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Summation sign for composition

  1. Oct 15, 2015 #1

    disregardthat

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    I was wondering if anyone knew the standard notation for the following. Suppose I have functions ##f_1,f_2 \ldots,f_n##, is there a compact way of writing ##f_1 \circ f_2 \circ \ldots \circ f_n## ? I am debating whether ##\bigcirc^n_{i=1} f_i## is proper or good notation. Have anyone encountered this notation?
     
  2. jcsd
  3. Oct 15, 2015 #2

    Mark44

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    This seems fairly compact to me.
    IMO, no. I have never seen this used as notation (which doesn't mean that no one has ever used it). I recommend sticking with ##f_1 \circ f_2 \circ \ldots \circ f_n##, as its meaning is obvious while your ##\bigcirc## notation is not.
     
  4. Oct 15, 2015 #3

    disregardthat

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    In my case, each ##f_i## is a series of compositions ##g_{i1} \circ g_{i2} \ldots \circ g_{ir_i}## as well, (in fact, each ##g_{ij}## is equipped with detailed superscripts). Furthermore, it is not one such ##f_1 \circ \ldots f_n## I want to write out, it is a composition of three such expressions, ##f_1 \circ \ldots \circ f_n##, ##h_1 \circ \ldots \circ h_m##, and ##t_1 \circ \ldots \circ t_k##, each ##f_i, h_i## and ##t_i## on the form above, each with detailed superscripts. Writing it out in full will take too much space and look too cluttery.
     
  5. Oct 15, 2015 #4

    Mark44

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    How about this? ##f_1 \circ f_2 \circ \ldots \circ f_n##, where ##f_i = g_{i1} \circ g_{i2} \circ \ldots \circ g_{ik}##
     
  6. Oct 15, 2015 #5

    disregardthat

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    That is what I wanted to avoid. Just like ##\sum_{ij} a_{ij}## is neater than "##b_{1} + b_{2} + \ldots##, where ##b_i = a_{i1} + a_{i2} + \ldots##"
     
  7. Oct 15, 2015 #6

    Fredrik

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    I haven't seen it.

    It's not obvious what ##\bigcirc_{i=1}^3 f_i## should mean.

    \begin{align*}
    &f_1\circ(f_2\circ f_3)\\
    &(f_1\circ f_2)\circ f_3\\
    &f_3\circ(f_2\circ f_1)\\
    &(f_3\circ f_2)\circ f_1
    \end{align*} When you're dealing with linear operators, the notation ##AB## is preferred over ##A\circ B##, and you could write ##\prod_{i=1}^n A_i##.
     
  8. Oct 15, 2015 #7
    Obviously, if you clearly define it and use it repeatedly, readers will be able to cope, but stylistically, it comes across as a circle as used in geometry. Why not define a functional notation using subscript? ## C(f)_{1,n} := f_1 \circ f_2 \circ \ldots \circ f_n ## While personally I like the idea of having a consensually acceptable operator in symmetry to sigma notation, you probably are better off sticking with something a little more conventional.

    $0.02,
    jtv
     
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