Probabilistic lower bounds on maximal determinants of binary matrices

General Lower Bounds on Maximal Determinants of Binary Matrices

We prove general lower bounds on the maximal determinant of n× n {+1,−1}matrices, both with and without the assumption of the Hadamard conjecture. Our bounds improve on earlier results of de Launey and Levin (2010) and, for certain congruence classes of n mod 4, the results of Koukouvinos, Mitrouli and Seberry (2000). In an Appendix we give a new proof, using Jacobi’s determinant identity, of a...

Lower bounds on maximal determinants of binary matrices via the probabilistic method

Let D(n) be the maximal determinant for n × n {±1}-matrices, and R(n) = D(n)/n be the ratio of D(n) to the Hadamard upper bound. We give several new lower bounds on R(n) in terms of d, where n = h + d, h is the order of a Hadamard matrix, and h is maximal subject to h ≤ n. A relatively simple bound is R(n) ≥ ( 2 πe )d/2( 1− d ( π 2h )1/2) for all n ≥ 1. An asymptotically sharper bound is R(n) ≥...

Lower bounds on maximal determinants of ±1 matrices via the probabilistic method

We show that the maximal determinant D(n) for n × n {±1}matrices satisfies R(n) := D(n)/nn/2 ≥ κd > 0. Here nn/2 is the Hadamard upper bound, and κd depends only on d := n − h, where h is the maximal order of a Hadamard matrix with h ≤ n. Previous lower bounds on R(n) depend on both d and n. Our bounds are improvements, for all sufficiently large n, if d > 1. We give various lower bounds on R(n...

Lower bounds of Copson type for the transpose of matrices on weighted sequence spaces

Lower Bounds of Copson Type for Hausdorff Matrices on Weighted Sequence Spaces

Let = be a non-negative matrix. Denote by the supremum of those , satisfying the following inequality: where , , and also is increasing, non-negative sequence of real numbers. If we used instead of The purpose of this paper is to establish a Hardy type formula for , where is Hausdorff matrix and A similar result is also established for where In particular, we apply o...