Related papers: Polyiamonds Attaining Extremal Topological Propert…
Consider the pairs $(f,G)$ with $f = f(x_1,\dots,x_N)$ being a polynomial defining a quasihomogeneous singularity and $G$ being a subgroup of ${\rm SL}(N,\mathbb{C})$, preserving $f$. In particular, $G$ is not necessary abelian. Assume…
For various sets of tiles, we count the ways to tile an Aztec diamond of order $n$ using tiles from that set. The resulting function $f(n)$ often has interesting behavior when one looks at $n$ and $f(n)$ modulo powers of 2.
A tiling of a topological disc by topological discs is called monohedral if all tiles are congruent. Maltby (J. Combin. Theory Ser. A 66: 40-52, 1994) characterized the monohedral tilings of a square by three topological discs. Kurusa,…
We introduce the general notions of an overconvergent site and a constructible crystal on an overconvergent site. We show that if $V$ is a geometric materialization of a locally noetherian formal scheme $X$ over an analytic space $O$…
We study ideal-theoretic conditions for a monomial ideal to be Golod. For ideals in a polynomial ring in three variables, our criteria give a complete characterization. Over such rings, we show that the product of two monomial ideals is…
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…
If particles interact according to isotropic pair potentials that favor multiple length scales, in principle a large variety of different complex structures can be achieved by self-assembly. We present, motivate, and discuss a conjecture…
We consider staged self-assembly systems, in which square-shaped tiles can be added to bins in several stages. Within these bins, the tiles may connect to each other, depending on the glue types of their edges. Previous work by Demaine et…
A hinged dissection of a set of polygons S is a collection of polygonal pieces hinged together at vertices that can be folded into any member of S. We present a hinged dissection of all edge-to-edge gluings of n congruent copies of a…
This paper introduces a new type of thin-shell wormhole constructed from a special class of compact stellar objects rather than black holes. The construction and concomitant investigation of the stability to linearized radial perturbations…
Given a tiling of a 2D grid with several types of tiles, we can count for every row and column how many tiles of each type it intersects. These numbers are called the_projections_. We are interested in the problem of reconstructing a tiling…
The combinatorics of tilings of a hexagon of integer side-length $n$ by 120 degree - 60 degree diamonds of side-length 1 has a long history, both directly (as a problem of interest in thermodynamic models) and indirectly (through the…
In recent years, experimental and theoretical investigations have shown that anisotropic colloids can self-organise into ordered porous monolayers, where the interplay of localised bonding sites, so called patches, with the particle's shape…
This note discusses some examples showing that the crystalline cohomology of even very mildly singular projective varieties tends to be quite large. In particular, any singular projective variety with at worst ordinary double points has…
In this paper we introduce poly-Poisson structures as a higher-order extension of Poisson structures. It is shown that any poly-Poisson structure is endowed with a polysymplectic foliation. It is also proved that if a Lie group acts…
We prove that polyharmonic maps of arbitrary order from complete nonparabolic Riemannian manifolds to arbitrary Riemannian manifolds must be harmonic if certain smallness and integrability conditions hold.
We analyze the topology and geometry of a polyhedron of dimension 2 according to the minimum size of a cover by PL collapsible polyhedra. We provide partial characterizations of the polyhedra of dimension 2 that can be decomposed as the…
In this paper, we study the extremal behaviour of deep holes in polyominoes. We determine the maximum number, $h_n$ of deep holes that an $n$-omino can enclose, ensuring that the boundary of each hole is disjoint from the boundaries of any…
We use first-principles computational methods to examine hole trapping in organic molecular crystals. We present a computational scheme based on the tuning of the fraction of exact exchange in hybrid density functional theory to eliminate…
The number of holes in a connected component in 2D images is a basic invariant. In this note, a simple formula was proven using our previous results in digital topology (Chen 2004, Chen and Rong (2010). The new is: $h =1+ (|C_4|-|C_2|)/4$,…