Normal toric varieties over a field or a discrete valuation ring are classified by rational polyhedral fans. We generalize this classification to normal toric varieties over an arbitrary valuation ring of rank one. The proof is based on a generalization of Sumihiro’s theorem to this non-noetherian setting. These toric varieties play an important role for tropicalizations. 2010 Mathematics Subje...

Let M be a tropical matrix (k + x) × (k + x ′) for some k, x , x ′ ∈ N − {0} with tropical rank k. We show that Kapranov rank is k too if x and x ′ are not too big; namely if we are in one of the following cases: a) k ≥ 6 and x , x ′ ≤ 2 b) k = 4, 5, x ≤ 2 and x ′ ≤ 3 (or obviuosly the converse, that is x ≤ 3 and x ′ ≤ 2) c) k = 3 and either x , x ′ ≤ 3 or x ≤ 2 and x ′ ≤ 4 (or obviuosly the co...

We investigate relations between different width parameters of graphs, in particular balanced separator number, treewidth, and cycle rank. Our main result states that a graph with balanced separator number k has treewidth at least k but cycle rank at most k · ( 1 + log nk ) , thus refining the previously known bounds, as stated by Robertson and Seymour (1986) and by Bodlaender et al. (1995). Fu...

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...