In this paper we introduce some monomial orders for the class of closed path polyominoes and prove that set generators polyomino ideal attached to a forms reduced Gröbner basis with respect these orders. It is known containing an $L$-configuration or ladder at least three steps, equivalently having no zig-zag walks, prime. As consequence, obtain coordinate ring walks normal Cohen-Macaulay domain.

This paper describes the application of biologically-inspired algorithms and concepts to the design of wideband antenna arrays. In particular, we address two specific design problems. The first involves the design of a constrained-feed network for a Rotman-lens beamformer. We implemented two evolutionary optimization (EO) approaches, namely a simple genetic algorithm (SGA) and a competent genet...

What is the maximum number of holes enclosed by a $d$-dimensional polyomino built $n$ tiles? Represent this $f_d(n)$. Recent results show that $f_2(n)/n$ converges to $1/2$. We prove for all $d \geq 2$ we have $f_d(n)/n \to (d-1)/d$ as goes infinity. also construct polyominoes in tori with maximal possible per tile. In our proofs, use metaphors from error-correcting codes and dynamical systems.

A plane tiling consisting of congruent copies of a shape is isohedral provided that for any pair of copies, there exists a symmetry of the tiling mapping one copy to the other. We give a O(n log2 n)-time algorithm for deciding if a polyomino with n edges can tile the plane isohedrally. This improves on the O(n18)-time algorithm of Keating and Vince and generalizes recent work by Brlek, Provença...

For a set D of polyominoes, a packing of the plane with D is a maximal set of copies of polyominoes from D that are not overlapping. A packing with smallest density is called a clumsy packing. We give an example of a set D such that any clumsy packing is aperiodic. In addition, we compute the smallest possible density of a clumsy packing when D consists of a single polyomino of a given size and...

For a set D of polyominoes, a packing of the plane with D is a maximal set of copies of polyominoes from D that are not overlapping. A packing with smallest density is called clumsy packing. We give an example of a set D such that any clumsy packing is aperiodic. In addition, we compute the smallest possible density of a clumsy packing when D consist of a single polyomino of a given size and sh...