User:Francis Tyers/Maths

i = 0;
for(n = 0; n < 10; n++)
{
  i = i + i;
}

$\sum _{i=0}^{10}i$

Kronecker Delta and Iverson Notation

The Kronecker Delta function below allows you to transform logical expressions into 0 and 1 integer values available for summing over sets, or for removing terms from an equation where you multiply a given term by the delta to keep it in or not dependent on some logical restriction.

$\delta _{ij}=\left\{{\begin{matrix}1,&{\mbox{if }}i=j\\0,&{\mbox{if }}i\neq j\end{matrix}}\right.$

In computer science this function became so useful for expressing algorithms it's frequently described using a more lightweight and intuitive notation called the Iverson bracket popularised by Kenneth E. Iverson. The Iverson bracket is similarly defined as the Kronecker Delta:

$[f]=\left\{{\begin{matrix}1,&{\text{if }}f{\text{ is true;}}\\0,&{\text{otherwise.}}\end{matrix}}\right.$

Imagine we wish to count the total number of instances of "the" in an array of words from a piece of text:

count = 0
for word in words
    count += 1 if word == "the"
print count

With Iverson bracket notation we would write this thusly:

let W be a set of sequences of characters from the English alphabet
let c be the sum of sequences w that are equal to (t,h,e)

$c=\sum _{w\in {}W}^{}[w=(t,h,e)]$