CUT (_) The phrase u;. 1 y applies u to each of a set of intervals of items of y to produce the items of the result. Each interval begins at an occurrence of the delimiter 0{y. For example, if s=. 5 3:
<;.1 y=.' worlds on worlds'
+-------+---+-------+
| worlds| on| worlds|
+-------+---+-------+
$;.1 y s$i.9 +/;.1 s$i.9
7 0 1 2 9 12 15
3 3 4 5 3 5 7
7 6 7 8
0 1 2
3 4 5
The phrase u;._1 y differs only in that the delimiters are excluded from the intervals. In u;.2 and u;._2 the delimiter is the last item, and marks the ends of intervals.
The phrase u;.0 y applies u to y after reversing y along each axis, and is equivalent to (0 _1 */$y) u;.0 y.
The monads u;.3 and u;._3 apply u to tessellation by "maximal cubes", that is, they are defined by their dyadic cases using the left argument ($$y)$<./$y.
CUT (_ _) The dyads u;.1 and u;._1 and u;.2 and u;._2 differ from the monads in that the intervals are delimited by the ones in the boolean argument x. Thus:
x=. 0 1 0 0 1 [ y=. i. 5 3
y x+/;.1 y x+/;.2 y
0 1 2 18 21 24 3 5 7
3 4 5 12 13 14 27 30 33
6 7 8
9 10 11
12 13 14
The phrase 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 for which the index k{v is negative. Finally, the order of the selected elements is reversed along each axis k for which k{1{x is negative. For example:
(1 _2,:_2 3)+;.0 i. 4 5
11 12 13
6 7 8
The cases u;.3 and u;._3 provide (possibly overlapping) tessellations. The phrase x u;._3 y applies u to each complete rectangle of size |1{x obtained by beginning at all positions obtained as integer multiples of (each element of) the movement vector 0{x. As in u;.0, reversal of each piece occurs along an axis for which the dimension 1{x is negative.
The degenerate case of a list x is equivalent to the left argument 1,:x, and therefore provides a complete tessellation of size x.
The case u;.3 differs from u;._3 only in that any shards of sizes less than |1{x are included. For example:
