Related papers: Local Covering Optimality of Lattices: Leech Latti…
The lattices $D_4$ and $E_8$ are known to be the densest lattices in dimensions 4 and 8, respectively. In this paper, we employ tools from algebraic number theory to prove that the $D_4$-lattice arises from an infinite family of totally…
G.F. Voronoi (1868-1908) wrote two memoirs in which he describes two reduction theories for lattices, well-suited for sphere packing and covering problems. In his first memoir a characterization of locally most economic packings is given,…
A lattice is called periodic extreme if it cannot locally be modified to yield a better periodic sphere packing. It is called strict periodic extreme if its sphere packing density is an isolated local optimum among periodic point sets. In…
Implicational bases are a well-known representation of closure spaces and their closure lattices. This representation is not unique, though, and a closure space usually admits multiple bases. Among these, the canonical base, the canonical…
We consider two constructions of an envelope for a finite locally distributive strong upper semilattice. The first is based on Birkhoff's representation of finite distributive lattices and the second on valuations on lattices. We show that…
We propose a local reinforcement technique for lattices in the vicinity of a stress-raiser such as a notch, in order to elevate the macroscopic strength and ductility. A spatially non-uniform waviness distribution of sinusoidally-shaped…
We describe a classification of pairs $M, N$ of lattices isometric to $EE_8:=\sqrt 2 E_8$ such that the lattice $M + N$ is integral and rootless and such that the dihedral group associated to them has order at most 12. It turns out that…
The Leech lattice, $\Lambda_{24}$, is represented on the space of octonionic 3-vectors. It is built from two octonionic representations of $E_{8}$, and is reached via $\Lambda_{16}$. It is invariant under the octonion index cycling and…
Materials featuring touching points, localized states, and flat bands are of great interest in condensed matter and artificial systems due to their implications in topology, quantum geometry, superconductivity, and interactions. In this…
An optimization method for the design of multi-lattice structures satisfying local buckling constraints is proposed in this paper. First, the concept of free material optimization is introduced to find an optimal elastic tensor distribution…
Given a sphere of any radius $r$ in an $n$-dimensional Euclidean space, we study the coverings of this sphere with solid spheres of radius one. Our goal is to design a covering of the lowest covering density, which defines the average…
This paper investigates the decoding of a remarkable set of lattices: We treat in a unified framework the Leech lattice in dimension 24, the Nebe lattice in dimension 72, and the Barnes-Wall lattices. A new interesting lattice is…
We consider substitution tilings and Delone sets without the assumption of finite local complexity (FLC). We first give a sufficient condition for tiling dynamical systems to be uniquely ergodic and a formula for the measure of cylinder…
In this short note we give a construction of an infinite series of Delone simplices whose relative volume grows super-exponentially with their dimension. This dramatically improves the previous best lower bound, which was linear.
A discrete-to-continuum transformation to model 2-D discrete lattices as energetically equivalent 1-D continuum beams is developed. The study is initiated in a classical setting but results in a non-classical two-scale micropolar beam model…
Covering a graph or a lattice with non-overlapping dimers is a problem that has received considerable interest in areas such as discrete mathematics, statistical physics, chemistry and materials science. Yet, the problem of percolation on…
Two constructions of lattice packings of $ n $-dimensional cross-polytopes ($ \ell_1 $ balls) are described, the density of which exceeds that of any prior construction by a factor of at least $ 2^{\frac{n}{\ln n}(1 + o(1))} $ when $ n \to…
We discuss the exact solutions of various models of the statistics of dimer coverings of a Bethe lattice. We reproduce the well-known exact results for noninteracting hard-core dimers by both a very simple geometrical argument and a general…
Besides all the attention given to lattice constructions, it is common to find some very interesting nonlattice constellations, as Construction C, for example, which also has relevant applications in communication problems (multi-level…
We affirm a conjecture of Sacks [1972] by showing that every countable distributive lattice is isomorphic to an initial segment of the hyperdegrees, $\mathcal{D}_{h}$. In fact, we prove that every sublattice of any hyperarithmetic lattice…