In this paper, we study the following problem: given a polytope P with Chvátal rank 1, does P contain an integer point? Boyd and Pulleyblank observed that this problem is in the complexity class NP ∩ co-NP, an indication that it is probably not NP-complete. We solve this problem in polynomial time for polytopes arising from the satisfiability problem of a given formula with at least three liter...

We show that the hypercube has a face-unfolding that tiles space, and that unfolding has an edge-unfolding that tiles the plane. So the hypercube is a “dimension-descending tiler.” We also show that the hypercube cross unfolding made famous by Dali tiles space, but we leave open the question of whether or not it has an edge-unfolding that tiles the plane.

In this article we present a method for automatic integration of parametric integrals over the unit hypercube using neural network. The fits network to primitive integrand loss function designed minimize difference between multiple derivatives and be integrated. We apply two example resulting from sector decomposition one-loop two-loop scalar integrals. Our can achieve per-mil percent accuracy ...

In this paper, we focus on a hypercube-like structure, the folded hypercube, which is basically a standard hypercube with some extra links between its nodes. Let f be a faulty vertex in an n-dimensional folded hypercube FQn. We show that FQn−{f } contains a faultfree cycle of every even length from 4 to 2n−2 if n ≥ 3 and, furthermore, every odd length from n+ 1 to 2n − 1 if n ≥ 2 and n is even....

We present embeddings of generalized ladders as subgraphs into the hypercube. By embedding caterpillars into ladders, we obtain embeddings of caterpillars into the hypercube. In this way we obtain almost all known results concerning the em-beddings of caterpillars into the hypercube. In addition we construct embeddings for some new types of caterpillars.