Related papers: Geometric implications of weak tiling

A famous problem in discrete geometry is to find all monohedral plane tilers, which is still open to the best of our knowledge. This paper concerns with one of its variants that to determine all convex polyhedra whose every cross-section…

Fuglede's conjecture states that for a subset $\Omega$ of a locally compact abelian group $G$ with positive and finite Haar measure, there exists a subset of the dual group of $G$ which is an orthogonal basis of $L^{2}(\Omega)$ if and only…

A fundamental theorem in discrete convex analysis states that a set function is M$^\natural$-concave if and only if its conjugate function is submodular. This paper gives a new proof to this fact.

We study tilings of polygons $R$ with arbitrary convex polygonal tiles. Such tilings come in continuous families obtained by moving tile edges parallel to themselves (keeping edge directions fixed). We study how the tile shapes and areas…

We show that the weakly \'etale morphisms, used to define the pro-\'etale site of a scheme, are characterized by a lifting property similar to the one which characterizes formally \'etale morphisms. In order to prove this, we prove a…

Many algebras are expected to have the Weak Lefschetz property though this is often very difficult to establish. We illustrate the subtlety of the problem by studying monomial and some closely related ideals. Our results exemplify the…

A combinatorial substitution is a map over tilings which allows to define sets of tilings with a strong hierarchical structure. In this paper, we show that such sets of tilings are sofic, that is, can be enforced by finitely many local…

Monotonicity with respect to all arguments is fundamental to the definition of aggregation functions. It is also a limiting property that results in many important non-monotonic averaging functions being excluded from the theoretical…

Let $\Delta=\Delta_1\times\ldots\times \Delta_d\subseteq\mathbb{R}^n$, where $\mathbb{R}^n=\mathbb{R}^{n_1}\times\cdots\times\mathbb{R}^{n_d}$ with each $\Delta_i\subseteq\mathbb{R}^{n_i}$ a non-degenerate simplex of $n_i$ points. We prove…

In this paper, we study superficial elements of an ideal with respect to a module from a geometrical point of view, using blowing-ups. The notion of weak transform is particularly relevant to this study. We use this viewpoint to get a…

In this note we study the weak topology on paired modules over a (not necessarily commutative) ground ring. Over QF rings we are able to recover most of the well known properties of this topology in the case of commutative base fields. The…

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 study the problem of determining whether a given frame is scalable, and when it is, understanding the set of all possible scalings. We show that for most frames this is a relatively simple task in that the frame is either not scalable or…

The $k$-tiling problem for a convex polytope $P$ is the problem of covering $\mathbb R^d$ with translates of $P$ using a discrete multiset $\Lambda$ of translation vectors, such that every point in $\mathbb R^d$ is covered exactly $k$…

Let (X,dX) and (Y,dY) be semimetric spaces with distance sets D(X) and, respectively, D(Y). A mapping F : X \to Y is a weak similarity if it is surjective and there exists a strictly increasing f : D(Y) \to D(X) such that dX = f \circ dY…

Given a finite point set $P\subset\mathbb{R}^d$, we call a multiset $A$ a one-sided weak $\varepsilon$-approximant for $P$ (with respect to convex sets), if $|P\cap C|/|P|-|A\cap C|/|A|\leq\varepsilon$ for every convex set $C$. We show…

Fuglede's conjecture in $\mathbb{Q}_p$ is proved. That is to say, a Borel set of positive and finite Haar measure in $\mathbb{Q}_p$ is a spectral set if and only if it tiles $\mathbb{Q}_p$ by translation.

Assume that a bounded scatterer is embedded into an infinite homogeneous isotropic background medium in two dimensions. The refractive index function is supposed to be piecewise constant. If the scattering interface contains a weakly or…

Many recent studies have demonstrated that scaling arguments, such as the so-called hierarchical {\em ansatz}, are extremely useful in understanding the statistical properties of weak gravitational lensing. This is especially true on small…

Covering a convex body by its homothets is a classical notion in discrete geometry that has resulted in a number of interesting and long-standing problems. Swanepoel introduced the covering parameter of a convex body as a means of…