Related papers: Towards Resolving Keller's Cube Tiling Conjecture …

Let $M^d$ denote the $d$-dimensional Euclidean, hyperbolic, or spherical space. The $r$-dual set of given set in $M^d$ is the intersection of balls of radii $r$ centered at the points of the given set. In this paper we prove that for any…

In this work, we consider tilings of the Hamming cube and look for metrics which turn the tilings into a perfect code. We consider the family of metrics which are determined by a weight and are compatible with the support of vectors…

We study self-similar attractors in the space $\mathbb{R}^d$, i.e., self-similar compact sets defined by several affine operators with the same linear part. The special case of attractors when the matrix $M$ of the linear part of affine…

Answering a recent question of Patchell and Spiro, we show that when a $d$-dimensional cube of side length $n$ is filled with letters, the word $\mathsf{CAT}$ can appear contiguously at most $(3^{d-1}/2)n^d$ times (allowing diagonals); we…

We measure whether there are numerous pairs of twin primes (hereafter referred to as twin prime pairs) according to the prime number inferred by sieve of Eratosthenes. In this study, we reveal at least three additional twin prime pairs…

We consider domino tilings of $3$-dimensional cubiculated regions. A three-dimensional domino is a 2x2x1 rectangular cuboid. We are particularly interested in regions of the form $R_N = D \times [0,N]$ where $D$ is a fixed quadriculated…

The preprints arXiv:math/0610728 and arXiv:math/0612451 are withdrawn due to a problem with Theorem 2.2 in arXiv:math/0610728. The theorem claims that for certain triangulated categories with finitely many indecomposable objects, the…

We study infinite dimensional tilting modules over a concealed canonical algebra of domestic or tubular type. In the domestic case, such tilting modules are constructed by using the technique of universal localization, and they can be…

For any ring $R$, we investigate balanced pairs of classes of modules and their relations to cotorsion triples. We characterize the case when a balanced pair generates a tilting cotorsion pair, and dually, when it cogenerates a cotilting…

The Erd\H{o}s-Mollin-Walsh conjecture, asserting the nonexistence of three consecutive powerful integers, remains a celebrated open problem in number theory. A natural line of inquiry, following recent work by Chan (2025), is to investigate…

We investigate tiling properties of spectra of measures, i.e., sets $\Lambda$ in $\br$ such that $\{e^{2\pi i \lambda x}: \lambda\in\Lambda\}$ forms an orthogonal basis in $L^2(\mu)$, where $\mu$ is some finite Borel measure on $\br$. Such…

We prove the conjecture made by G.Wegner in 1977 that the square of every planar, cubic graph is $7$-colorable. Here, $7$ cannot be replaced by $6$.

The purpose of this note is to attract attention to the following conjecture (metastable $r$-fold Whitney trick) by clarifying its status as not having a complete proof, in the sense described in the paper. Assume that…

It is proved that a multiset of permissible arcs over a tiling is uniquely determined by its intersection vector under a mild condition. This generalizes a classical result over marked surfaces with triangulations. We apply this result to…

The middle levels conjecture asserts that there is a Hamiltonian cycle in the middle two levels of $2k+1$-dimensional hypercube. The conjecture is known to be true for $k \leq 17$ [I.Shields, B.J.Shields and C.D.Savage, Disc. Math., 309,…

A pair of orthogonal latin cubes of order $q$ is equivalent to an MDS code with distance $3$ or to an ${\rm OA}_1(3,5,q)$ orthogonal array. We construct pairs of orthogonal latin cubes for a sequence of previously unknown orders…

We study certain structural properties of fine zonotopal tilings, or cubillages, on cyclic zonotopes $Z(n,d)$ of an arbitrary dimension $d$ and their relations to $(d-1)$-separated collections of subsets of a set $\{1,2,\ldots,n\}$.…

We show that three dimensional cubes of any size can be tiled with trominoes and, when necessary, one or two singletons in any positions. Cubes of side length a multiple of three can always be tiled with trominoes (known), cubes of side…

Motzkin and Taussky (and independently, Gerstenhaber) proved that the unital algebra generated by a pair of commuting $d\times d$ matrices over a field has dimension at most $d$. Since then, it has remained an open problem to determine…

A subset V of GF(2)^n is a tile if GF(2)^n can be covered by disjoint translates of V. In other words, V is a tile if and only if there is a subset A of GF(2)^n such that V+A = GF(2)^n uniquely (i.e., v + a = v' + a' implies that v=v' and…