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Consider the ring of polynomials $R:=\mathbb{Z}[x_1,x_2,x_3]$. Define the operators $E, I:R\rightarrow R$ by $Ef(x_1,x_2,x_3)=f(x_1-1,x_2,x_3)$ and the identity $If=f$. Let $\mathcal{L}:R\rightarrow R …

Let $\mathfrak{S}_n$ be the permutation group on $[n]$. Denote the cardinality of $\{\pi\in\mathfrak{S}_n: \pi^2=id\}$, the set of involutions, by $I(n)$. It is well-known that these numbers have the …

The Hardy-Littlewood maximal operator $$Mf(x)=\sup_{x\in B}\frac1{\vert B\vert}\int_B\vert f(y)\vert dy$$ where the supremum is taken over all balls $B\subset\mathbb{R}^n$ which contain $x$. It is w …

In my earlier MO question, I was seeking for a proof for $\det A_{\infty}:=\det(I_{\infty}-M_{\infty}^2) =\sqrt[4]{1-x^2}$ where $M_n$ is the $n\times n$ matrix: $$M_n =\left[\frac{2i+1}{2(i+j+1)}\bin …

Let $A$ and $B$ be two matrices of eigenvalues $\lambda_i$ and $\mu_i$, respectively. The spectral variation of $B$ w.r.t. $A$ and the eigenvalue variation of $B$ and $A$ are, respectively, \begin{a …

Let $M_n$ be an $n\times n$ matrix defined as $$M_n =\left[\frac{2i+1}{2(i+j+1)}\binom{i-1/2}i\binom{j-1/2}jx^{i+j+1}\right]_{i,j=0}^n.$$ With $I_n$ the identity matrix, consider $A_n:=I_n-M_n^2$. Whe …