Polyomino convolutions and tiling problems

We define a convolution operation on the set of polyominoes and use it to obtain a criterion for a given polyomino not to tile the plane (rotations and translations allowed). We apply the criterion to several families of polyominoes, and show that the criterion detects some cases that are not detectable by generalized coloring arguments.Comment: 8 pages, 8 figures. To appear in \emph{J. of Combin. Theory Ser. A

Modelling quasicrystals at positive temperature

We consider a two-dimensional lattice model of equilibrium statistical mechanics, using nearest neighbor interactions based on the matching conditions for an aperiodic set of 16 Wang tiles. This model has uncountably many ground state configurations, all of which are nonperiodic. The question addressed in this paper is whether nonperiodicity persists at low but positive temperature. We present arguments, mostly numerical, that this is indeed the case. In particular, we define an appropriate order parameter, prove that it is identically zero at high temperatures, and show by Monte Carlo simulation that it is nonzero at low temperatures

Aperiodic Tilings: Breaking Translational Symmetry

Classical results on aperiodic tilings are rather complicated and not widely understood. Below, an alternative approach is discussed in hope to provide additional intuition not apparent in classical works.Comment: 4 pages, 2 figures, minor change

X-ray - Infrared relation of AGNs and search for highly obscured accretion in the AKARI NEP Field

The infrared Astronomical Satellite AKARI conducted deep ($\sim$ 0.4 deg$^2$) and wide ($\sim 5.4$ deg$^2$) surveys around the North Ecliptic Pole (NEP) with its InfraRed Camera (IRC) with nine filters continuously covering the 2-25 $\mu$m range. These photometric bands include three filters that fill the `Spitzer gap' between the wavelength coverages of IRAC and MIPS. This unique feature has enabled us to make sensitive mid-infrared detection of AGN candidates at z$\sim$ 1-2, based on the Spectral Energy Distribution (SED) fitting including hot dust emission in the AGN torus. This enables us to compare X-rays and the AGN torus component of the infrared emission to help us identify highly absorbed AGNs, including Compton-thick ones. We report our results of the Chandra observation of the AKARI NEP Deep Field and discuss the prospects for upcoming Spectrum-RG (eROSITA+ART-XC) on the AKARI Wide field.Comment: 4 pages, 2 figures. Submitted for publication in the proceedings of the IAU Symp. 341 "PanModel2018 : Challenges in Panchromatic Galaxy Modelling with Next Generation Facilities

Sudoku Variants on the Torus

This paper examines the mathematical properties of Sudoku puzzles defined on a Torus. We seek to answer the questions for these variants that have been explored for the traditional Sudoku. We do this process with two such embeddings. The end result of this paper is a deeper mathematical understanding of logic puzzles of this type, as well as a fun new puzzle which could be played

Periodic Tilings and Auxetic Deployments

We investigate geometric characteristics of a specific planar periodic framework with three degrees of freedom. While several avatars of this structural design have been considered in materials science under the name of chiral or missing rib models, all previous studies have addressed only local properties and limited deployment scenarios. We describe the global configuration space of the framework and emphasize the geometric underpinnings of auxetic deformations. Analogous structures may be considered in arbitrary dimension