Robert M. Erdahl

Zonotopes, Dicings, and Voronoi's Conjecture on Parallelohedra

In 1909, Voronoi conjectured that if some selection of translates of a polytope forms a facet-to-facet tiling of euclidean space, then the polytope is affinely equivalent to the Voronoi polytope for a lattice. He referred to polytopes with this tiling property as parallelohedra, but they are now frequently called parallelotopes. I show that Voronoi?s conjecture holds for the special case where the parallelotope is a zonotope. I also show that the Voronoi polytope for a lattice is a zonotope if and only if the Delaunay tiling for the lattice is a dicing (defined at the beginning of Section 3).

Two algorithms for the lower bound method of reduced density matrix theory

Abstract We analyze the lower bound method of reduced density matrix theory, a method which obtains a lower bound to the ground state energy of a many-fermion system as well as an approximation to the corresponding reduced density matrix. Our main result is a theorem giving necessary and sufficient conditions for the optimum for the central optimization problem of this method. Based on this theorem we have developed two algorithms for solving this optimization problem. We consider their convergence properties.

On lattice dicing

Abstract A lattice dicing is an arrangement of hyperplanes with sufficient regularity so that the vertices of the resulting partition form a lattice. In this paper we introduce the notion of lattice dicing and point out how lattice dicings relate to geometry of numbers. We give a complete description of the possible lattice dicings of R n and n ⩽ 5.

A cone of inhomogeneous second-order polynomials

AbstractLet ℘n be the cone of quadratic function % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVy0df9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lqpe0x% c9q8qqaqFn0dXdir-xcvk9pIe9q8qqaq-xir-f0-yqaqVeLsFr0-vr% 0-vr0xc8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaacaWFgb% GaaGymaiaac6cacaWFGaGaamOzaiabg2da9iaadAgadaWgaaWcbaGa% aGimaaqabaGccqGHRaWkdaaeabqaaiaadAgadaWgaaWcbaGaamyAaa% qabaGccaWG4bWaaSbaaSqaaiaadMgaaeqaaaqabeqaniabggHiLdGc% cqGHRaWkdaaeabqaaiaadAgadaWgaaWcbaGaamyAaiaadQgaaeqaaO% GaamiEamaaBaaaleaacaWGPbaabeaaaeqabeqdcqGHris5aOGaamiE% amaaBaaaleaacaWGQbaabeaakiaacYcacaWGMbWaaSbaaSqaaiaadM% gacaWGQbaabeaakiabg2da9iaadAgadaWgaaWcbaGaamOAaiaadMga% aeqaaOGaaiilaaaa!59ED!

Spectral properties of cones of approximately representable reduced density matrices

F1. f = f_0 + \sum {f_i x_i } + \sum {f_{ij} x_i } x_j ,f_{ij} = f_{ji} ,

Voronoi and Delaunay Tilings for Lattices

on ℝn that satisfy the additional condition % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVy0df9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lqpe0x% c9q8qqaqFn0dXdir-xcvk9pIe9q8qqaq-xir-f0-yqaqVeLsFr0-vr% 0-vr0xc8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaacaWFgb% Gaa8Nmaiaac6cacaWFGaGaamOzaiaacIcacaWG6bGaaiykaeXafv3y% SLgzGmvETj2BSbacfaGae4xzImRaaGimaiaacYcacaWG6bGaeyicI4% 8efv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiyqacqqFKeIw% daahaaWcbeqaaiaad6gaaaGccaGGSaaaaa!570C!

Voronoi-Dickson Hypothesis on Perfect Forms and L-types

F2. f(z) \geqslant 0,z \in \mathbb{Z}^n ,