this is one of those things that I know how to do in a programming environment but not sure how it translates into mathematics. I'm trying to express a sum so that it is easily visible that certain elements can be seen to be equal. Here is the original sum:
$$ s=\sum_{\alpha = 0}^{3}\sum_{\beta = 0}^{3}M_{\alpha \beta}(\Delta x ^ \alpha)(\Delta x ^ \beta) $$
where the greek characters are indices not exponents. Here is my reformulated version:
$$ s = \sum_{\alpha=0}^{3} M_{\alpha\alpha}(\Delta x^\alpha)^2 + \sum_{\alpha=0}^{3}\sum_{\beta = \alpha + 1}^{3} (M_{\alpha\beta} + M_{\beta\alpha}) (\Delta x^\alpha)(\Delta x^\beta) $$
This would make sense in something like a for loop where the order of summation is clear, but I'm not really sure how to express it in the context of the summation symbols. As a specific question, could someone tell me if this is mathematically valid?