Let X be a smooth projective curve of genus g ≥ 2 defined over an algebraically closed field k of characteristic p > 0. Given a semistable vector bundle E over X , we show that its direct image F∗E under the Frobenius map F of X is again semistable. We deduce a numerical characterization of the stable rank-p vector bundles F∗L, where L is a line bundle over X .

The compressibility of Schur complement matrices is the essential ingredient for H-matrix techniques, and is well understood for Laplace type problems. The Helmholtz case is more difficult: there are several theoretical results which indicate when good compression is possible with additional techniques, and in practice sometimes basic H-matrix techniques work well. We investigate the compressib...

It has been shown that a best rank-R approximation of an order-k tensor may not exist when R ≥ 2 and k ≥ 3. This poses a serious problem to data analysts using tensor decompositions. It has been observed numerically that, generally, this issue cannot be solved by consecutively computing and subtracting best rank-1 approximations. The reason for this is that subtracting a best rank-1 approximati...

A transversal of a hypergraph is a set of vertices intersecting each hyperedge. We design and analyze new exponential-time algorithms to enumerate all inclusion-minimal transversals of a hypergraph. For each fixed k ≥ 3, our algorithms for hypergraphs of rank k, where the rank is the maximum size of a hyperedge, outperform the previous best. This also implies improved upper bounds on the maximu...

A bounded linear operator acting on a Hilbert space is a generalized quadratic operator if it has an operator matrix of the form [ aI cT dT ∗ bI ] . It reduces to a quadratic operator if d = 0. In this paper, spectra, norms, and various kinds of numerical ranges of generalized quadratic operators are determined. Some operator inequalities are also obtained. In particular, it is shown that for a...

Randomized sampling has recently been demonstrated to be an efficient technique for computing approximate low-rank factorizations of matrices for which fast methods for computing matrix vector products are available. This paper describes an extension of such techniques to a wider class of matrices that are not themselves rankdeficient, but have off-diagonal blocks that are. Such matrices arise ...

We introduce a randomized procedure that, given an m×n matrix A and a positive integer k, approximates A with a matrix Z of rank k. The algorithm relies on applying a structured l×m random matrix R to each column of A, where l is an integer near to, but greater than, k. The structure of R allows us to apply it to an arbitrary m× 1 vector at a cost proportional to m log(l); the resulting procedu...

An n×n correlation matrix has k factor structure if its off-diagonal agrees with that of a rank k matrix. Such correlation matrices arise, for example, in factor models of collateralized debt obligations (CDOs) and multivariate time series. We analyze the properties of these matrices and, in particular, obtain an explicit formula for the rank in the one factor case. Our main focus is on the nea...

Animal feed is commonly formulated with vitamin and mineral premixes to supply the micronutrient requirements of animal. Premixes typically require carriers which act as a delivery system for micronutrients aid in uniformity, dilution, dispersibility across entire mix. However, little information available regarding how different might influence dispersion nutrients throughout premix. Therefore...

We address the problem of computing a low-rank estimate Y of the solution X of the Lyapunov equation AX + XA′ + Q = 0 without computing the matrix X itself. This problem has applications in both the reduced-order modeling and the control of large dimensional systems as well as in a hybrid algorithm for the rapid numerical solution of the Lyapunov equation via the alternating direction implicit ...