Notation: Concatenate two functions (piecewise) / Concatenate two vectors, lists or tuples Functions
Given two functions $f: F\rightarrow F'$ and $g: G\rightarrow G'$ with disjount domains ($G\cap F=\emptyset$), what are common ways to denote the following function
$ h: F \cup G \rightarrow F' \cup G'$
$ h(x) = \left\{ \begin{array}{ll}f(x)&x\in F\\g(x)&x\in G\end{array} \right. $
Maybe, $h=g||f$ ?
Vectors / Tuples / Lists
Given $x=(a_1,\dots,a_m)$ and $y=(a_{m+1},\dots,a_n)$, is there a standard notation for
$z=(a_1,\dots,a_n)$
Again, what about $z=x||y$?
 A: for Functions:
Wikipedia calls it an override or overriding union and uses
$ h=f\oplus g$
http://en.wikipedia.org/wiki/Overriding_(mathematics)#Restrictions_and_extensions
A: If you take the set-theoretic view of functions as sets of ordered pairs, the function defined on the union is literally just the union of the functions.
The tendency to have a notion of codomain of a function that is distinct from its image means this view is perhaps not pedantically correct, but it is nevertheless fairly clear (at least to set theorists) what you mean when you write $f \cup g$.
(I can't necessarily vouch that anyone actually does this, mind!)
A: For vectors, one would write
$$\mathbf{z} = [\mathbf{x}\ \mathbf{y}]$$
or
$$\mathbf{z} = \begin{pmatrix} \mathbf{x} \\ \mathbf{y}\end{pmatrix}$$
depending on whether the vectors are row/column vectors. You don't often see this, because concatenating the vectors in this way doesn't necessarily take the properties of the vector spaces to which $\mathbf{x}$ and $\mathbf{y}$ belong and carry them over to the vector space to which $\mathbf{z}$ belongs.
However, this notation is common when describing, for instance, numerical algorithms, etc. It's common in engineering, where one creates state space models in this form. For example, one might write a structural dynamics controllability model as:
$$\frac{d}{dt}\begin{pmatrix} \mathbf{x}(t) \\ \dot{\mathbf{x}}(t) \\ \mathbf{\lambda}(t) \end{pmatrix} = A\begin{pmatrix} \mathbf{x}(t) \\ \dot{\mathbf{x}}(t) \\ \mathbf{\lambda}(t) \end{pmatrix} + B\mathbf{u}(t).$$
Note, however, that "stacking" the vectors doesn't really mean anything mathematically; it's just a notational shortcut.
