Arrangements, circular arrangements and the crossing number of C7×Cn

PC trees and circular-ones arrangements

A 0-1 matrix has the consecutive-ones property if its columns can be ordered so that the ones in every row are consecutive. It has the circular-ones property if its columns can be ordered so that, in every row, either the ones or the zeros are consecutive. PQ trees are used for representing all consecutive-ones orderings of the columns of a matrix that has the consecutive-ones property. We give...

Crossing edges and faces of line arrangements in the plane

For any natural number n we define f(n) to be the minimum number with the following property. Given any arrangement A(L) of n blue lines in the real projective plane one can find f(n) red lines different from the blue lines such that any edge in the arrangement A(L) is crossed by a red line. We define h(n) to be the minimum number with the following property. Given any arrangement A(L) of n blu...

Sparse Arrangements and the Number of Views of Polyhedral Scenes

A New Statistic on Linear and Circular r-Mino Arrangements

We introduce a new statistic on linear and circular r-mino arrangements which leads to interesting polynomial generalizations of the r-Fibonacci and r-Lucas sequences. By studying special values of these polynomials, we derive periodicity and parity theorems for this statistic.

Dynamics of circular arrangements of vorticity in two dimensions.

The merger of two like-signed vortices is a well-studied problem, but in a turbulent flow, we may often have more than two like-signed vortices interacting. We study the merger of three or more identical corotating vortices initially arranged on the vertices of a regular polygon. At low to moderate Reynolds numbers, we find an additional stage in the merger process, absent in the merger of two ...