# A question regarding case notation

Consider the parameter $P$ and a property $S$ associated with this parameter represented as $P_S$. For example, consider that the height of a tree (“H”) is my parameter and the property is the tree’s name. Therefore, $H_{T1}$ refers to the height of the tree named $T1$.

Now, consider that $H_{T1}$ depends on 2 inputs, named $A$ and $B$ respectively. So, for the above example we have:

$$H_{T1}^{A,B}$$

My first concern is whether the above is a valid notation or should I have written something else like $H_{T1}(A,B)$?

I am then using the cases environment in order to define the different value that $H_{T1}$ takes for a number of differetn $A, B$ inputs:

$$\begin{eqnarray} H_{T1}^{A,B} = \begin{cases} H_{T1}^{5,8} & \text{if A=5, B=8}\\ H_{T1}^{9,1} & \text{if A=5, B=1}\\ H_{T1}^{3,3} & \text{if A=3, B=3}\\ \end{cases} \end{eqnarray}$$

For a particular set of $A,B$ inputs, e.g. for the set $(A=3, B=3)$, $H_{T1}$ depends on $2$ extra inputs, named $C$ and $D$ respectively like that:

$$\begin{eqnarray} H_{T1}^{3,3,C,D} = \begin{cases} H_{T1}^{3,3,1,1} & \text{if A=3, B=3, C=1, D=1}\\ H_{T1}^{3,3,4,1} & \text{if A=3, B=3, C=4, D=1}\\ H_{T1}^{3,3,6,2} & \text{if A=3, B=3, C=6, D=2}\\ \end{cases} \end{eqnarray}$$

Since only for $H_{T1}^{3,3}$ I have the 2 extra inputs, what would be the best way of writing that? I thought of writing something like that:

$$\begin{eqnarray} H_{T1}^{A,B} = \begin{cases} H_{T1}^{5,8} & \text{if A=5, B=8}\\ H_{T1}^{9,1} & \text{if A=5, B=1}\\ H_{T1}^{3,3,C,D} & \text{if A=3, B=3}\\ \end{cases} \end{eqnarray}$$

where

$$\begin{eqnarray} H_{T1}^{3,3,C,D} = \begin{cases} H_{T1}^{3,3,1,1} & \text{if A=3, B=3, C=1, D=1}\\ H_{T1}^{3,3,4,1} & \text{if A=3, B=3, C=4, D=1}\\ H_{T1}^{3,3,6,2} & \text{if A=3, B=3, C=6, D=2}\\ \end{cases} \end{eqnarray}$$

but I am not sure whether it’s correct or not.

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You are very free to define your notation the way you like, even indices and exponents to the left are possible. However, some of these notations bear connotations that may not match your application, e.g. covariant and contravariant dependencies. –  Hagen von Eitzen Dec 17 '12 at 22:38
Reminds me of a book that, after talking about the spectral sequence ${}^{II}E^r_{p,q}$ apologized, and remarked the notation could be worse by putting something on the remaining corner. –  Hurkyl Dec 17 '12 at 22:53
@Hagen: Thanks for your input. I was just wondering whether there was an alternative way of writing it that could potentially be more universal or more readable probably...If you look at my attempt of writing it in the end, I don't know how good it looks to say that $H_{T1}^{A,B}$ has only two inputs and then write that in one case is equal to $H_{T1}^{A,B,C,D}$....@Hurkyl: Defining a complicated notation while it could have been simpler is what I am trying to avoid ;) Would you happen to have an alternative to the above that potentially looks more attractive or makes more sense? Thanks –  limp Dec 18 '12 at 8:55

You could use $H_{T1,A,B}$, or $H_{T1;A,B}$. Your idea for your function is almost right, but you've included some superfluous information. Try

$H_{T1,A,B,C,D}=\cases{a, \text{if } A=1,B=1,∀C,D,\\b, \text{if }A=2,B=2, ∀ C,D,\\c, \text{if } A=3,B=3,C=1,D=2,\\d, \text{if } A=3,B=3,C=1,D=3}$

or use your simplifying trick and say

$H_{T1,A,B,C,D}=\cases{a, \text{if } A=1,B=1,\\b, \text{if }A=2,B=2, \\ H_{T1,3,3,C,D}, \text{if } A=3,B=3, \text{where}}$

$H_{T1,3,3,C,D}=\cases{c, \text{if } C=1,D=2,\\d, \text{if } C=1,D=3.}$

However, you can make clearer what is going on by presenting $H_{T1,A,B},A,B\not=3$ as a special case of $H_{T1,A,B,C,D}. Thus,$H_{T1,A,B,C,D}=\cases{H_{T1,A,B}, \text{if } (A,B)\not=(3,3),\\c, \text{if } A=B=3,C=1,D=2,\\d, \text{if } A=B=3,C=1,D=3.}\$

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