Abstract. The numerical range of a matrix is studied geometrically via the cone of positive semidefinite matrices (or semidefinite cone for short). In particular, it is shown that the feasible set of a two-dimensional linear matrix inequality (LMI), an affine section of the semidefinite cone, is always dual to the numerical range of a matrix, which is therefore an affine projection of the semid...

Replacement of corn by citrus pulp or coffee hulls explores the potential of dairy cows to digest fiber-rich feedstuffs. However, for the State of Minas Gerais, Brazil, replacing citrus pulp by coffee hulls may reduce milk production costs, since citrus pulp needs to be imported from another state, while coffee hulls are highly available at essentially no cost. The objective of this experiment ...

We describe an algorithm for the numerical computation of the rank-one convex envelope of a function f : Mm×n → R. We prove its convergence and an error estimate in L∞.

We study matrix sketching methods for regularized variants of linear regression, low rank approximation, and canonical correlation analysis. Our main focus is on sketching techniques which preserve the objective function value for regularized problems, which is an area that has remained largely unexplored. We study regularization both in a fairly broad setting, and in the specific context of th...

In this paper we study a generalization of Kruskal’s permutation lemma to partitioned matrices. We define the k’-rank of partitioned matrices as a generalization of the k-rank of matrices. We derive a lower-bound on the k’-rank of Khatri–Rao products of partitioned matrices. We prove that Khatri–Rao products of partitioned matrices are generically full column rank.

This paper presents a new variation of Tverberg’s theorem. Given a discrete set S of R, we study the number of points of S needed to guarantee the existence of an m-partition of the points such that the intersection of the m convex hulls of the parts contains at least k points of S. The proofs of the main results require new quantitative versions of Helly’s and Carathéodory’s theorems.

We show that, if k and ` are positive integers and r is sufficiently large, then the number of rank-k flats in a rank-r matroid M with no U2,`+2-minor is less than or equal to number of rank-k flats in a rank-r projective geometry over GF(q), where q is the largest prime power not exceeding `.

We give algorithms for approximation by low-rank positive semidefinite (PSD) matrices. For symmetric input matrix A ∈ Rn×n, target rank k, and error parameter ε > 0, one algorithm finds with constant probability a PSD matrix Ỹ of rank k such that ‖A− Ỹ ‖2F ≤ (1+ε)‖A−Ak,+‖ 2 F , where Ak,+ denotes the best rank-k PSD approximation to A, and the norm is Frobenius. The algorithm takes time O(nnz(A...

The canonical height ĥ on an abelian variety A defined over a global field k is an object of fundamental importance in the study of the arithmetic of A. For many applications it is required to compute ĥ(P ) for a given point P ∈ A(k). For instance, given generators of a subgroup of the Mordell-Weil group A(k) of finite index, this is necessary for most known approaches to the computation of gen...