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I came across this expression in the documentation for some software:

$\displaystyle\sum\limits_{j=1}^c\alpha_{i*(j)k}q_j=Q_k$

Here, $\alpha$ is the element of a matrix with the rather curious index "$i*(j)k$", which the documentation describes as being a mapping function from the elements of one matrix to another. I have never seen this notation before in my life, specifically the use of the multiplication operator and the parentheses. I am familiar with Einstein notation for matricies, but this does not seem to be related to that. It is possible that this is just a typo in the documentation, but in case anyone can make sense of it, please help me understand! Thanks.

math_man
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Carlton
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  • A further perplexity of that definition is the meaning of "$i$" in there. Is there context for that? – Semiclassical Sep 25 '14 at 15:56
  • The software is for analyzing chemical reactions; "$i$" in this case is the index of a chemical element that appears in the reaction. For example, $i=1$ is carbon, $i=2$ is hydrogen, etc. – Carlton Sep 25 '14 at 16:03
  • Gotcha. Best I can guess is that they intend it as $\alpha_{i(j),k}$ i.e. $\alpha$ as an array. But that still leaves the $i(j)$ mysterious. – Semiclassical Sep 25 '14 at 16:13
  • Yes, I forgot to mention that $\alpha$ is a 2D matrix/array, so it definitely needs a comma in there somewhere to denote the two indicies. I think you're correct in that there's a missing comma. – Carlton Sep 25 '14 at 16:19
  • I'm guessing the star is there in order to visually separate $a_i$, the array for a particular chemical element, from the indices $j$ and $k$? But this still doesn't explain the parentheses on $(j)$, so who knows... – alexis Sep 25 '14 at 16:29
  • I'm emailing the lead developer now. I'll report back with his response. BTW, is it uncouth to answer my own question? – Carlton Sep 25 '14 at 16:35
  • Not at all. In fact, I encourage people to do so when they can. – Semiclassical Sep 25 '14 at 16:41
  • I have a guess that it means the $k$th column of $Q$ is given by the matrix-vector product of the matrix $\alpha_i$ with the column vector $q$. If the parenthesis were on the $k$ instead, I think it may have been clearer. – amcalde Sep 25 '14 at 18:18

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As far as I can tell, this obscure notation was invented by the authors of the paper. In short, there are two vectors $I$ and $J$, and by definition $J$ is a subset of $I$. Therefore, any element $J_j$ has a corresponding element $I_i$, and the mapping function between these two elements is denoted by "$i*(j)$". In practice, this is used to say that when the index $j$ is used (such as in the summation in my original post), we can instead map it to a corresponding index $i$ and pull the value of interest from $I$ instead of $J$.

The paper in which I found this is publically available if anyone cares to read it. It is technical report number AD/A-002 791 by Stanford Research Institute, from March 1974. The title is "Tiger Computer Program Documentation". A Google search for the title should give you a link to the report as hosted on www.dtic.mil. The equation of interest is on page I-B-4, second full paragraph.

Thanks to all who helped.

Carlton
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