We show that in every arrangement of n red and blue lines — in general position and not all of the same color — there is a path through a linear number of cells where red and blue lines are crossed alternatingly (and no cell is revisited). When all lines have the same color, and hence the preceding alternating constraint is dropped, we prove that the dual graph of the arrangement always contain...

This paper is about two arrangements of hyperplanes. The first — the Shi arrangement — was introduced by Jian-Yi Shi to describe the Kazhdan-Lusztig cells in the affine Weyl group of type A. The second — the Ish arrangement — was recently defined by the first author who used the two arrangements together to give a new interpretation of the q, t-Catalan numbers of Garsia and Haiman. In the prese...

By means of an apparatus featuring a set of suspended baited containers, search abilities of 4 capuchin monkeys (Cebus apeUa) were evaluated. The experiment featured different spatial configurations of the search space. Results showed that monkeys exhaustively searched 9 containers spatially distributed as a 3 × 3 matrix, a cross, a line, or a circle. Search efficiency was higher when the searc...

order type, 17amoeba, 102 balanced pair, 38basis, 73Baxter, 53bipolar orientation, 47 chirotope, 122, 125closed sector, 104coloring graph, 144complete order type extension, 15completion property, 59convex body, 124convex polyhedron, 127convex polytope, 127convex position, 117crossing family, 20crosspolytope, 79density, 63, 64determ...

Let A be a hyperplane arrangement, and k a field of arbitrary characteristic. We show that the projective degree-one resonance variety R(A, k) of A over k is ruled by lines, and identify the underlying algebraic line complex L(A, k) in the Grassmannian G(2, kn), n = |A|. L(A, k) is a union of linear line complexes corresponding to the neighborly partitions of subarrangements of A. Each linear l...

Given a collection L of line segments, we consider its arrangement and study the problem of covering all cells with line segments of L. That is, we want to find a minimum-size set L′ of line segments such that every cell in the arrangement has a line from L′ defining its boundary. We show that the problem is NP-hard, even when all segments are axis-aligned. In fact, the problem is still NP-hard...

For most algorithms dealing with sets of points in the plane, the only relevant information carried by the input is the combinatorial configuration of the points: the orientation of each triple of points in the set (clockwise, counterclockwise, or collinear). This information is called the order type of the point set. In the dual, realizable order types and abstract order types are combinatoria...

We consider the inverse boundary value problem in the case of discrete electrical networks containing nonlinear (non-ohmic) conductors. For a fixed nonlinear electrical network, we show that under reasonable assumptions, there are well-defined Dirichletto-Neumann and Neumann-to-Dirichlet maps relating boundary voltages and boundary currents. We also generalize work of Curtis, Morrow, and others...