APT Graph Colouring

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In order to directly assess DNA strands going through the nanopore, two major problems needed solving. The first was that DNA moved too fast to reliably detect; the second was that individual bases still could not be differentiated, just purines and pyrimidines. In 2005, Bayley (who by then had moved to Oxford) made progress on the first issue, working with scientists at the Scripps Institute to slow the template DNA down by adding short “hairpin” structures that partially blocked off the pore. That year, Bayley co-founded Oxford Nanopore Technologies (ONT) to develop the emerging sequencing method. ONT quickly brought together various technologies, licensing IP from the labs of Bayley, Deamer, Branton, and others.

We show this in a couple of steps. First, as a consequence of how we constructed these fractional parts, we can rewrite any xk as the difference between the k-th multiple of r and the associated integer part vk. After making these substitutions for indices h and g, we get:

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