# Operations on two conditions

Hi this is an extension of my previous question: Combining two equations for two conditions

I was wanted to know if operations which were to be carried out on both conditions could be placed outside the piecewise brackets. i.e. would the following be valid

$$y = \sum\limits_{i=1}^3 A.\begin{cases} x^2 + b_i &\text{if }x \leq M,\\ x^3 + c_i &\text{if }x \gt M. \end{cases}$$

-

I have seen similar things before, but it's ugly. Here's why I would avoid writing it like this.

First, it appears that only the constant terms $b_i$ and $c_i$ are affected by the summation. Burying everything under the sum symbol leads me to believe that the values of $A$, $M$ or $x$ may also depend on $i$. In short, the notation leads to to the question, "are these really invariant, or did the author forget a subscript?"

I would break this down into two equations, really:

$$y = Au,$$ $$u = \left\{ \begin{array}{cc} x^2+\sum_{i=1}^3 b_i, & \mathrm{if}\ x \le M, \\ x^3+\sum_{i=1}^3 c_i, & \mathrm{if}\ x > M.\end{array}\right.$$

In my personal opinion, this is clearer: it indicates that the structure of $y$ is $A$ times some other term (scalar, vector, whatever) $u$; furthermore, it makes it clear that this term $u$ has a piecewise structure.

Note that the piecewise structure of $u$ need not imply that $y$ also has a piecewise structure; your proposed notation may lead the reader to believe that such is the case. (For instance, in Hammerstein or Wiener non-linear control problems, you generally have a control vector or scalar with some piecewise nonlinearity such as a saturation, backlash, hysteresis, or dead-zone nonlinearity, being fed into a linearized model; the resulting solution of the complete system is not piecewise, despite the piecewise structure of the control term).

-