The solution of linear systems arising from the linear stability analysis of solutions of the Navier–Stokes equations is considered. Due to indefiniteness of the submatrix corresponding to the velocities, these systems pose a serious challenge for iterative solution methods. In this paper, the augmented Lagrangian-based block triangular preconditioner introduced by the authors in [2] is extende...

A problem of paramount importance in both pure (Restricted Invertibility problem) and applied mathematics (Feature extraction) is the one of selecting a submatrix of a given matrix, such that this submatrix has its smallest singular value above a specified level. Such problems can be addressed using perturbation analysis. In this paper, we propose a perturbation bound for the smallest singular ...

Definition 1.1. A matrix A ∈ Zm×n is totally unimodular if the determinant of every square submatrix B ∈ Zk×k equals either −1, 0, or +1. Alternatively, by Cramer’s rule, A ∈ Zm×n is totally unimodular if every nonsingular submatrix B ∈ Zk×k has an integral inverse B−1 ∈ Zk×k. Recall: B−1 = 1 det B B ∗, where B∗ is the adjugate matrix (transpose of the matrix of cofactors) of B. One important c...

Suppose λ1 ≥ · · · ≥ λn ≥ 0 are the eigenvalues of an n × n totally nonnegative matrix, and λ̃1 ≥ · · · ≥ λ̃k are the eigenvalues of a k × k principal submatrix. A short proof is given of the interlacing inequalities: λi ≥ λ̃i ≥ λi+n−k, i = 1, . . . , k. It is shown that if k = 1, 2, n− 2, n− 1, λi and λ̃j are nonnegative numbers satisfying the above inequalities, then there exists a totally nonneg...

We study a block triangular preconditioner for finite element approximations of the linearized Navier–Stokes equations. The preconditioner is based on the augmented Lagrangian formulation of the problem and was introduced by the authors in [SIAM J. Sci. Comput., 28 (2006), pp. 2095–2113]. In this paper we prove field-of-values type estimates for the preconditioned system which lead to optimal c...

We present efficient data structures for submatrix maximum queries in Monge matrices and Monge partial matrices. For n× n Monge matrices, we give a data structure that requires O(n) space and answers submatrix maximum queries in O(logn) time. The best previous data structure [Kaplan et al., SODA‘12] required O(n logn) space and O(log n) query time. We also give an alternative data structure wit...

Syndrome-Trellis Code (STC) is a near-optimal convolutional method for adaptive steganography. Hitherto, the existing steganography commonly depends on carefully designed distortion cost function, which controls embedding position of message in cover signal. From another point view, we implement by improving STC coding process (named Adaptive-STC). The parity-check matrix key to encoding and ex...

We develop the quantum inverse scattering method for the one-dimensional Hubbard model on the infinite interval at zero density. R-matrix and monodromy matrix are obtained as limits from their known counterparts on the finite interval. The R-matrix greatly simplifies in the considered limit. The new R-matrix contains a submatrix which turns into the rational R-matrix of the XXX-chain by an appr...

We propose a convex optimization formulation with the Ky Fan 2-k-norm and `1-norm to find k largest approximately rank-one submatrix blocks of a given nonnegative matrix that has low-rank block diagonal structure with noise. We analyze low-rank and sparsity structures of the optimal solutions using properties of these two matrix norms. We show that, under certain hypotheses, with high probabili...

7 Given an n × n matrix, its principal rank characteristic sequence is a sequence of 8 length n + 1 of 0s and 1s where, for k = 0, 1, . . . , n, a 1 in the kth position indicates 9 the existence of a principal submatrix of rank k and a 0 indicates the absence of such 10 a submatrix. The principal rank characteristic sequences for symmetric matrices over 11 various fields are investigated, with ...