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5. Forks

As illustrated above, an isolated sequence of three verbs is called a fork; its monadic and dyadic cases are defined by:

       g
 / \
f   h
|   |
y   y 		         g
   / \
  f   h
 / \ / \
x  y x  y

The diagrams above provide visual images of the fork.

Before reading the notes at the right (and by using the facts that %: denotes the root function and ] denotes the identity function), try to state in English the significance of each of the following sentences:

   a=: 8 7 6 5 4 3
   b=: 4 5 6 7 8 9
   2 %: b                     Square root of b
2 2.23607 2.44949 2.64575 2.82843 3
                
   3 %: b                     Cube root of b
1.5874 1.70998 1.81712 1.91293 2 2.08008
                
   (+/ % #) b                 Arithmetic mean or average
6.5
           
   (# %: */) b                Geometric mean
6.26521

   (] - (+/ % #)) b           Centre on mean (two forks)
_2.5 _1.5 _0.5 0.5 1.5 2.5

   (] - +/ % #) b             Two forks (fewer parentheses)
_2.5 _1.5 _0.5 0.5 1.5 2.5

  
   a (+ * -) b                Dyadic case of fork
48 24 0 _24 _48 _72

   (a^2)-(b^2)
48 24 0 _24 _48 _72
                
   a (< +. =) b               Less than or equal
0 0 1 1 1 1
  
   a<b
0 0 0 1 1 1

   a=b
0 0 1 0 0 0

   a (<: = < +. =) b          A tautology (<: is less than or equal)
1 1 1 1 1 1
  
   2 ([: ^ -) 0 1 2           Cap yields monadic case
7.38906 2.71828 1 

   evens=: [: +: i.           +: is double
   evens 7
0 2 4 6 8 10 12

   odds=: [: >: evens         >: is increment
   odds 7
1 3 5 7 9 11 13

Exercises

5.1   Enter 5#3 and similar expressions to determine the definition of the dyad # and then state the meaning of the following sentence: 
(# # >./) b=: 2 7 1 8 2
Answer: #b repetitions of the maximum over b

5.2   Cover the comments on the right, write your own interpretation of each sentence, and then compare your statements with those on the right:
(+/ % #) b                    Mean of b
(# # +/ % #) b                (n=:#b) repetitions of mean
+/(##+/%#) b                  Sum of n means
(+/b)=+/(##+/%#) b            Tautology
(*/b)= */(###%:*/) b          The product over b is the product over n
                              repetitions of the geometric mean of b


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