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I am trying to solve the following integral $$ \int_{-1}^{1}\;db\;||[t_{b}(A),J]||_{F}^{2} $$ where $t_{b}$ is the entrywise threshold of the matrix A ($0$ if $a_{ij}<b$, $a_{ij}$ if $a_{ij}>b$, with $|a_{i,j}|\in [0,1]$ ). Further $A$ is a symmetric positive definite matrix, $J$ is the all ones matrix and $F$ indicates the Frobenius norm (the zero or one norm will work as well).

Any hope?

Thanks a lot!

Fabio

Note: an equivalent problem would be if $t_{b}(A)$ is the hadamard $p$-power of $A$ and we substitute the integral with the sum $\sum_{p=1}^{\infty}$. I also tried to solve the integral in this last formulation but without success.

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  • $\begingroup$ I am curious to know why this has a "finite groups" tag? $\endgroup$
    – Geoff Robinson
    Commented Sep 28, 2015 at 23:42
  • $\begingroup$ Could you rewrite the equivalent problem formally. Is it of the form $\sum_{p \ge 1} \|[A^{\circ p}-J]\|^2$? $\endgroup$
    – Suvrit
    Commented Sep 29, 2015 at 0:42
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    $\begingroup$ @GeoffRobinson Hi Geoff, the matrix A should be coming from a multiplication table of a group. $\endgroup$
    – Fabio
    Commented Sep 29, 2015 at 14:11
  • $\begingroup$ @Survit Hi Survit, I can rewrite it as $\sum_{p=1}^{\infty} ||A^{\circ p}J- JA^{\circ p} ||_{F}^{2}$. Thanks. $\endgroup$
    – Fabio
    Commented Sep 29, 2015 at 14:14

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