What does it mean to have a subscript and superscript?

Question

What does it mean to have a subscript and superscript ?

I came across this equation from a paper:

$$ \phi(w,\xi) = \mathrm{min} \frac{1}{2}\sum_{m=1}^{k}(w_m \cdot w_m) + C\sum_{i=1}^{\gamma}\sum_{m \neq y_i} \xi_i^m $$

I'm having trouble understanding what is going on in the last summation.

Answer 1

In this case, it seems that it is just another index. In the second line of Eq. (3), you can see that $m$ can be any of $\{1,\dots,k\}$ as long as $m\neq y_i.$ Since $\xi$ has two indexes, you can think of it as a matrix.

Note that in tensor-notation superscripts vs. subscripts have a very specific meaning, but this paper does not seem to use tensors, so no worries here!

Answer 2

Usually the subscript i represents a counter, here going from 1 to $\gamma$ in the outer sum, and the superscript $m$ represents as usual an exponent $(\zeta_1)^2$ squared and so on.