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Radical initiator
From formulasearchengine
Revision as of 16:44, 20 January 2014 by en>Smokefoot(mention ATRP, reference March which discusses this area extensively)
When we apply the pairing function to and we often denote the resulting number as
This definition can be inductively generalized to the Cantor tuple function
as
Inverting the Cantor pairing function
Suppose we are given z with
and we want to find x and y. It is helpful to define some intermediate values in the calculation:
where t is the triangle number of w. If we solve the quadratic equation
for w as a function of t, we get
which is a strictly increasing and continuous function when t is non-negative real. Since
we get that
and thus
.
where is the floor function.
So to calculate x and y from z, we do:
.
Since the Cantor pairing function is invertible, it must be one-to-one and onto.
References
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