For a 3-connected planar graph G with circumference c ≥ 44 it is proved that G has a cycle of length at least 1 36 c + 20 3 through any four vertices of G.

In this paper, we classify the perfect lattices in dimension 8. There are 10916 of them. Our classification heavily relies on exploiting symmetry in polyhedral computations. Here we describe algorithms making the classification possible.

Under the assumption of polyhedron, the Demyanov difference of convex compact sets is computed via some linear programmings by C procedure. The difference from the existing results is also compared, and some examples with the corresponding numerical results are given.

In this paper, necessary and sufficient conditions for the positive invariance of convex polyhedra with respect to linear discrete-time singular systems subject to bounded additive disturbances are established. New notions of ∆-invariance under different assumptions on the initial conditions are defined. Specifically, the notions of simple and weak ∆invariance are considered. They can be seen a...

For a d-dimensional polyhedral complex P , the dimension of the space of piecewise polynomial functions (splines) on P of smoothness r and degree k is given, for k sufficiently large, by a polynomial f(P, r, k) of degree d. When d = 2 and P is simplicial, in [1] Alfeld and Schumaker give a formula for all three coefficients of f . However, in the polyhedral case, no formula is known. Using loca...

One way of computing the probability of a specific voting situation under the Impartial Anonymous Culture assumption is via counting integral points in polyhedra. Here, Ehrhart theory can help, but unfortunately the dimension and complexity of the involved polyhedra grows rapidly with the number of candidates. However, if we exploit available polyhedral symmetries, some computations become poss...

It is proved that the decision problem about the existence of an embedding of face-width 3 of a given graph is NP-complete. A similar result is proved for some related decision problems. This solves a problem raised by Neil Robertson.

For D, D analytic polyhedra in Cn, it is proven that a biholomorphic mapping f : D→ D extends holomorphically to a dense boundary subset under certain condition of general position. This result is also extended to a more general class of domains with no smoothness condition on the boundary.

We study low-order reconstruction operators on polyhedral meshes, providing a unified framework for degrees of freedom attached to vertices, edges, faces, and cells. We present two equivalent sets of design properties and draw links with the literature. In particular, the two-level construction based on a P0consistent and a stabilization part provides a systematic way of designing these operato...

This paper presents a general treatment of the combinatorial approach to the scheduling problem, enhancing previous formulations in the literature. The focus of this paper is a formal analysis of the ILP approach, which we use to evaluate the structure of our formulation. Polyhedral theory and duality theory are used to demonstrate that e cient solutions of the scheduling problem can be expecte...