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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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