A total order of the facets of a polytope is a geometric shelling if there exists a combinatorially equivalent polytope in which the corresponding order of facets becomes a line shelling. The subject of this paper is (geometric) shellings of 3-polytopes. Recently, a graph theoretical characterization of geometric shellings of 3-polytopes were given by Holt & Klee and Mihalisin & Klee. We rst gi...

A flexible birdcage-type resonant RF detector for magnetic resonance imaging is described. The circuit consists of a polygonal ring of magnetically coupled L–C resonators, a periodic structure supporting backward magnetoinductive waves. The elements are mechanically linked to allow relative rotation, and the pivot point is optimised to hold the nearest-neighbour coupling coefficient invariant t...

In this paper, we will study the h-vectors of a slightly more general class of polytopes. A d-polytope is called simple in edges if each its edge is incident exactly to d − 1 facets. We will prove that for any polytope simple in edges all numbers h[d/2], h[d/2]+1,. . ., hd are nonnegative and hk 6 hd−k for k 6 d/2. Polytopes simple in edges appear for instance as (closures of) fundamental polyh...

In his book on convex polytopes [2] A.D. Aleksandrov raised a general question of finding variational formulations and solutions to geometric problems of existence of convex polytopes in Rn+1, n ≥ 2, with prescribed geometric data. Examples of such problems for closed convex polytopes for which variational solutions are known are the celebrated Minkowski problem [2] and the Gauss curvature [18]...

The aim of the paper is to calculate face numbers of simple generalized permutohedra, and study their f -, hand γvectors. These polytopes include permutohedra, associahedra, graphassociahedra, simple graphic zonotopes, nestohedra, and other interesting polytopes. We give several explicit formulas for h-vectors and γ-vectors involving descent statistics. This includes a combinatorial interpretat...

Identifying legal and illegal states significantly reduces computational complexity of ATPG. A unified framework for identification of the legal and illegal states is presented. Most known methods for identification of the legal and illegal states are interpretable within this framework. New theorems and the resulting procedures for identifying exact collection of legal or illegal states of a c...