Lek-Heng Lim University of Chicago 15.1 Hypermatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-2 15.2 Tensors and Multilinear Functionals. . . . . . . . . . . . . . . . . 15-6 15.3 Tensor Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-12 15.4 Border Rank . . . . . . . . . . . . . . . . . . . . . . . ....

We prove that a simple geometric theory of SU-rank 1 is n-ample if and only if the associated theory equipped with an predicate for an independent dense subset is n-ample for n at least 2.

We explore the size of the largest (permuted) triangular submatrix of a random matrix, and more precisely its asymptotical behavior as the size of the ambient matrix tends to infinity. The importance of such permuted triangular submatrices arises when dealing with certain combinatorial algebraic settings in which these submatrices determine the rank of the ambient matrix, and thus attract a spe...

Motivated by work of Stembridge, we study rank functions for Viennot’s heaps of pieces. We produce a simple and sufficient criterion for a heap to be a ranked poset and apply the results to the heaps arising from fully commutative words in Coxeter groups. To appear in the Journal of Combinatorial Theory, Series A

We provide a general theorem implying that for a (strongly) dependent theory T the theory of su ciently well-behaved pairs of models of T is again (strongly) dependent. We apply the theorem to the case of lovely pairs of thorn-rank one theories as well as to a setting of dense pairs of rst-order topological theories.

We describe the system that won Track 1 of the Yahoo! Learning to Rank Challenge.

In real applications, our observations are often noisy, or even grossly corrupted, and some observations may be missing. This fact naturally leads to the problem of recovering a low-rank matrix X from a corrupted observation matrix X = X + E (each column of X is an observation vector), with E being the unknown noise. Due to the low-rank property of X, it is straightforward to consider the follo...

In this paper, using the Hojman construction, we give examples of various Poisson brackets which differ from those which are usually analyzed in Hamiltonian mechanics. They possess a nonmaximal rank, and in the general case an invariant measure and Casimir functions can be globally absent for them.

In this paper we further develop the theory of matrices over the extended tropical semiring. Introducing a notion of tropical linear dependence allows for a natural definition of matrix rank in a sense that coincides with the notions of tropical regularity and invertibility.

Manually assigned keywords provide a valuable means for accessing large document collections. They can serve as a shallow document summary and enable more efficient retrieval and aggregation of information. In this paper we investigate keywords in the context of the Dutch Folktale Database, a large collection of stories including fairy tales, jokes and urban legends. We carry out a quantitative...