The cubic chessboard

From chessboard tweed to chessboard nanowire structure during pseudospinodal decomposition.

Microstructure evolution in complex nonlinear systems is of great interest from both scientific and engineering viewpoints. Here, we consider an important case of such an evolution, a coherent decomposition of a homogeneous parent phase involving the symmetry-lifting crystal lattice rearrangement of the product phase. It is shown that under certain conditions the transformation develops as a ps...

Enumerating Distinct Chessboard Tilings

Counting the number of distinct colorings of various discrete objects, via Burnside’s Lemma and Pólya Counting, is a traditional problem in combinatorics. Motivated by a method for proving upper bounds on the order of the minimal recurrence relation satisfied by a set of tiling instances, we address a related problem in a more general setting. Given an m× n chessboard and a fixed set of (possib...

Combinatorial Chessboard Rearrangements

Many problems concerning tilings of rectangular boards are of significant combinatorial interest. In this paper we introduce a similar type of counting problem based on game piece rearrangements. Many of these rearrangements satisfy recurrence relations which can be computed using various combinatorial techniques. We also present the solution to a rearrangement counterpart to the well–known kni...

Chessboard complexes indomitable

We give a simpler, degree-theoretic proof of the striking new Tverberg type theorem of Blagojević, Ziegler and Matschke, arXiv:0910.4987v2. Our method also yields some new examples of “constrained Tverberg theorems” including a simple colored Radon’s theorem for d + 3 points in Rd. This gives us an opportunity to review some of the highlights of this beautiful theory and reexamine the role of c...

The Rook on the Half - Chessboard , or