notation for summation of subsets of time data

Question

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?

Answer 1

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."