- if A in K
_{n}is partly decomposable then φ(A)<φ(J_{n}); - if the zeroes in A in K
_{n}form a block then A is not a φ-maximising matrix; - φ(A)<φ(J
_{n}) unless δ:=per(J_{n})-per(A)≤ O(n^{4}e^{-2n}) and|k-Σ

_{i inα}r_{i}| < √2δk, |k-Σ_{i inβ}c_{i}| < √2δk and Σ_{i in α,j in β}a_{ij}< k+√δkfor all sets α,β of k integers chosen from {1,2,...,n}.

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