Let exq(G; n) be the maximum number of points in a rank-n geometry (simple matroid) that is representable over GF (q) and that has no restriction isomorphic to the geometry G. We find exq(G; n) for several infinite families of geometries G, and we show that if G is a binary affine geometry, then lim n→∞ ex2(G; n) 2n − 1 = 0.

We propose an algorithm for solving optimization problems defined on a subset of the cone of symmetric positive semidefinite matrices. This algorithm relies on the factorization X = Y Y T , where the number of columns of Y fixes an upper bound on the rank of the positive semidefinite matrix X. It is thus very effective for solving problems that have a low-rank solution. The factorization X = Y ...

We consider the problem of finding the spectrum of an operator taking the form of a low-rank (rank one or two) non-normal perturbation of a well-understood operator, motivated by a number of problems of applied interest which take this form. We use the fact that the system is a low-rank perturbation of a solved problem, together with a simple idea of classical differential geometry (the envelop...

If F is a geometry for this rank 3 diagram and F' is its shadow geometry with respect to the central node of the diagram, then f satisfies the Intersection Property (IP), it belongs to the following diagram where L and L* denote the classes of linear spaces and dual linear spaces, respectively. According to a well established habit, we only consider residually connected and firm geometries. Tha...

Basic constructions: The completeness and compactness theorems. The Löwenheim-Skolem theorem. Omitting types. Ultraproducts. Indiscernible sequences. Nice properties: Saturation, homogeneity, and the monster model. Elimination of imaginaries and Meq. Quantifier elimination. Model completeness. Countably categorical theories. Stable theories: Characterizations of a stable formula. Stable, supers...

The robust estimation of multivariate location and shape is one of the most challenging problems in statistics and crucial in many application areas. The objective is to find highly efficient, robust, computable and affine equivariant location and covariance matrix estimates. In this paper three different concepts of multivariate sign and rank are considered and their ability to carry informati...