A Definition of Type Domain of a Parallelotope

Each convex polytope P = P(α) can be described by a set of linear inequalities determined by vectors p and right hand sides α(p). For a fixed set of vectors p, a type domain D(P₀) of a polytope P₀ and, in particular, of a parallelotope P₀ is defined as a set of parameters α(p) such that polytopes P(α) have the same combinatorial type as P₀ for all α ∈ D(P₀).In the second part of the paper, a facet description of zonotopes and zonotopal parallelotopes are given.The article is published in the author’s wording

Significant Conditions on the Two-electron Reduced Density Matrix from the Constructive Solution of N-representability

We recently presented a constructive solution to the N-representability problem of the two-electron reduced density matrix (2-RDM)---a systematic approach to constructing complete conditions to ensure that the 2-RDM represents a realistic N-electron quantum system [D. A. Mazziotti, Phys. Rev. Lett. 108, 263002 (2012)]. In this paper we provide additional details and derive further N-representability conditions on the 2-RDM that follow from the constructive solution. The resulting conditions can be classified into a hierarchy of constraints, known as the (2,q)-positivity conditions where the q indicates their derivation from the nonnegativity of q-body operators. In addition to the known T1 and T2 conditions, we derive a new class of (2,3)-positivity conditions. We also derive 3 classes of (2,4)-positivity conditions, 6 classes of (2,5)-positivity conditions, and 24 classes of (2,6)-positivity conditions. The constraints obtained can be divided into two general types: (i) lifting conditions, that is conditions which arise from lifting lower (2,q)-positivity conditions to higher (2,q+1)-positivity conditions and (ii) pure conditions, that is conditions which cannot be derived from a simple lifting of the lower conditions. All of the lifting conditions and the pure (2,q)-positivity conditions for q>3 require tensor decompositions of the coefficients in the model Hamiltonians. Subsets of the new N-representability conditions can be employed with the previously known conditions to achieve polynomially scaling calculations of ground-state energies and 2-RDMs of many-electron quantum systems even in the presence of strong electron correlation

A Definition of Type Domain of a Parallelotope

Each convex polytope P = P(α) can be described by a set of linear inequalities determined by vectors p and right hand sides α(p). For a fixed set of vectors p, a type domain D(P₀) of a polytope P₀ and, in particular, of a parallelotope P₀ is defined as a set of parameters α(p) such that polytopes P(α) have the same combinatorial type as P₀ for all α ∈ D(P₀).In the second part of the paper, a facet description of zonotopes and zonotopal parallelotopes are given.The article is published in the author’s wording.</p

L-polytopes, Even Unimodular Lattices, and Perfect Lattices

It is shown here that every L-polytope of an even unimodular lattice does not generate the lattice. It is given here the corrected formulation of a previous result of the author [3] on relations between extreme L-polytopes and perfect lattices. We prove here the following special case. If the square radius of the circumscribing sphere of an extreme L-polytope P of a lattice L is less than the minimal norm m of L, then the m-extension of P generates a perfect lattice. 1 Introduction Recall some notions of integral lattices and L-polytopes. Details see in [1]. An L-polytope of a lattice L is the convex hull of all lattice points lying on an empty sphere. An empty sphere in a lattice L of dimension n is such a sphere that there is no lattice point inside the sphere, and the lattice points lying on the sphere affinely generate the n-dimensional space. We call the radius of the empty sphere a radius of the inscribed Lpolytope. Let P be an L-polytope of a lattice L. We take the center of..

Определение области типа параллелоэдра

Each convex polytope P = P(α) can be described by a set of linear inequalities determined by vectors p and right hand sides α(p). For a fixed set of vectors p, a type domain D(P₀) of a polytope P₀ and, in particular, of a parallelotope P₀ is defined as a set of parameters α(p) such that polytopes P(α) have the same combinatorial type as P₀ for all α ∈ D(P₀).In the second part of the paper, a facet description of zonotopes and zonotopal parallelotopes are given.The article is published in the author’s wording.Любой выпуклый многогранник P = P(α) может быть описан системой линейных неравенств, определяемых векторами p и правыми частями α(p). Для фиксированного множества векторов p определяется область типа D(P₀) многогранника P₀, и в частности параллелоэдра P₀, как такое множество параметров α(p), что много- гранники P(α) имеют тот же комбинаторный тип, что и P₀ для всех α ∈ D(P₀). Во второй части статьи дается фасетное описание зонотопов и зонотопных параллелоэдров. Статья публикуется в авторской редакции