11,639 research outputs found
Flat Foldings of Plane Graphs with Prescribed Angles and Edge Lengths
When can a plane graph with prescribed edge lengths and prescribed angles
(from among \}) be folded flat to lie in an
infinitesimally thin line, without crossings? This problem generalizes the
classic theory of single-vertex flat origami with prescribed mountain-valley
assignment, which corresponds to the case of a cycle graph. We characterize
such flat-foldable plane graphs by two obviously necessary but also sufficient
conditions, proving a conjecture made in 2001: the angles at each vertex should
sum to , and every face of the graph must itself be flat foldable.
This characterization leads to a linear-time algorithm for testing flat
foldability of plane graphs with prescribed edge lengths and angles, and a
polynomial-time algorithm for counting the number of distinct folded states.Comment: 21 pages, 10 figure
Type II Superstring Field Theory: Geometric Approach and Operadic Description
We outline the construction of type II superstring field theory leading to a
geometric and algebraic BV master equation, analogous to Zwiebach's
construction for the bosonic string. The construction uses the small Hilbert
space. Elementary vertices of the non-polynomial action are described with the
help of a properly formulated minimal area problem. They give rise to an
infinite tower of superstring field products defining a
generalization of a loop homotopy Lie algebra, the genus zero part generalizing
a homotopy Lie algebra. Finally, we give an operadic interpretation of the
construction.Comment: 37 pages, 1 figure, corrected typos and added reference
Yangian Symmetry for Bi-Scalar Loop Amplitudes
We establish an all-loop conformal Yangian symmetry for the full set of
planar amplitudes in the recently proposed integrable bi-scalar field theory in
four dimensions. This chiral theory is a particular double scaling limit of
gamma-twisted weakly coupled N=4 SYM theory. Each amplitude with a certain
order of scalar particles is given by a single fishnet Feynman graph of disc
topology cut out of a regular square lattice. The Yangian can be realized by
the action of a product of Lax operators with a specific sequence of
inhomogeneity parameters on the boundary of the disc. Based on this
observation, the Yangian generators of level one for generic bi-scalar
amplitudes are explicitly constructed. Finally, we comment on the relation to
the dual conformal symmetry of these scattering amplitudes.Comment: 40 pages, 20 figure
TS2PACK: A Two-Level Tabu Search for the Three-dimensional Bin Packing Problem
Three-dimensional orthogonal bin packing is a problem NP-hard in the strong sense where a set of boxes must be orthogonally packed into the minimum number of three-dimensional bins. We present a two-level tabu search for this problem. The first-level aims to reduce the number of bins. The second optimizes the packing of the bins. This latter procedure is based on the Interval Graph representation of the packing, proposed by Fekete and Schepers, which reduces the size of the search space. We also introduce a general method to increase the size of the associated neighborhoods, and thus the quality of the search, without increasing the overall complexity of the algorithm. Extensive computational results on benchmark problem instances show the effectiveness of the proposed approach, obtaining better results compared to the existing one
Small grid embeddings of 3-polytopes
We introduce an algorithm that embeds a given 3-connected planar graph as a
convex 3-polytope with integer coordinates. The size of the coordinates is
bounded by . If the graph contains a triangle we can
bound the integer coordinates by . If the graph contains a
quadrilateral we can bound the integer coordinates by . The
crucial part of the algorithm is to find a convex plane embedding whose edges
can be weighted such that the sum of the weighted edges, seen as vectors,
cancel at every point. It is well known that this can be guaranteed for the
interior vertices by applying a technique of Tutte. We show how to extend
Tutte's ideas to construct a plane embedding where the weighted vector sums
cancel also on the vertices of the boundary face
- …