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Although not significant under our cut-off, a higher than expected proportion of these genes lies on chromosome 2 L (Odds-ratio = 1.47, P = 0.045).
We prove that when l- odd, k, s- even (the other cases are similar and will be omitted).
We see from Equation (1) when l- odd, k- even that p = a q + b p + c q d p + e q, and q = a p + b q + c p d q + e p. Then d p 2 + e p q = a d p q + a e q 2 + b p + c q, (9).
Specifically, given a window size L (L is an odd number), the centroid coverage of the nucleotide i in the middle of the window is defined as: (2) Centroid i = log 10 9 N * 1 L ∑ k = i - L - 1 / 2 i + L - 1 / 2 Coverage k + 1, Where i is the i-th position (nucleotide) on the chromosome.
If l is odd, size of the feature vector will be 4 l /2, and if l is even, the size will be (4 l +4 l /2 /2.
So the mapping L is odd.
Thus, for convenience, we write G ∗ as G σ when l is odd.
If l is odd, then a 2 i ( G σ ) = m ( G, i ).
For the UCA with 2L-1 eLements (L assumed odd), the array position vectors are, (4.11).
Let μ ∈ R be a characteristic value of L of odd multiplicity.
From Lemma 4.1, we can see that the skew energy of G σ is independent of the orientation σ when l is odd.
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