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What is the difference between the summations using $i<j$ and $i\neq j$ in the formula below: $$\sigma^{2}(\boldsymbol{w})=\sum_{i} \tilde{w}_{i}^{2}+2 \sum_{i<j} \tilde{w}_{i} \tilde{w}_{j} \rho_{i, j}=\sum_{i} \tilde{w}_{i}^{2}+\rho(\boldsymbol{w}) \sum_{i \neq j} \tilde{w}_{i} \tilde{w}_{j}$$ Screenshot here.

  1. Are both summations operationally equivalent?
  2. If so, why break consistency and have two competing representations?
  3. Which summation is more correct, or which to use for which situations?

One of the non-best answers here seem to apply, but not sure how in my case.

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    $\begingroup$ $\sum_{i\ne j}=2\sum_{I<j}$ here $\endgroup$
    – J. W. Tanner
    Commented Nov 10, 2020 at 12:47
  • $\begingroup$ @CalvinKhor how did you automatically convert the screenshot to maths? $\endgroup$
    – develarist
    Commented Nov 10, 2020 at 12:47
  • $\begingroup$ they are not equivalent, one is more general than the other but if there is commutativity, you multiply by $2$. What are the variabes? $\endgroup$
    – Phicar
    Commented Nov 10, 2020 at 12:48
  • $\begingroup$ I'd say there's not enough context to be sure, but note that $i\neq j$ iff ($i<j$ OR $j<i$) $\endgroup$
    – Calvin Khor
    Commented Nov 10, 2020 at 12:48
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    $\begingroup$ @develarist its not so hard to type the Mathjax of a single line, which you can learn from this link, but I did do it automatically with a software called `mathpix' $\endgroup$
    – Calvin Khor
    Commented Nov 10, 2020 at 12:50

1 Answer 1

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$\sum_{i<j}$ sums over all the possible pairs $(i,j)$ for which $i<j$ holds. Similarly, $\sum_{i\neq j}$ sums over all the possible pairs $(i,j)$ for which $i\neq j$ holds.

For example, if $i$ and $j$ can take values in $\{1,2,3\}$, then

$$\sum_{i<j}a_{i,j}=a_{1,2}+a_{1,3}+a_{2,3},$$

whereas

$$\sum_{i\neq j}a_{i,j}=a_{1,2}+a_{1,3}+a_{2,1}+a_{2,3}+a_{3,1}+a_{3,2}.$$

If the summand is symmetric, i.e., $a_{i,j}=a_{j,i}$ holds for all $i$ and $j$, these two quantities are related by

$$\sum_{i\neq j}a_{i,j}=2\sum_{i<j}a_{i,j}.$$

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    $\begingroup$ is it more correct to have double summation symbol, or should they be combined into one: $\sum_{i\neq j}^N w_i w_j$ or $\sum_{i=1}^N \sum_{i\neq j}^N w_i w_j$? and are they equivalent $\endgroup$
    – develarist
    Commented Nov 10, 2020 at 13:05
  • $\begingroup$ @develarist, Both sound equally correct to me, so it might be better to choose a more economical convention. But if you feel that the double summation notation seems ambiguous, it might be good to write $$\sum_{i,j \mathop{:} i\neq j} w_i w_j.$$ By doing so, we can clarify the summation variables. $\endgroup$
    – Sangchul Lee
    Commented Nov 10, 2020 at 13:35

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