0

In the specified equation, the lower limit of the inner summation ('$i$' with numerical subscript) seems to have all the values as same, in each loop of summation, which it clearly can't be. I'm not able to understand how to expand this summation for given $n$, say $n = 5$; Can someone please write first few terms of the expanded summation with proper and simple explanation. Thanks in advance.

$$ \Delta W\left(c_{1}, c_{2}, \ldots, c_{n}\right)=\sum_{\mathbf{k}=2}^{n} \sum_{i_{1}, \ldots i_{k}=1 \atop i_{1}<\ldots<i_{k}}^{n} \Delta \tilde{W}\left(c_{i_{1}}, c_{i_{2}}, \ldots, c_{i_{k}}\right) $$

Link to equation image hosted in ImgBB: https://ibb.co/d7yZCgm

Source of equation: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1459517/

Equation number: (6)

Wolgwang
  • 1,563
Justin K
  • 3

1 Answers1

0

I'd say it starts with $k=2$ and the inner summation is for $i_1 = 1$ up to $n$ then $i_2 = 2$ up to $n$ and so on for all combinations of these indices beginning at $1$ and also satisfying the condition $i_1 < \cdots < i_k$.

The expansion would begin

$\Delta \tilde{W}(c_1, c_2) + \Delta \tilde{W}(c_1, c_3) + \cdots + \Delta \tilde{W}(c_1, c_n) + \cdots +\Delta \tilde{W}(c_{n-1}, c_n) + \Delta \tilde{W}(c_1, c_2, c_3) + \cdots + \Delta \tilde{W}(c_{n-2}, c_{n-1}, c_n) + \cdots$

The last term would be $\Delta \tilde{W}(c_1, \cdots ,c_n)$.

DavidW
  • 572
  • I hope that helps and makes sense in the context of the paper. – DavidW Oct 22 '20 at 13:20
  • Hi, thankyou for replying. Your explanation so far makes sense in context, but I have one doubt. Why did you not consider ΔW(c2,c2) or ΔW(cn,cn), i.e. equal terms. These equal terms are actually not considered in context of paper (like you did correctly) but I wanted to know how you actually reached to that conclusion, by expanding the summation. Thanks for the help. – Justin K Oct 23 '20 at 03:30
  • That's good to read. With $\Delta \tilde{W}(c_2, c_2)$, for example, the condition $i_1 < i_2 < \cdots < i_k$ would not be satisfied because the subscripts are equal and not strictly ascending. – DavidW Oct 23 '20 at 03:46
  • Thankyou very much for the answer. I missed that detail. – Justin K Oct 23 '20 at 04:22
  • No problem. The notation is very compressed and there were no examples of equation 6 in the paper. – DavidW Oct 23 '20 at 04:24