We provide a complete description of normal affine algebraic varieties over the real numbers endowed with an effective action circle, that is, form complex multiplicative group whose locus consists unitary circle in plane. Our approach builds on geometrico-combinatorial actions split tori terms proper polyhedral divisors semiprojective due to Altmann and Hausen.

Underwater bubbles are extremely good acoustic resonators, but freely evolving and dissolving. Recently it was found that can be stabilized in frames, the influence of frame shape is still undocumented. Here we first explore vibration polyhedral with a low number faces, shaped as five Platonic solids. Their resonance frequency well approximated by formula for spherical same volume. Then extend ...

A new class of polyhedra is discovered—bipyramids of arbitrarily prescribed genus. A two-dimensional generalization of Fáry’s Theorem is established. A purely combinatorial definition of a polyhedral suspension is given. A new regular 2-dimensional polyhedron is constructed in four dimensions. Short running title: Polyhedral suspensions. 2000 AMS Subject Classification: Primary: 57M20; Secondar...

We present eecient polynomial time algorithms that place bn=2c vertex guards which cover the surface of an n-vertex polyhedral terrain, and similarly, bn=3c edge guards which cover the surface of an n-vertex polyhedral terrain. The time complexity of both algorithms, dominated by the cost of nding a maximum matching in a graph, is O(n 3=2).

The (axial) three index assignment problem, also known as the threedimensional matching problem, is the problem of assigning one item to one job at one point or interval of time in such a way as to minimize the total cost of the assignment. Until now the most efficient algorithms explored for solving this problem are based on polyhedral combinatorics. So far, four important facet classes Q, P ,...

Introduction. It is the purpose of this paper to give some weaker analogues for polyhedral manifolds of well-known properties of combinatorial manifolds. This is now possible primarily because of the recent work of Mazur [3 ; 4] and M. Brown [l ]. A crucial issue in this area is the extent to which these analogues may be improved for arbitrary triangulated manifolds. We shall prove a theorem wh...

We define certain natural finite sums of n’th roots of unity, called GP (n), that are associated to each convex integer polytope P , and which generalize the classical 1-dimensional Gauss sum G(n) defined over Z/nZ, to higher dimensional abelian groups and integer polytopes. We consider the finite Weyl group W, generated by the reflections with respect to the coordinate hyperplanes, as well as ...