Related papers: Lattice packings through division algebras
We consider Euclidean lattices spanned by images of algebraic conjugates of an algebraic number under Minkowski embedding, investigating their rank, properties of their automorphism groups and sets of minimal vectors. We are especially…
In this article, we generalise a result of Pottmeyer from the multiplicative group of the algebraic numbers to almost split semiabelian varieties defined over number fields. This concerns a consequence of R\'emond's generalisation of…
In this note we give a simple proof of the classical fact that the hexagonal lattice gives the highest density circle packing among all lattices in $R^2$. With the benefit of hindsight, we show that the problem can be restricted to the…
A classical theorem of Ingham extended Parseval's formula of the trigonometrical system to arbitrary families of exponentials satisfying a uniform gap condition. Later his result was extended to several dimensions, but the optimal…
We construct lattice gauge field theory based on a quantum group on a lattice of dimension 1. Innovations include a coalgebra structure on the connections, and an investigation of connections that are not distinguishable by observables. We…
It is shown why the discriminant of a maximal order within a cyclic division algebra must be minimized in order to get the densest possible matrix lattices with a prescribed nonvanishing minimum determinant. Using results from class field…
The intrinsic connection between lattice theory and topology is fairly well established, For instance, the collection of open subsets of a topological subspace always forms a distributive lattice. Persistent homology has been one of the…
MacMahon proved a simple product formula for the generating function of plane partitions fitting in a given box. The theorem implies a $q$-enumeration of lozenge tilings of a semi-regular hexagon on the triangular lattice. In this paper we…
Toric (or sparse) elimination theory is a framework developped during the last decades to exploit monomial structures in systems of Laurent polynomials. Roughly speaking, this amounts to computing in a \emph{semigroup algebra}, \emph{i.e.}…
We propose a unified framework that synthesizes advances in high-dimensional lattice theory with novel computational algorithms for the shortest vector problem (SVP) to model pure root lattices and compute sphere packing densities. Building…
In this paper, we introduce the concept of a (lattice) skew Hilbert algebra as a natural generalization of Hilbert algebras. This notion allows a unified treatment of several structures of prominent importance for mathematical logic, e.g.…
In this paper, we introduce a novel generalization of the classical property of algebras known as "being alternative," which we term "partially alternative." This new concept broadens the scope of alternative algebras, offering a fresh…
It is well-known that the densest lattice sphere packings also typically have large kissing numbers. The sphere packing density maximization problem is known to have a solution among well-rounded lattices, of which the integer lattice…
We state the formula for the critical number of vertices of a convex lattice polygon that guarantees that the polygon contains at least one point of a given sublattice and give a partial proof of the formula. We show that the proof can be…
We construct a bounded and symmetric convex body in $\ell_2(\Gamma)$ (for certain cardinals $\Gamma$) whose translates yield a tiling of $\ell_2(\Gamma)$. This answers a question due to Fonf and Lindenstrauss. As a consequence, we obtain…
We prove Manin's conjecture concerning the distribution of rational points of bounded height, and its refinement by Peyre, for wonderful compactifications of semi-simple algebraic groups over number fields. The proof proceeds via the study…
For any cluster algebra whose underlying combinatorial data can be encoded by a bordered surface with marked points, we construct a geometric realization in terms of suitable decorated Teichmueller space of the surface. On the geometric…
We treat problems of fair division, their various interconnections, and their relations to Sperner's lemma and the KKM theorem as well as their variants. We prove extensions of Alon's necklace splitting result in certain regimes and relate…
Let $L$ denote a finite lattice with at least two points and let $A$ denote the incidence algebra of $L$. We prove that $L$ is distributive if and only if $A$ is an Auslander regular ring, which gives a homological characterisation of…
In this work, we establish theoretical and practical connections between vertex indexing for sparse graph/network compression and matrix ordering for sparse matrix-vector multiplication and variable elimination. We present a fundamental…