Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

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

share|cite|improve this question
up vote 0 down vote accepted

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).

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.