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Is there a closed form expression for the inverse of the matrix with elements $A_{i,j}=x_i$ for $i=j$ and $A_{i,j}=1$ for $i\neq j$?

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Variance of exponential random variable

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Expectation of $(c+e^{N(0,\sigma^2)})^{-n},\, n>0$

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Maximum-bend TSP