The aim of the paper is to give an effective formula for the calculation of the probability that a random subset of an affine geometry AG(r−1, q) has rank r. Tables for the probabilities are given for small ranks. The expected time to the first moment at which a random subset of an affine geometry achieves the rank r is derived.

Introduction 1 1. Representations of the fundamental group 3 2. Abelian groups and rank one Higgs bundles 5 3. Stable vector bundles and Higgs bundles 6 4. Hyperbolic geometry: G = PSL(2,R) 8 5. Moduli of hyperbolic structures and representations 13 6. Rank two Higgs bundles 19 7. Split R-forms and Hitchin’s Teichmüller component 21 8. Hermitian symmetric spaces: Maximal representations 24 Refe...

in this paper we enumerate fuzzy subgroups, up to a natural equivalence, of some finite abelian p-groups of rank two where p is any prime number. after obtaining the number of maximal chains of subgroups, we count fuzzy subgroups using inductive arguments. the number of such fuzzy subgroups forms a polynomial in p with pleasing combinatorial coefficients. by exploiting the order, we label the s...

Abstract We construct wonderful compactifications of the spaces linear maps and symmetric a given rank as blowups secant varieties Segre Veronese varieties. Furthermore, we investigate their birational geometry relations with some degree two stable maps.

This is a foundational paper in tropical linear algebra, which is linear algebra over the min-plus semiring. We compare three natural definitions of the rank of a matrix, called the Barvinok rank, the Kapranov rank and the tropical rank, and we show that they differ in general. Connections to polyhedral geometry, particularly to subdivisions of products of simplices, are emphasized.

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 consider the problem of determining the symmetric tensor rank for symmetric tensors with an algebraic geometry approach. We give algorithms for computing the symmetric rank for 2 × · · · × 2 tensors and for tensors of small border rank. From a geometric point of view, we describe the symmetric rank strata for some secant varieties of Veronese varieties.