If there exist indices (i_) and (j_) such that the maximum is negative then stop.
If < i k ′ > is distinct from < i k > and ∑ k = 1 n i k = ∑ k = 1 n i ′ k, there exist indices j and s such that i j ≥ i j ′ + 1 and i s ′ ≥ i s + 1.
By Proposition 2, if ((x,y) in mathcal{S}_{2}), then there exist indices (m,k in mathbb{N}) such that (T^{k} x,y) notin mathcal{B}_{delta } cap mathcal{Q}_{1}) and (T^{-m} x,y) noT^{-m} xcal{B}_{delta }cap mathcal{Q}_{1}).
A is called weakly chained diagonally dominant if (sigma_{i}leq1), (J(A)={iin N: sigma_{i}<1 } neqvarnothing) and for all (iin N/J(A)), there exist indices (i_{1},ldots,i_{k},in{k}) iN N with (a_{i_{l}i_{l+1}}neq0), (0leq lleq k-1), where (i_{0}=i) and (i_{k}in J(A)).
There must exist indices of both S i and S j in z that precede d, are contained in d, and succeed d.
Formally, S contains s i such that may exist indices i, j with i< j implying pos(s i)+ len(s i) > pos(s i +1).
Suppose that there exist indices i ≠ j such that S i ∩ d is a non-prefix/suffix substring of S i and S j ∩ d is a non-prefix/suffix substring of S j.
Two subsequences S = (s1, s2,…, s l s ) and T = (t1, t2,…, t l t ) of a string X overlap if either (i) there exist indices i : 1 ≤ i < l s and j : 1 ≤ j < l t such that i = j, or (ii) there exist indices i, i′ : 1 ≤ i < i′ < l s and a j, j′ : 1 ≤ j < j′ < l t such that either i < j < i′ < j′ or j < i < j′ < i′.
Definition 2. Subsequences S = (s1,..., s k ) and T = (t1,..., t l ) of a string x are overlapping in x if there exist indices i, i' and j, j' such that 1 ≤ i < i' ≤ k, 1 ≤ j < j' ≤ l, and (s i, s i ') and (t j, t j ') are alternating in x.
In order to describe the impact of disagreement between observed and predicted diabetes risk as a function of the proportion of the population where that disagreement exists, an index called the population disagreement index (PDI) was developed.
