We begin by defining and studying tightness and the two-piece property for smooth and polyhedral surfaces in three-dimensional space. These results are then generalized to surfaces with boundary and with singularities, and to surfaces in higher dimensions. Later sections deal with generalizations to the case of smooth and polyhedral submanifolds of higher dimension and codimension, in particula...

To understand the hierarchical self-organization behaviors of soft materials as well as their dependence on molecular geometry, a series of AB n dendron-like molecules based on polyhedral oligomeric silsesquioxane (POSS) nanoparticles were designed and synthesized. The apex of these molecules is a hydrophilic POSS cage with 14 hydroxyl groups (denoted DPOSS). At its periphery, there are differe...

Two algorithms for producing polyhedral representations for 3-connected planar graphs are discussed in the paper. One of them uses Tutte’s drawing algorithm [11] to produce 2D drawing. Then the drawing is lifted into 3D space obtaining a polyhedral embedding. The other is simple algorithm by G. Hart [4] for drawing canonical polyhedral representations. Some alternative aspects (physical model, ...

Polyhedral embeddings of cubic graphs by means of certain operations are studied. It is proved that some known families of snarks have no (orientable) polyhedral embeddings. This result supports a conjecture of Grünbaum that no snark admits an orientable polyhedral embedding. This conjecture is verified for all snarks having up to 30 vertices using computer. On the other hand, for every nonorie...

Earth abundant element clathrate phases are of interest for a number of applications ranging from photovoltaics to thermoelectrics. Silicon-containing type I clathrate is a framework structure with the stoichiometry A8-xSi46 (A = guest atom such as alkali metal) that can be tuned by alloying and doping with other elements. The type I clathrate framework can be described as being composed of two...

Abstract This paper aims to show that there exists a triangulation of the Heisenberg group $$\mathbb {H}^n$$ H n into singular simplexes with regularity properties on both low-dimensional and high-dimensional layers. For low dimensions, we request...

Abstract Crystallographic distortions in the alternating aluminium and silicon tetrahedral framework of sodalite (Na 8 Al 6 Si O 24 Cl 2 ), beryllium helvine (Mn Be S (framework designated SOD) are described terms a set condensed normal mode amplitudes phases derived from an ideal tetrahedron theoretical aristotype phase. For sodalite-structured hettotype phase space group $P{\overline 4} 3n$ ,...

This paper presents deployment strategies for a team of multiple mobile robots with line-of-sight visibility in 1.5D and 2.5D terrain environments. Our objective is to distributively achieve full visibility of a polyhedral environment. In the 1.5D polyhedral terrain, we achieve this by determining a set of locations that the robots can distributively occupy. In the 2.5D polyhedral terrain, we a...

In the last decade our capability of solving integer programming problems has increased dramatically due to the effectiveness of cutting plane methods based on polyhedral investigations. Polyhedral cutting planes have become central features in optimization software packages for integer programming. Here we present some of the important polyhedral methods used in discrete optimization. We discu...

This paper presents a new algorithm for correcting incorrect projections of a polyhedral scene. Such projections arise in applications where an image of a polyhedral world is taken and its edges and vertices are extracted. Along the way, the true positions of the vertices in the 2D projection are perturbed due to digitization errors and the preprocessing. As most available algorithms for recons...