectors satisfying the sparsity estimate $$ \max\{\|u\|_0,\|v\|_0\}\leq d-s, $$ where, as usual, for any ...

Given an integer matrix $M\in\mathbb Z^{n\times n}$ of real rank $k$ what is the minimum and maximum number of rank $1$ matrices $B_1$ to $B_t$ we require so that $M\equiv\sum_{i=1}^tB_i\bmod 2$? If $...

Consider the set $\mathcal S(p)$ of symmetric matrices $A$ of size $p\times p$ with bounded spectrum, say, $\lVert A\rVert_\text{op}\le 10$ and $\lVert A^{-1}\rVert_\text{op}\le 10$. Let $\alpha>0$ ...

It is well-known that in the classic coupon collecting problem (CCP), the expected waiting time is \begin{equation*} T_n(x_1,\ldots,x_n) = \sum_{k=1}^n (-1)^{k+1}\sum_{1\le i_1 < \cdots < i_k \...

For $s=0,1,\dots$ and $n=1,2,\dots$, denote $r_{n,s}=\sum_{k=1}^n k^s$. It is well-known that $r_{n,s}$ are polynomials in $n$ with leading term $\frac{1}{s+1}n^{s+1}$. Let $R_{n,s}$ be the $(s+1)\...