# Notation for denoting one argument fixed and acting upon the other

Given a function $f_1:\mathbb R^n\mapsto \mathbb R$, and a fixed vector $v\in \mathbb R^n$ I construct another function $f_2:\mathbb R^n \times \mathbb R^n\mapsto \mathbb R$ such that $f_2(d,v):=f_1(T(v) \cdot d)$ where $T(v)$ is some transformation matrix depending on $v$. I need to compute two quantities which are characteristic of the functions themselves, namely their global sensitivity: $$S(f_1) = \max\limits_{d,d^\prime} |f_1(d)-f_1(d^\prime)|$$ and for a fixed $v$: $$S(f_2) = \max\limits_{d,d^\prime} |f_2(d,v)-f_2(d^\prime,v)|$$ where $d$ and $d^\prime$ differ on at most one item.

I am looking for a notation to make clear that $v$ is being held constant when I refer to $S(f_2)/S(f_1)$ (as this is a critical fact to highlight and I refer to that quantity quite a lot in my article). I have came up with several alternatives (dropping the numeric subscript), including $S(f_v)$, $S_d(T(v)\cdot d)$, $S(f(d;v))$ but none of them seems suggestive enough (at least for me).

I am pretty sure someone has met this problem before (the problem of denoting one argument of being fixed and acting upon the other) and if such a notation was notorious (although I haven't heard of it) I'd like to stick to it instead of making up a notation of my own. Hence I am asking this question here.

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## 2 Answers

I'd introduce a map $R$ that sends a function $f\colon\mathbb R^n\mapsto \mathbb R$ and a vector $v\in\mathbb R^n$ to the function $R(f,v)\colon\mathbb R^n\mapsto\mathbb R$ with $R(f,v)(d)=f(T(v)\cdot d)$. Then the ratio you want to express is $S(R(f,v))/S(f)$, and the dependence on $v$ is explicit in the notation.

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This is great! Thank you :) –  M. Alaggan Mar 21 '12 at 15:45

Another way is to use lower-indices: given a function $f:\mathbb R^n\to\mathbb R$ define $f_v:\mathbb R^n\to\mathbb R$ as $$f_v(x) = f(T(v)\cdot x).$$ Then in $$\frac{S(f_v)}{S(f)}$$ the dependence on $v$ is explicit.

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