# Notation: Elementwise cropping for vectors

I am looking for a notation for following function: A vector $\vec{a}$ is normalized using L2-norm. But if the value of one element is bigger then a hysteresis an other value $h$ is used. For example after normalization $\vec{a} = [1, 2, 3, 4]$ and the hysteresis is $3$ then the final vector looks like this: $\vec{a} = [1, 2, 3, 3]$. How can I write this down in a correct mathematical notation. It is basically a "element wise" defined function for a vector. I guess it should somehow look like this:

$$\vec{a} = \left\{ \begin{array}{ll} h & element > h \\ \frac{\vec{a}}{\sqrt{||\vec{a}||^2_2 + \epsilon^2}} & else wise\\ \end{array} \right.$$

• "min" notation is classical but you should add that is to be understood elementwise ; here it would be: $\min((1,2,3,4),3*(1,1,1,1))$ – Jean Marie Apr 18 '17 at 9:12

Use an index: $$a_i'=\min(a_i,h)$$