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20. Rank
The shape`` (`$`),`` tally`` (`#`),`` and rank`` (`#@$`),`` of a noun are illustrated by the noun`report` ,`` which may be construed as a report covering two years of four quarters of three months each:

       ]report=: i. 2 4 3
     0  1  2
     3  4  5
     6  7  8
     9 10 11

    12 13 14
    15 16 17
    18 19 20
    21 22 23

       ($ ; # ; #@$) report        Shape, Number of items, Rank
    +-----+-+-+
    |2 4 3|2|3|
    +-----+-+-+

The last k axes determine a k-cell of a noun; the 0-cells of`report` are the atoms (such as`4` and`14`),`` the 1-cells are the three-element quarterly reports, and the two-cells (or major cells or items) are the two four-by-three yearly reports.
The rank conjunction`"` is used in the phrase`f"k` to apply a function`f` to each of the`k-`cells of its argument. For example:

       ,report
    0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

       ,"2 report
     0  1  2  3  4  5  6  7  8  9 10 11
    12 13 14 15 16 17 18 19 20 21 22 23

       <@i. s=: 2 5
    +---------+
    |0 1 2 3 4|
    |5 6 7 8 9|
    +---------+
       <@i."0 s
    +---+---------+
    |0 1|0 1 2 3 4|
    +---+---------+

Both the left and right ranks of a dyad may be specified. Thus:

       10 11 12 (,"0 1 ; ,"1 1 ; ,"1) 0 1 2
    +--------+--------------+--------------+
    |10 0 1 2|10 11 12 0 1 2|10 11 12 0 1 2|
    |11 0 1 2|              |              |
    |12 0 1 2|              |              |
    +--------+--------------+--------------+

The basic characteristics adverb`b.` is very useful in analyzing functions (or expressions that define them) with respect to their ranks. For example:

       (# b. 0) ; (+/\ b. 0) ; (+/\ % #) b. 0
    +-----+-----+-----+
    |_ 1 _|_ 0 _|_ _ _|
    +-----+-----+-----+

Exercises

20.1 | Observe the results of the following uses of the monads produced by the rank conjunction, and comment on them:
    a=: i. 3 4 5
    <"0 a
    <"1 a
    <"2 a
    <"3 a
    < a
    <"_1 a
    <"_2 a
    mean=: +/ % #
    mean a
    mean"1 a
    mean"2 a
Answer:`<"k` applies`<` to each cell of rank`k` ,`` with`<"(#$a) a` being equivalent to`<a` .`` Moreover, a negative value of`k` specifies a complementary rank that is effectively`|k` less than the rank of the argument`a` .
20.2 | Use the results of the following experiments to state the relation between the conjunctions`@` (Atop)`` and`@:` (At), and compare your conclusions with the dictionary definition:
    (g=: <"2) a=: i. 3 4 5
    |. @: g a
    |. @ g a
    |: @: (<"1) a
    |: @ (<"1) a
Answer: The rank of the function`|. @: g` is itself infinite and`|.` therefore applies to the entire list result of`g a` ,`` consequently reversing it. On the other hand, the function`f @ g` inherits the rank of`g` ,`` and`|.` therefore applies individually to the atoms produced by`g` ,`` producing no effect.
20.3 | Use the results of the following experiments to comment on the use of the rank conjunction in dyads:
    b=: 'ABC'
    c=: 3 5 $ 'abcdefghijklmno'
    c
    b,c
    b ,"0 1 c
    b ,"1 1 c
    b ,"1 c

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