Related papers: The Pattern Complexity of the Squiral Tiling

We aim at studying collections of algebraic structures defined over a commutative ring and investigating the complexity of significant constructions carried out on these objects. The assignment of measures of size, via a multiplicity…

We determine all tight Lagrangian surfaces in $S^2 \times S^2$. In particular, globally tight Lagrangian surfaces in $S^2 \times S^2$ are nothing but real forms.

We generalise our earlier work on the number of squares in binary recurrence sequences, $\left\{ y_{k} \right\}_{k \geq -\infty}$. In the notation of our previous papers, here we consider the case when $N_{\alpha}$ is any negative integer…

Predicting the optical response of macroscopic arrangements of individual scatterers is a computational challenge, as the problem involves length scales across multiple orders of magnitude. We present a full-wave optical method to highly…

We consider the problem of finding 4 rational squares, such that the product of any two plus the sum of the same two always gives a square. We give some historical background and exhibit one such quadruple.

A formula for the number of toroidal m x n binary arrays, allowing rotation of the rows and/or the columns but not reflection, is known. Here we find a formula for the number of toroidal m x n binary arrays, allowing rotation and/or…

A method to define the complex structure and separate the conformal mode is proposed for a surface constructed by two-dimensional dynamical triangulation. Applications are made for surfaces coupled to matter fields such as $n$ scalar fields…

In this paper we provide a unified combinatorial approach to establish a connection between Stirling permutations, cycle structures of permutations and perfect matchings. The main tool of our investigations is MY-sequences. In particular,…

Given a graph $G$ and collection of subgraphs $T$ (called tiles), we consider covering $G$ with copies of tiles in $T$ so that each vertex $v\in G$ is covered with a predetermined multiplicity. The multinomial tiling model is a natural…

This article is dedicated to domino tilings of square grids. In each of these grids domino tilings are represented using linear-recurrent sequences. For different grids are determined new dependencies.

We give an explicit formula for the number of permutations avoiding cyclically a consecutive pattern in terms of the spectrum of the associated operator of the consecutive pattern. As an example, the number of cyclically consecutive…

In this article we give explicit descriptions of the multiplicities of some classes of monomial ideals. For instance, we give a formula for the multiplicities of all codimension 1 monomial ideals, and another formula for the multiplicities…

We consider a special type of self-similar sets, called fractal squares, and give a brief review on recent results and unsolved issues with an emphasis on their topological properties.

We propose a simple formula for multichannel resonant scattering with parameters related to physical resonant properties. It can be used to predict residue phase from other resonant parameters and describe the shape of scattering amplitudes…

Suppose $P$ is a symmetric convex polygon in the plane. We give a polynomial time algorithm that decides if $P$ can tile the plane by transations at some level (not necessarily at level one; this is multiple tiling). The main technical…

We solve the Picard number problem for complex quintic surfaces by proving that every number between 1 and 45 occurs as Picard number of a quintic surface over the rationals. Our main technique consists in arithmetic deformations of…

We derive a general recurrence relation for squares of Fibonacci-like numbers. Various properties are developed, including double binomial summation identites.

We determine the crossing number of polynomial size curve systems on standard surfaces, in terms of the genus, up to high precision.

Tiling planar regions with dominoes is a classical problem in which the decision and counting problems are polynomial. We prove a variety of hardness results (both NP- and #P-completeness) for different generalizations of dominoes in three…

This paper addresses the question of whether a single tile with nearest neighbor matching rules can force a tiling in which the tiles fall into a large number of isohedral classes. A single tile is exhibited that can fill the Euclidean…