A polyhedral map is called {p, q}-equivelar if each face has p edges and each vertex belongs to q faces. In [12], it was shown that there exist infinitely many geometrically realizable {p, q}-equivelar polyhedral maps if q > p = 4, p > q = 4 or q − 3 > p = 3. It was shown in [6] that there exist infinitely many {4, 4}and {3, 6}-equivelar polyhedral maps. In [1], it was shown that {5, 5}and {6, ...

Let X be a polyhedral complex with finitely many isometry classes of links. We establish a restriction on the covolumes of uniform lattices acting on X. When X is two-dimensional and has all links isometric to either a complete bipartite graph or the building for a Chevalley group of rank 2 over a field of prime order, we obtain further restrictions on covolumes.

We present a new library for extracting a polyhedral model from C source. The library is based on clang, the LLVM C frontend, and isl, a library for manipulating quasi-affine sets and relations. The use of clang for parsing the C code brings advanced diagnostics and full support for C99. The use of isl allows for an easy construction and a powerful and compact representation of the polyhedral m...

The polyhedral model provides a powerful mathematical abstraction to enable effective optimization of loop nests with respect to a given optimization goal, e.g., exploiting parallelism. Unexploited reduction properties are a frequent reason for polyhedral optimizers to assume parallelism prohibiting dependences. To our knowledge, no polyhedral loop optimizer available in any production compiler...

We define higher polyhedral K-groups for commutative rings, starting from the stable groups of elementary automorphisms of polyhedral algebras. Both Volodin’s theory and Quillen’s + construction are developed. In the special case of algebras associated with unit simplices one recovers the usual algebraic K-groups, while the general case of lattice polytopes reveals many new aspects, governed by...

We define higher polyhedral K-groups for commutative rings, starting from the stable groups of elementary automorphisms of polyhedral algebras. Both Volodin’s theory and Quillen’s + construction are developed. In the special case of algebras associated with unit simplices one recovers the usual algebraic K-groups, while the general case of lattice polytopes reveals many new aspects, governed by...

In this paper we propose a new algorithm for learning polyhedral classifiers. In contrast to existing methods for learning polyhedral classifier which solve a constrained optimization problem, our method solves an unconstrained optimization problem. Our method is based on a logistic function based model for the posterior probability function. We propose an alternating optimization algorithm, na...