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I have 5 robots and for each new job, one of them finish last. What is the correct notation for getting the maximum of the times of the 5 robots for a job.

is below acceptable?

$MaxTime = \max\{t_1,t_2,\ldots,t_n\} $

Thanks.

Jack
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  • Looks good to me. Sometimes, people even write $\max(t_1,\dots,t_n)$, which is sloppy, but acceptable. Your notation is completely correct. – 5xum Feb 25 '15 at 09:41
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    This is correct. – Claude Leibovici Feb 25 '15 at 09:42
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    I have seen this notation many times so it might be correct. But is it also correct to write $\max\left({t_1,t_2,...,t_n}\right)$. That means a map which has a set as argument? Look for me personal more convenient. – Matthias Feb 25 '15 at 09:45
  • @5xum $\max(t_1,\ldots,t_n)$ does not need to by sloppy for appropriately defined $\max$. – dtldarek Feb 25 '15 at 09:49
  • @dtldarek Sure, but then $\max$ is, technically speaking, not a function because it has a varying amount of arguments. – 5xum Feb 25 '15 at 09:51
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    @5xum Not really, perhaps it is a shorthand for $\max\big((t_1,\ldots,t_n)\big)$, observe that you can write $\sin \alpha$ without the parentheses, similarly there is notation $\max_{k}t_k$ which also does not use parentheses. – dtldarek Feb 25 '15 at 09:53
  • @dtldarek Good point. – 5xum Feb 25 '15 at 09:54

3 Answers3

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Depending on context, there a number of things you could write (and this list is not exhaustive):

\begin{align} &\max\{t_1,\ldots,t_n\} \quad\text{ or }\quad \sup\big\{t_1,\ldots,t_n\big\}\\ &\max(t_1,\ldots,t_n)\\ &\max\big(\{t_1,\ldots,t_n\}\big)\\ &\max\big((t_1,\ldots,t_n)\big)\\ &\max\big(\langle t_1,\ldots,t_n\rangle\big)\\ &\max_{i = 1}^{n} t_i \quad\text{ or }\quad \max_{i = 1,\ldots,n} t_i\\ &\max\Big((t_i)_{i = 1,\ldots,t_n}\Big)\\ &\sup\big\{t_i \mid 1 \leq i \leq n\big\}.\\ \end{align} If you want to complicate things and $t_i \geq 0$, there's also the $p$-norm: $$\lim_{p \to \infty}\left(\sum_{i = 0}^{n}|t_i|^p\right)^{\frac{1}{p}}.$$

My personal preference is the first or the second from the list, depending on what is more readable in the context.

I hope this helps $\ddot\smile$

Edit:

Please note that the supremum operator $\sup$ is most often used in a context when we don't know or don't care if the maximum is attained by any element it the set, e.g. you cannot write $\max\left\{-\frac{1}{n} \mid n \geq 1\right\}$ because the set does not contain $0$, yet $\sup\left\{-\frac{1}{n} \mid n \geq 1\right\} = 0$. On the other hand, it is certainly not wrong to write $\sup A$ for a finite set $A$ or any other that contains its supremum, and authors often switch between them depending on whether they want to stress or deemphasize that property.

dtldarek
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  • ...is $\sup$ a bit redundant for a finite list? Perhaps an improvement to the answer could be to explain where you would use $\sup$. – JP McCarthy Feb 25 '15 at 11:05
  • @JpMcCarthy I would not say it is redundant, and even if it is, it is certainly not wrong. Although a comment explaining the difference is a great idea, fixed now. Thanks for your suggestion! – dtldarek Feb 25 '15 at 11:20
  • ... oh certainly not wrong. +1 for the edit! – JP McCarthy Feb 25 '15 at 11:20
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You can also write $t_1 \vee \cdots \vee t_n$ (notation from lattice theory) or $t_1 \oplus \cdots \oplus t_n$ (notation used in max-plus theory).

polmath
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$t_r \in \{t_1, t_2, ..., t_n \} $ s.t. $t_r>t_i \ \ \forall t_i \in \{t_1, t_2, ..., t_n\}\setminus \{t_r\}$.

tomi
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