The polyhedral model is an algebraic framework for affine program representations and transformations for enhancing locality and parallelism. Compared with traditional AST-based transformation frameworks, the polyhedral model can easily handle imperfectly nested loops and complex data dependences within and across loop nests in a unified framework. On the other hand, AST-based transformation fr...

The ability to transform two-dimensional (2D) materials into a three-dimensional (3D) structure while preserving their unique inherent properties might offer great enticing opportunities in the development of diverse applications for next generation micro/nanodevices. Here, a self-assembly process is introduced for building free-standing 3D, micro/nanoscale, hollow, polyhedral structures config...

We prove a result relating the Euler characteristic of a polyhedral closed homology manifold to the finite space associated with a triangulation of the manifold. We then give a new proof that polyhedral closed homology manifolds have Euler characteristic 0.

The goal of this thesis is to design algorithms that run with better complexity when compiling or parallelizing loop programs. The framework within which our algorithms operate is the polyhedral model of compilation which has been successful in the design and implementation of complex loop nest optimizers and parallelizing compilers. The algorithmic complexity and scalability limitations of the...

We present an algorithm for computing exact shortest paths, and consequently distances, from a generalized source (point, segment, polygonal chain or polygonal region) on a polyhedral terrain in which polygonal chain or polygon obstacles are allowed. We also present algorithms for computing discrete Voronoi diagrams of a set of generalized sites (points, segments, polygonal chains or polygons) ...

Automated program verification often proceeds by exhibiting inductive invariants entailing the desired properties. For numerical properties, a classical class of invariants is convex polyhedra: solution sets of system of linear (in)equalities. Forty years of research on convex polyhedral invariants have focused, on the one hand, on identifying “easier” subclasses, on the other hand on heuristic...

The 3-D reconstruction of visible polyhedral faces from a pair of general perspective views with the aid of a calibration plate is addressed. A polyhedron is placed on a planar calibration plate and two side views of both the polyhedron and the calibration plate are taken. Through proper arrangements we may assume that in the two views a number of polyhedral edges lying on the calibration plate...

We study rigidity of polyhedral surfaces and the moduli space of polyhedral surfaces using variational principles. Curvature like quantities for polyhedral surfaces are introduced. Many of them are shown to determine the polyhedral metric up to isometry. The action functionals in the variational approaches are derived from the cosine law and the Lengendre transformation of them. These include e...

This paper is concerned with the unique determination of a threedimensional polyhedral bi-periodic diffraction grating by the scattered electromagnetic fields measured above the grating. It is shown that the uniqueness by any given incident field fails for seven simple classes of regular polyhedral gratings. Moreover, if a regular bi-periodic polyhedral grating is not uniquely identifiable by a...

We prove that, both in the Lobachevskij and spherical 3-spaces, there exist nonconvex compact boundary-free polyhedral surfaces without selfintersections which admit nontrivial continuous deformations preserving all dihedral angles and study properties of such polyhedral surfaces. In particular, we prove that the volume of the domain, bounded by such a polyhedral surface, is necessarily constan...