Polyhedral Omega is a new algorithm for solving linear Diophantine systems (LDS), i.e., for computing a multivariate rational function representation of the set of all non-negative integer solutions to a system of linear equations and inequalities. Polyhedral Omega combines methods from partition analysis with methods from polyhedral geometry. In particular, we combine MacMahon’s iterative appr...

We give an elementary introduction to some recent polyhedral techniques for understanding and solving systems of multivariate polynomial equations. We provide numerous concrete examples and assume no background in algebraic geometry. Highlights include the following: (1) A completely self-contained proof of an extension of Bernstein’s Theorem. Our extension relates volumes of polytopes with the...

We consider linear optimization problems over the cone of copositive matrices. Such conic optimization problems, called copositive programs, arise from the reformulation of a wide variety of difficult optimization problems. We propose a hierarchy of increasingly better outer polyhedral approximations to the copositive cone. We establish that the sequence of approximations is exact in the limit....

Despite researchers’ efforts in the last couple of decades, reachability analysis is still a challenging problem even for linear hybrid systems. Among the existing approaches, the most practical ones are mainly based on bounded-time reachable set over-approximations. For the purpose of unbounded-time analysis, one important strategy is to abstract the original system and find an invariant for t...

The ability to reason about polyhedral forms is arguably an important element of spatial (and more broadly mathematical) visualization. This paper describes our progress toward creating real-time software "advisors" whose goal is to enhance this type of spatial reasoning. Our advisors are procedures built to augment a program named HyperGami (an educational application for the creation of paper...

Let P be a polyhedron in H$ of finite volume such that the group Γ(P) generated by reflections in the faces of P is a discrete subgroup of IsomH$. Let Γ+(P) denote the subgroup of index 2 consisting entirely of orientation-preserving isometries so that Γ+(P) is a Kleinian group of finite covolume. Γ+(P) is called a polyhedral group. As discussed in [12] and [13] for example (see §2 below), asso...

I’m trying to understand which Riemannian manifolds can be Lipschitz approximated by polyhedral spaces of the same dimension with curvature bounded below. The necessary conditions I found consist of some special inequality for curvature at each point (the geometric curvature bound). This inequality is also sufficient condition for local approximation. I conjecture that it is also a sufficient c...

Ambiguity, and its resolution, are key elements of communication; and, in particular, of music (See, e.g., the examples discussed by L. Bernstein in bis f amous Harvard lectures [1). Besides rhythmical ambiguity, it is particularly harmonic ambiguity of both simultaneaus notes, i.e., chords, and of sequential notes, i.e., rnelodies, that play a major role. Auditory pitch comes into play through...

We study a principal-agent problem with multiple identical agents, where the action-dependent stochastic relationship between actions and output is perceived to be ambiguous, and agents are ambiguity averse. We argue that ambiguity, and particularly ambiguity aversion, make it more attractive for the principal to choose a tournament. If agents are risk neutral, but ambiguity averse, we show tha...