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I am analysing electricity consumption by checking firstly if the demand (say beta) is greater than a certain constant limit (baseline, say alpha) for every hour of the year, and then summing this difference (beta - alpha) each hour over a day for those hours greater than the baseline (ignoring hours where beta < alpha). There may be e.g. 10 days in the year where the consumption exceeds the baseline and I would like to represent the largest of these summations. Is there notation for describing the maximum value of this daily summation?

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Let $D(t): \lbrace 1,2,\dots,365 \rbrace \to \mathbb{R}$ be the function that takes as input the day of the year and as output the summation over the hours that you described. Then, one would write $$ \max \lbrace D(t) ~\vert~ 1 \leq t \leq 365 \rbrace.$$

You would read this as "the maximum value of the function $D(t)$ over the domain of integers between 1 and 365 inclusively."

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thanks - so, if beta is the hourly demand, and k is the baseline constant, then something like: $\max \lbrace D(t) ~\vert~ 1 \leq t \leq 365 \rbrace$ where $D(t) = \sum_{t=0}^{24} (\beta_{(t)} -k) \text{ for }\beta_{(t)} > k $ (I only want to sum the exceedances) –  icenov Aug 28 '12 at 10:19

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