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Let $X$ and $Y$ be two ordered pairs $X = (A,B)$ and $Y = (A,B)$. Then let $A$ and $B$ be two sequences "nested" in the pairs $A = \langle A_1,A_2,...,A_n \rangle$ and $B = \langle B_1,B_2,...B_n \rangle$ (the elements of the sequences in $X$ and $Y$ can be different). Finally, let each element of $A$ and $B$ also be a sequence $A_i = \langle a_1,a_2,...,a_n \rangle$ and $B_i = \langle b_1,b_2,...b_n \rangle$.

1) Which is the best notation to reference the elements inside the sequence of each pair? $X_{A_{1}}$?

2) Which is the best notation to reference the elements in the sequences of the sequences $A$ and $B$ ? $X_{{A_{1_{a_1}}}}$ and $X_{{B_{1_{b_1}}}}$?

I would like to avoid the excessive use of subscripts that makes it nearly impossible to read because of the size. Any suggestion?

copenhagen
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2 Answers2

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It depends very much on what your purposes are. I would be disinclined to 'share' letters between the ordered pairs in e.g. a proof and would much prefer something like 'let $X=(A,B)$ and $Y=(C,D)$'; this saves you from having to disambiguate between $X$ and $Y$ in your subsequences in the first place. Then I would say $A=\langle A_1, A_2, \ldots, A_n\rangle$ where $A_1=\langle a_{1,1}, a_{1,2}, \ldots, a_{1,m}\rangle$, etc; if the two indices need to be more strongly distinguished then $a_{i;j}$ is also fine notation. Ideally, though, I would encourage restructuring your argument so that you don't need quite so much nesting to whatever extent possible — it may be inevitable that you have to compare, e.g., $a_{2,4}$ with $d_{5,3}$ but it's generally a sign that the overall structure of whatever you're talking about could be substantially cleaned up.

Steven Stadnicki
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  • I liked the second part of your suggestion, thanks. I'm just not sure if the double index would not cause confusion on the reader. About the first part, I could not change the letters of the ordinate pair because A and B are two sets of different types of things (ofc the type os elements of A is the same in X and Y).

    And the comparsion $a_{i,j}$ of $X$ with $a_{i,j}$ of $Y$ is something that certainly will happen in my case!

     – copenhagen Dec 22 '13 at 18:44
  • That $A$ and $C$ are of the same 'type' of thing doesn't mean that they have to share a letter; I don't think there's so much confusion that you couldn't use $A$ for one and $C$ for the other, if clearly notated. – Steven Stadnicki Dec 22 '13 at 19:22
  • Also, you wouldn't compare individual samples from multiple test runs against each other, for the most part - that's what aggregate statistics are for! I think the fact that it's hard to find a real-world example suggests that this isn't a situation that 'should' happen if you've structured your work correctly. – Steven Stadnicki Dec 22 '13 at 19:24
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1) Nested subscripts as you've suggested.

2) You can write it out more in-line using projection functions. E.g. $\pi_n (\pi_A (X))$. Though this seems annoying order-reversing on the indices.

3) Define a function or notation that maps the index references to the right thing. E.g. $\pi:\{X,Y\}\times\{A,B\}\times\mathbb{N}\times\mathbb{N}\to$whatever the range of those interior sequences is. So then you could just write $\pi(X,A,n,k)$ or $X_{A,n,k}$ and avoid subscripts after the initial definition.

matt
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  • Thanks. The second option would be very annoying to read! The third one is a good solution, the con is the need to define another function that the reader would have to remember. But if I don't find a better way it is a good possibility! – copenhagen Dec 22 '13 at 18:57