I am trying to understand one research paper and it have some sigma symbols in formula ,
∑ signs I can see denote a sum over all "relevant" values of t
But in implementation they are using reduce sum function that's where i am confuse. Can anyone explain what actually first sigma represent and what second sigma represent ?
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The lack of an upper limit in summation symbol ∑ generally denotes that it is an unbound method, i.e. summing over all values of the variable, t in the above example. Note, 'all' does not necessarily mean infinity. As far as reduce sum function you mentioned, it simply means taking a summation to get a single value. Just another terminology for the function.
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It means you sum it over all the values of $t$.
Which raises the questions what are the values of $t$ that we are supposed to sum?
That can be given in context. The most common method is $\sum\limits_{t=a}^b$ which means $t =a,a+1,a+2,...., b=$. But that is not the only possible method. $\sum\limits_{t =n^2;n\in\mathbb N}$ would be, for example, that $t = 1,4,9,16,....$. We can even do $\sum\limits_{t|56}$ which means $t=1,2,4,7,8,14,28,56$.
So what is $t$ in this case? I have no idea. It is probably something that is clear in context. I notice you have a variable $\alpha_{it}$. That implies that $t$ has already been introduced as an indexing variable. So I'd assume $t$ are the possible values of the index.
I can't see what's in the numerator of your formula for $\alpha_{it}$, but both $\sum$ signs I can see denote a sum over all "relevant" values of $t$. For brevity, authors write like that whenever the context makes the set of relevant indices obvious. However, the explanation that makes it obvious is probably not in the algebra, but in the text between equations.