Extending a matroid by a cocircuit

AbstractOur main result describes how to extend a matroid so that its ground set is a modular hyperplane of the larger matroid. This result yields a new way to view Dowling lattices and new results about line-closed geometries. We complement these topics by showing that line-closure gives simple geometric proofs of the (mostly known) basic results about Dowling lattices. We pursue the topic of line-closure further by showing how to construct some line-closed geometries that are not supersolvable

Minimal blocks of binary even-weight vectors

AbstractOdd circuits are minimal 1-blocks over GF(2) and the odd circuit of size 2t+1 can be represented by the vectors of Hamming weight 2t in a (2t+1)-dimensional vector space over GF(2). This is the tip of an iceberg. Let f(2t,k,2) be the maximum number of binary k-dimensional column vectors such that for all s, 1⩽s⩽t, no 2s columns sum to the zero vector. If k=2, k=3, k=4, or k⩾5 and 2t is sufficiently large (for example, 2t⩾2k−k+1 suffices), then the set of vectors of weight 2t in a (f(2t,k,2)+2t−1)-dimensional vector space over GF(2) is a minimal k-block over GF(2)

The Contributions of Dominic Welsh to Matroid Theory

Dominic Welsh began writing papers in matroid theory nearly forty years ago. Since then, he has made numerous important contributions to the subject. This chapter reviews Dominic Welsh\u27s work in and influence on the development of matroid theory

Polynomial Equivalence of Complexity Geometries

This paper proves the polynomial equivalence of a broad class of definitions of quantum computational complexity. We study right-invariant metrics on the unitary group -- often called `complexity geometries' following the definition of quantum complexity proposed by Nielsen -- and delineate the equivalence class of metrics that have the same computational power as quantum circuits. Within this universality class, any unitary that can be reached in one metric can be approximated in any other metric in the class with a slowdown that is at-worst polynomial in the length and number of qubits and inverse-polynomial in the permitted error. We describe the equivalence classes for two different kinds of error we might tolerate: Killing-distance error, and operator-norm error. All metrics in both equivalence classes are shown to have exponential diameter; all metrics in the operator-norm equivalence class are also shown to give an alternative definition of the quantum complexity class BQP. My results extend those of Nielsen et al., who in 2006 proved that one particular metric is polynomially equivalent to quantum circuits. The Nielsen et al. metric is incredibly highly curved. I show that the greatly enlarged equivalence class established in this paper also includes metrics that have modest curvature. I argue that the modest curvature makes these metrics more amenable to the tools of differential geometry, and therefore makes them more promising starting points for Nielsen's program of using differential geometry to prove complexity lowerbounds.Comment: v2: minor improvements and enhancement