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I have a summation which should go over a set, excluding one of the elements which returns the maximum value. What I first wrote was:

$$ \sum_{\substack {j \in M \\ j \neq \arg\max_{j} \{f_j\}}} f_j $$

but since $\arg \max$ may return more than one element, and so it can be a set, I changed my notation to:

$$ \sum_{\substack {j \in M \\ j \notin \arg\max_{j} \{f_j\}}} f_j $$

Still, this is not correct since only one $j$ should be excluded from the summation, not all of them (if there exist more than one element which maximizes $f$).

My question is, how to write it correctly, to mean that only one of the elements which maximizes $f$ should be excluded from the sum function. In my case, it does not matter which one.

Mostafa
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    Good question. How about $\sum_{j\in M}f_j-\max_jf_j$ instead? – Karl Dec 23 '22 at 02:24
  • @Karl Thank you. Indeed that would my last resort if I cannot do it with $\arg\max$. – Mostafa Dec 23 '22 at 02:30
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    Do it in a way that is easiest for your reader. @Karl 's suggestion works. So would a description in words rather than symbols. – Ethan Bolker Dec 23 '22 at 02:41
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    The set of values you want to sum over is hard to write down because it isn't well-defined (which element do you pick from the argmax to exclude?). Of course the sum itself is still well-defined, but I don't think there's a notational construct that communicates this. – Karl Dec 23 '22 at 04:25
  • Thank you, Karl and Ethan. As I said it does not matter which element, as my intention is to deduct the max value only (it is part of a larger statement to show that a this sum function is monotnically non-increasing in some constant). Anyway, I will follow your suggestion, Karl. – Mostafa Dec 23 '22 at 09:23

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