tailieunhanh - Nanotechnology Science and Computation part 5

Tham khảo tài liệu 'nanotechnology science and computation part 5', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 94 . Reif S. Sahu P. Yin Lemma 2. Suppose that the neighborhood tiles independent of tile T2 i j have correctly computed V and U . If there is a single pad mismatch between tile T fi j and another tile just below or to its immediate right then there are at least two further pad mismatches between pairs of adjacent tiles in the immediate neighborhood of tile T fi j . Furthermore given the location of the initial mismatch the location of the second mismatch can be determined among at most three locations in the neighborhood of T2 i j given the location of the initial and the second mismatches the location of the third mismatch can be determined among at most five locations. Proof. Suppose a pad binding error occurs on a bottom pad or right pad of tile T2 i j but no further pad mismatch occurs between two neighborhood tiles which are independent of T2 i j . We now consider a case-by-case analysis of possible pad mismatches. Case Tfii j 1 V i-1 j xx V 3rd mismatch V i-1 j-1 V i-1 j-2 V i-1 j-1 Mismatch h-4- A Ad Tfii j - 1 V i j-1 Fig. 8. Case in the proof of Lemma 2. 1 First consider the case where the pad binding error occurs on the V i 1 j 2 portion of the bottom pad of tile T2 i j . Consider the case where the pad binding error is due to the incorrect value of the right portion V i 1 j 2 of the bottom pad of tile T2 i j there may also be the incorrect value of the other portions of the bottom pad of tile T2 i j . Further consider case Fig. 8 when there is no mismatch on the bottom portion U i 2 j 1 of the right pad. Immediately we have a mismatch on the portion V i 1 j 1 of the right pad of T2 i j since V i 1 j 1 U i 2 j 1 OP1 V i 1 j 2 and OP1 is XOR. Furthermore tile T2 i j will determine an incorrect value for the V i 1 j 1 portion of Error-Resilient DNA Tilings 95 Case Tz i j 1 V i-1 j V i-1 j-1 T2 i j - 1 Fig. 9. Case in the proof of Lemma 2 its top pad resulting in a mismatch either on the bottom or on the right pad of T2 i