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I read a book on Kernels, See the following example. Why the authors take square two times here? what is the logic?

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It's a typo, the kernel is $$K(x_1,x_2)=\exp\left(-||x_1-x_2||^2 \over 2\sigma^2\right)$$

Note that if you do the calculation w/o squaring again, the result rounded upto two digits will be the same.

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