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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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