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Insert m/ u/  _ _ _ Table

u/y applies the dyad u between the items of y . Thus:
   m=: i. 3 2
   m;(+/m);(+/"1 m);(+/2 3 4)
+---+---+-----+-+
|0 1|6 9|1 5 9|9|
|2 3|   |     | |
|4 5|   |     | |
+---+---+-----+-+
m/y inserts successive verbs from the gerund m between items of y, extending m cyclically as required. Thus, +`*/i.6 is 0+1*2+3*4+5 .
 		   If x and y are numeric lists, then x */ y is their multiplication table. Thus:
   1 2 3 */ 4 5 6 7
 4  5  6  7
 8 10 12 14
12 15 18 21
In general, each cell of x is applied to the entire of y . Thus x u/ y is equivalent to x u"(lu,_) y where lu is the left rank of u .

The case */ is called outer product in tensor analysis.
 

If y has no items (that is, 0=#y), the result of u/y is the neutral or identity element of the function u . A neutral of a function u is a value e such that x u e ↔ x or e u x ↔ x, for every x in the domain (or some significant sub-domain such as boolean) of u . This definition of insertion over an argument having zero items extends partitioning identities of the form u/y ↔ (u/k{.y) u (u/k}.y) to the cases k e. 0,#y .

The identity function of u is a function ifu such that ifu y ↔ u/y if 0=#y . The identity functions used are:

Identity function For
 
0  $~ }.@$ <  >  +  -  +.  ~:  |  (2 4 5 6 b.)
1  $~ }.@$ =  <:  >:  *  %  *.  %:  ^  !  (1 9 11 13 b.)
_  $~ }.@$ <.
__ $~ }.@$ >.
(v^:_1 ifu$0) $~ }.@$   u&.v
i.@(0&,)@(2&}.)@$ ,
/:@{. C.  {
=@/:@{. %.  +/ . *
ifu@# u/



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