Related papers: Rigid polyboxes and Keller's conjecture

Let $E$ be an elliptic curve with complex multiplication by a ring $R$, where $R$ is an order in an imaginary quadratic field or quaternion algebra. We define sesquilinear pairings ($R$-linear in one variable and $R$-conjugate linear in the…

A Keller map is a counterexample to the Jacobian Conjecture. In dimension two every such map, if exists, leads to a complicated set of conditions on the map between the Picard groups of suitable compactifications of the affine plane. This…

A tiling of the sphere by triangles, squares, or hexagons is convex if every vertex has at most 6, 4, or 3 polygons adjacent to it, respectively. Assigning an appropriate weight to any tiling, our main result is explicit formulas for the…

We classify the convex polytopes whose symmetry groups have two orbits on the flags. These exist only in two or three dimensions, and the only ones whose combinatorial automorphism group is also two-orbit are the cuboctahedron, the…

In 1946 Fine and Niven posed problem E724, asking to demonstrate that every hypercube can be tiled by any number of hypercubic tiles larger than some value. This requires only basic number theory, but the problem of finding the smallest…

Two-dimensional random tilings of rhombi can be seen as projections of two-dimensional membranes embedded in hypercubic lattices of higher dimensional spaces. Here, we consider tilings projected from a $D$-dimensional space. We study the…

On smooth projective varieties of dimension $d$, $d$-tilting bundles are important in both geometry and representation theory, since they provide a bridge from the geometry of such varieties to the derived McKay correspondence and to higher…

Let $S$ be an upper cluster algebra, which is a subalgebra of $R$. Suppose that there is some cluster variable $x_e$ such that ${R}_{{x}_e} = S[{x}_e^{\pm 1}]$. We try to understand under which conditions ${R}$ is an upper cluster algebra,…

We consider tilings of deficient rectangles by the set $\mathcal{T}_4$ of ribbon $L$-tetrominoes. A tiling exists iff the rectangle is a square of odd side. The missing cell is on the main NW--SE diagonal, in an odd position if the square…

A dyadic tile of order n is any rectangle obtained from the unit square by n successive bisections by horizontal or vertical cuts. Let each dyadic tile of order n be available with probability p, independently of the others. We prove that…

This paper studies two families of constraints for two-dimensional and multidimensional arrays. The first family requires that a multidimensional array will not contain a cube of zeros of some fixed size and the second constraint imposes…

Given a partition $h_1+h_2+\dots+h_k = n$, a latin square of order $n$ with pairwise disjoint subsquares of orders $h_1,\dots ,h_k$ is called a realization. When the values $h_i$ are of at most two sizes, the existence of a realization has…

A Ryser design $\mathcal{D}$ on $v$ points is a collection of $v$ proper subsets (called blocks) of a point-set with $v$ points such that every two blocks intersect each other in $\lambda$ points (and $\lambda < v$ is a fixed number) and…

The study of tilings is a major problem in many mathematical instances, which is studied in two main different approaches: when considering the existence (or obstructions to the existence) of a tiling with a given tile and the other…

We study the dissection of a square into congruent convex polygons. Yuan \emph{et al.} [Dissecting the square into five congruent parts, Discrete Math. \textbf{339} (2016) 288-298] asked whether, if the number of tiles is a prime number…

A $d$-dimensional Latin hypercube of order $n$ is a $d$-dimensional array containing symbols from a set of cardinality $n$ with the property that every axis-parallel line contains all $n$ symbols exactly once. We show that for $(n, d)…

We investigate the combinatorics of quivers that arise from triangulations of even-dimensional cyclic polytopes. Work of Oppermann and Thomas pinpoints such quivers as the prototypes for higher-dimensional cluster theory. We first show that…

We propose a unified perspective on two sets of objects that usually arise in the study of bipartite field theories. Each of the sets consists of a polytope, or equivalently a toric Calabi-Yau, and a quiver theory. We refer to the two sets…

Problem 4.19 in Ziegler's "Lectures on Polytopes" asserts that every simple $3$-dimensional polytope has the property that its dual can be constructed as the convex hull of a subset of the vertices of the original simple polytope. In this…

In a region $R$ consisting of unit squares, a domino is the union of two adjacent squares and a (domino) tiling is a collection of dominoes with disjoint interior whose union is the region. The flip graph $\mathcal{T}(R)$ is defined on the…