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I have seen notation like this, commonly in regard to matrix multiplication:

$$ b_i = \sum_{j \ne i} A_{ij} x_j $$

So this is a matrix multiplication that excludes the diagonal ($j=i$).

I'm kind of confused when it's ok to just leave off specification of the upper limit or lower limit from the sum. From here, I'm assuming if you have an exclusion, you're supposed to sum over all elements not excluded?

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    It's common to omit when understood. The fact there's an $i$ on the LHS, makes it clear that the sum on RHS is over $j$ (and $i$ is fixed), where $j \in [a, b]$ for some $a,b$ inferred from the context.2012-08-22
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    Speaking of summation notations, you can also have things like $\large\sum\limits_{d \mid n} d,$ or any other predicate.2012-08-23

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