2
$\begingroup$

Excuse me for this canonical or simple question. But someone, please, can explain me, also with simplest example for a teacher of an high school, the significance of these symbols/operators?

$$\sum_{\text{cyc}}, \qquad \prod_{\text{cyc}}, \qquad \color{red}{?}$$

PS: I require the simplest explanation.

$\endgroup$
6
  • 2
    $\begingroup$ In what context have you seen these symbols? I'm inclined to think that these are sums and products over elements in cyclic groups, or permutation groups potentially. $\endgroup$
    – Osama Ghani
    Commented Dec 26, 2020 at 23:16
  • $\begingroup$ @OsamaGhani Hi, in some questions of algebra-precalculus where there are inequalities: math.stackexchange.com/questions/1775572/… $\endgroup$
    – Sebastiano
    Commented Dec 26, 2020 at 23:18
  • 5
    $\begingroup$ $\sum\limits_{cyc} a^3+b^2+c=(a^3+b^2+c)+(b^3+c^2+a)+(c^3+a^2+b)$, for example $\endgroup$
    – J. W. Tanner
    Commented Dec 26, 2020 at 23:20
  • 2
    $\begingroup$ Sebastiano--- Tanner's comment is already an example of cyclic sum. To get one for cyclic product just change to $\Pi a^3b^2c$ (with cyc under the big Pi). $\endgroup$
    – coffeemath
    Commented Dec 27, 2020 at 0:00
  • 1
    $\begingroup$ @coffeemath Thank you also for you. But..I wanted give also an upvote and a check mark$\ddot\smile$. I like give the upvotes. $\endgroup$
    – Sebastiano
    Commented Dec 27, 2020 at 10:52

1 Answer 1

Reset to default
4
$\begingroup$

A cyclic summation cycles through all the variables.

For example, if there are three variables, $a, b,$ and $c$,

then $\sum\limits_{cyc} (a^3+b^2c)$ is the sum of $a^3+b^2c$ and that expression with the variables cycled through

(i.e., $a\mapsto b, b\mapsto c, c\mapsto a$, and also $a\mapsto c, b\mapsto a, c\mapsto b$):

$(a^3+b^2c)+(b^3+c^2a)+(c^3+a^2b)$.

Similarly, the cyclic product $\prod\limits_{cyc}(a^3+b^2c)=(a^3+b^2c)(b^3+c^2a)(c^3+a^2b).$

$\endgroup$
3
  • $\begingroup$ Hi, do you know the reason for my actual downvote? $\endgroup$
    – Sebastiano
    Commented Jan 11, 2021 at 20:21
  • 1
    $\begingroup$ Sorry, @Sebastiano, I do not know why your question was down-voted $\endgroup$
    – J. W. Tanner
    Commented Jan 11, 2021 at 20:33
  • 1
    $\begingroup$ Boh, sometimes I have not understood the users. :-( $\endgroup$
    – Sebastiano
    Commented Jan 11, 2021 at 20:35

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .