120,485 research outputs found
Decomposition of Geometric Set Systems and Graphs
We study two decomposition problems in combinatorial geometry. The first part
deals with the decomposition of multiple coverings of the plane. We say that a
planar set is cover-decomposable if there is a constant m such that any m-fold
covering of the plane with its translates is decomposable into two disjoint
coverings of the whole plane. Pach conjectured that every convex set is
cover-decomposable. We verify his conjecture for polygons. Moreover, if m is
large enough, we prove that any m-fold covering can even be decomposed into k
coverings. Then we show that the situation is exactly the opposite in 3
dimensions, for any polyhedron and any we construct an m-fold covering of
the space that is not decomposable. We also give constructions that show that
concave polygons are usually not cover-decomposable. We start the first part
with a detailed survey of all results on the cover-decomposability of polygons.
The second part investigates another geometric partition problem, related to
planar representation of graphs. The slope number of a graph G is the smallest
number s with the property that G has a straight-line drawing with edges of at
most s distinct slopes and with no bends. We examine the slope number of
bounded degree graphs. Our main results are that if the maximum degree is at
least 5, then the slope number tends to infinity as the number of vertices
grows but every graph with maximum degree at most 3 can be embedded with only
five slopes. We also prove that such an embedding exists for the related notion
called slope parameter. Finally, we study the planar slope number, defined only
for planar graphs as the smallest number s with the property that the graph has
a straight-line drawing in the plane without any crossings such that the edges
are segments of only s distinct slopes. We show that the planar slope number of
planar graphs with bounded degree is bounded.Comment: This is my PhD thesi
Deformation of the Cubic Open String Field Theory
We study a consistent deformation of the cubic open bosonic string theory in
such a way that the non-planar world sheet diagrams of the perturbative string
theory are mapped onto their equivalent planar diagrams of the light-cone
string field theory with some length parameters fixed. An explicit evaluation
of the cubic string vertex in the zero-slope limit yields the correct
relationship between the string coupling constant and the Yang-Mills coupling
constant. The deformed cubic open string field theory is shown to produce the
non-Abelian Yang-Mills action in the zero-slope limit if it is defined on
multiple D-branes. Applying the consistent deformation systematically to
multi-string world sheet diagrams, we may be able to calculate scattering
amplitudes with an arbitrary number of external open strings.Comment: 10 pages, 10 figure
Planar Drawings with Few Slopes of Halin Graphs and Nested Pseudotrees
The of a planar graph is the
minimum number of edge slopes in a planar straight-line drawing of . It is
known that for every planar graph of maximum
degree . This upper bound has been improved to if has
treewidth three, and to if has treewidth two. In this paper we
prove when is a Halin graph, and thus has
treewidth three. Furthermore, we present the first polynomial upper bound on
the planar slope number for a family of graphs having treewidth four. Namely we
show that slopes suffice for nested pseudotrees.Comment: Extended version of "Planar Drawings with Few Slopes of Halin Graphs
and Nested Pseudotrees" appeared in the Proceedings of the 17th Algorithms
and Data Structures Symposium (WADS 2021
A Universal Slope Set for 1-Bend Planar Drawings
We describe a set of Delta-1 slopes that are universal for 1-bend planar drawings of planar graphs of maximum degree Delta>=4; this establishes a new upper bound of Delta-1 on the 1-bend planar slope number. By universal we mean that every planar graph of degree Delta has a planar drawing with at most one bend per edge and such that the slopes of the segments forming the edges belong to the given set of slopes. This improves over previous results in two ways: Firstly, the best previously known upper bound for the 1-bend planar slope number was 3/2(Delta-1) (the known lower bound being 3/4(Delta-1)); secondly, all the known algorithms to construct 1-bend planar drawings with O(Delta) slopes use a different set of slopes for each graph and can have bad angular resolution, while our algorithm uses a universal set of slopes, which also guarantees that the minimum angle between any two edges incident to a vertex is pi/(Delta-1)
A new continuous planar fit method for calculating fluxes in complex, forested terrain.
The planar fit method is often recommended for long-term eddy covariance flux measurements since it offers a number of advantages over rotating into streamwise coordinates. For sites over complex, forested terrain a single planar fit may not account for complex variations in slope and canopy cover with wind direction. An alternative to the planar fit method is presented where the tilt angle is fitted as a continuous function of the wind direction. This retains many of the benefits of the planar fit method, while at the same time better representing local variations in tilt with wind direction
Grain Boundary Scars and Spherical Crystallography
We describe experimental investigations of the structure of two-dimensional
spherical crystals. The crystals, formed by beads self-assembled on water
droplets in oil, serve as model systems for exploring very general theories
about the minimum energy configurations of particles with arbitrary repulsive
interactions on curved surfaces. Above a critical system size we find that
crystals develop distinctive high-angle grain boundaries, or scars, not found
in planar crystals. The number of excess defects in a scar is shown to grow
linearly with the dimensionless system size. The observed slope is expected to
be universal, independent of the microscopic potential.Comment: 4 pages, 3 eps figs (high quality images available from Mark Bowick
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