Related papers: The optimal lattice quantizer in nine dimensions
In this paper we describe an algorithm that quickly computes a maximal a-valued lattice in an F-vector space equipped with a non-degenerate bilinear form, where a is a fractional ideal in a number field F. We then apply this construction to…
The main objective of this thesis is a classification project for integral lattices. Using Kneser's neighbour method we have developed the computer program tn to classify complete genera of integral lattices. Main results are detailed…
The goal of this work is to investigate the optimality of the $d$-dimensional rock-salt structure, i.e., the cubic lattice $V^{1/d}\mathbb{Z}^d$ of volume $V$ with an alternation of charges $\pm 1$ at lattice points, among periodic…
I present an overview of the calculations of the isovector axial vector form factor of the nucleon, $G_A(Q^2)$, using lattice QCD. Based on a comparison of results from various collaborations, a case is made that lattice results are now…
We consider a periodic lattice structure in $d=2$ or $3$ dimensions with unit cell comprising $Z$ thin elastic members emanating from a similarly situated central node. A general theoretical approach provides an algebraic formula for the…
For a lattice/linear code, we define the Voronoi spherical cumulative density function (CDF) as the CDF of the $\ell_2$-norm/Hamming weight of a random vector uniformly distributed over the Voronoi cell. Using the first moment method…
In this note we give a polynomial time algorithm for solving the closest vector problem in the class of zonotopal lattices. The Voronoi cell of a zonotopal lattice is a zonotope, i.e. a projection of a regular cube. Examples of zonotopal…
The lattice size of a lattice polytope is a geometric invariant which was formally introduced in the context of simplification of the defining equation of an algebraic curve, but appeared implicitly earlier in geometric combinatorics.…
The Erd\H{o}s distance problem concerns the least number of distinct distances that can be determined by $N$ points in the plane. The integer lattice with $N$ points is known as \textit{near-optimal}, as it spans $\Theta(N/\sqrt{\log(N)})$…
The highest possible minimal norm of a unimodular lattice is determined in dimensions n <= 33. There are precisely five odd 32-dimensional lattices with the highest possible minimal norm (compared with more than 8*10^20 in dimension 33).…
We describe algorithms which address two classical problems in lattice geometry: the lattice covering and the simultaneous lattice packing-covering problem. Theoretically our algorithms solve the two problems in any fixed dimension d in the…
The sphere packing problem is an old puzzle. We consider packings with m spheres in the unit cell (m-periodic packings). For the case m = 1 (lattice packings), Voronoi proved there are finitely many inequivalent local optima and presented…
We denote by Conc(L) the semilattice of all finitely generated congruences of a lattice L. For varieties (i.e., equational classes) V and W of lattices such that V is contained neither in W nor its dual, and such that every simple member of…
The lattice dimension of a graph G is the minimal dimension of a cubic lattice in which G can be isometrically embedded. We prove that the lattice dimension of a tree with n leaves is $\lceil n/2 \rceil$.
Given an arbitrary basis for a mathematical lattice, to find a ``good" basis for it is one of the classic and important algorithmic problems. In this note, we give a new and simpler proof of a theorem by Regavim (arXiv:2106.03183): we…
We show that the degree of a graded lattice ideal of dimension 1 is the order of the torsion subgroup of the quotient group of the lattice. This gives an efficient method to compute the degree of this type of lattice ideals.
A method is developed to compute minimal energy vortex lattices in a general Ginzburg-Landau model of a superconductor subjected to an applied magnetic field. The model may have any number of components and may be spatially anisotropic. The…
In this paper we present the first known deterministic algorithm for the construction of multiple rank-1 lattices for the approximation of periodic functions of many variables. The algorithm works by converting a potentially large…
We present an adaptation of Voronoi theory for imaginary quadratic number fields of class number greater than 1. This includes a characterisation of extreme Hermitian forms which is analogous to the classic characterisation of extreme…
In section 1 we give an improved lower bound on Hermite's constant $\delta_{2g}$ for symplectic lattices in even dimensions ($g=2n$) by applying a mean-value argument from the geometry of numbers to a subset of symmetric lattices. Here we…