Publication

Abstract : Let Ωn denote the set of all doubly stochastic matrices of order n. Lih and Wang conjectured that for n≥3, per(tJn+(1−t)A)≤tperJn+(1−t)perA, for all A∈Ωn and all t∈[0.5,1], where Jn is the n×n matrix with each entry equal to 1n. This conjecture was proved partially for n≤5. \\ \indent Let Kn denote the set of non-negative n×n matrices whose elements have sum n. Let ϕ be a real valued function defined on Kn by ϕ(X)=∏ni=1ri+∏nj=1cj - perX for X∈Kn with row sum vector (r1,r2,...rn) and column sum vector (c1,c2,...cn). A matrix A∈Kn is called a ϕ-maximizing matrix if ϕ(A)≥ϕ(X) for all X∈Kn. Dittert conjectured that Jn is the unique ϕ-maximizing matrix on Kn. Sinkhorn proved the conjecture for n=2 and Hwang proved it for n=3. \\ \indent In this paper, we prove the Lih and Wang conjecture for n=6 and Dittert conjecture for n=4.