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Cut m;.n  u;.n  _ 1/2 _ Cut

u;.0 y applies u to y after reversing y along each axis; it is equivalent to (0 _1 */$y) u;.0 y .

The fret 0{y (the leading item of y) marks the start of an interval of items of y ; the phrase u;.1 y applies u to each such interval. The phrase u;._1 y differs only in that frets are excluded from the result. In u;.2 and u;._2 the fret is the last item, and marks the ends of intervals.

The monads u;.3 and u;._3 apply u to tesselation by “maximal cubes”, that is, they are defined by their dyadic cases using the left argument ($$y)$<./$y .

m;.n y applies successive verbs from the gerund m to the cuts of y , extending m cyclically as required.
 
  x u;.0 y applies u to a rectangle or cuboid of y with one vertex at the point in y indexed by v=:0{x , and with the opposite vertex determined as follows: the dimension is |1{x , but the rectangle extends back from v along any axis j for which the index j{v is negative. Finally, the order of the selected items is reversed along each axis k for which k{1{x is negative. If x is a vector, it is treated as the matrix 0,:x .

The frets in the dyadic cases 1 , _1 , 2 , and _2 are determined by the 1s in boolean vector x ; an empty vector x and non-zero #y indicates the entire of y . If x is the atom 0 or 1 it is treated as (#y)#x . In general, boolean vector >j{x specifies how axis j is to be cut, with an atom treated as (j{$y)#>j{x .

u;.3 and u;._3 yield (possibly overlapping) tesselations. x u;._3 y applies u to each complete rectangle of size |1{x beginning at integer multiples of (each item of) the movement vector 0{x , with an infinite size being replaced by the signed length of the corresponding axis. As in u;.0 , reversal occurs along each axis for which the size 1{x is negative. The case of a list x is equivalent to 1,:x , and therefore provides a complete tesselation of size x . The case u;.3 differs in that shards of length less than |1{x are included.

x m;.n y applies successive verbs from the gerund m to the cuts of y , extending m cyclically as required.

The 0- and 3-cuts have a left rank of 2; the 1- and 2-cuts have a left rank of 1.
 

   y=: 'worlds on worlds '
   (<;.2 y) ; ($;._2 y) ; (3 5$i.10) ; (+/ ;.1 (3 5$i.10))
+---------------------+-+---------+-----------+
|+-------+---+-------+|6|0 1 2 3 4|5 7 9 11 13|
||worlds |on |worlds ||2|5 6 7 8 9|0 1 2  3  4|
|+-------+---+-------+|6|0 1 2 3 4|           |
+---------------------+-+---------+-----------+

   1 0 1 0 0 <;.1 i.5 7
+--------------------+
|0 1 2  3  4  5  6   |
|7 8 9 10 11 12 13   |
+--------------------+
|14 15 16 17 18 19 20|
|21 22 23 24 25 26 27|
|28 29 30 31 32 33 34|
+--------------------+
   ('';1 0 0 0 1 0 1) <;.1 i.5 7
+-----------+-----+--+
| 0  1  2  3| 4  5| 6|
| 7  8  9 10|11 12|13|
|14 15 16 17|18 19|20|
|21 22 23 24|25 26|27|
|28 29 30 31|32 33|34|
+-----------+-----+--+
   ('';1) <;.1 i.5 7
+--+--+--+--+--+--+--+
| 0| 1| 2| 3| 4| 5| 6|
| 7| 8| 9|10|11|12|13|
|14|15|16|17|18|19|20|
|21|22|23|24|25|26|27|
|28|29|30|31|32|33|34|
+--+--+--+--+--+--+--+
   (1 0 1 0 0;1 0 0 0 1 0 1) <;.1 i.5 7
+-----------+-----+--+
|0 1 2  3   | 4  5| 6|
|7 8 9 10   |11 12|13|
+-----------+-----+--+
|14 15 16 17|18 19|20|
|21 22 23 24|25 26|27|
|28 29 30 31|32 33|34|
+-----------+-----+--+

   x=:1 _2,:_2 3 [ z=: i. 5 5
   x ; (x ];.0 z) ; z
+-----+--------+--------------+
| 1 _2|11 12 13| 0  1  2  3  4|
|_2  3| 6  7  8| 5  6  7  8  9|
|     |        |10 11 12 13 14|
|     |        |15 16 17 18 19|
|     |        |20 21 22 23 24|
+-----+--------+--------------+

   (y=: a. {~ (a. i. 'a') + i. 4 4);(a=: 1 1 ,: 2 2)
+----+---+
|abcd|1 1|
|efgh|2 2|
|ijkl|   |
|mnop|   |
+----+---+

   (<;.3 y) ; ((($$y)$<./$y)<;.3 y) ; (a <;.3 y) ; <(a <;._3 y)
+---------------+---------------+------------+----------+
|+----+---+--+-+|+----+---+--+-+|+--+--+--+-+|+--+--+--+|
||abcd|bcd|cd|d|||abcd|bcd|cd|d|||ab|bc|cd|d|||ab|bc|cd||
||efgh|fgh|gh|h|||efgh|fgh|gh|h|||ef|fg|gh|h|||ef|fg|gh||
||ijkl|jkl|kl|l|||ijkl|jkl|kl|l||+--+--+--+-+|+--+--+--+|
||mnop|nop|op|p|||mnop|nop|op|p|||ef|fg|gh|h|||ef|fg|gh||
|+----+---+--+-+|+----+---+--+-+||ij|jk|kl|l|||ij|jk|kl||
||efgh|fgh|gh|h|||efgh|fgh|gh|h||+--+--+--+-+|+--+--+--+|
||ijkl|jkl|kl|l|||ijkl|jkl|kl|l|||ij|jk|kl|l|||ij|jk|kl||
||mnop|nop|op|p|||mnop|nop|op|p|||mn|no|op|p|||mn|no|op||
|+----+---+--+-+|+----+---+--+-+|+--+--+--+-+|+--+--+--+|
||ijkl|jkl|kl|l|||ijkl|jkl|kl|l|||mn|no|op|p||          |
||mnop|nop|op|p|||mnop|nop|op|p||+--+--+--+-+|          |
|+----+---+--+-+|+----+---+--+-+|            |          |
||mnop|nop|op|p|||mnop|nop|op|p||            |          |
|+----+---+--+-+|+----+---+--+-+|            |          |
+---------------+---------------+------------+----------+


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