196,538 research outputs found
Grid Representations and the Chromatic Number
A grid drawing of a graph maps vertices to grid points and edges to line
segments that avoid grid points representing other vertices. We show that there
is a number of grid points that some line segment of an arbitrary grid drawing
must intersect. This number is closely connected to the chromatic number.
Second, we study how many columns we need to draw a graph in the grid,
introducing some new \NP-complete problems. Finally, we show that any planar
graph has a planar grid drawing where every line segment contains exactly two
grid points. This result proves conjectures asked by David Flores-Pe\~naloza
and Francisco Javier Zaragoza Martinez.Comment: 22 pages, 8 figure
Steinitz Theorems for Orthogonal Polyhedra
We define a simple orthogonal polyhedron to be a three-dimensional polyhedron
with the topology of a sphere in which three mutually-perpendicular edges meet
at each vertex. By analogy to Steinitz's theorem characterizing the graphs of
convex polyhedra, we find graph-theoretic characterizations of three classes of
simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric
projection in the plane with only one hidden vertex, xyz polyhedra, in which
each axis-parallel line through a vertex contains exactly one other vertex, and
arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz
polyhedra are exactly the bipartite cubic polyhedral graphs, and every
bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of
a corner polyhedron. Based on our characterizations we find efficient
algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure
Quantum Structure in Competing Lizard Communities
Almost two decades of research on applications of the mathematical formalism
of quantum theory as a modeling tool in domains different from the micro-world
has given rise to many successful applications in situations related to human
behavior and thought, more specifically in cognitive processes of
decision-making and the ways concepts are combined into sentences. In this
article, we extend this approach to animal behavior, showing that an analysis
of an interactive situation involving a mating competition between certain
lizard morphs allows to identify a quantum theoretic structure. More in
particular, we show that when this lizard competition is analyzed structurally
in the light of a compound entity consisting of subentities, the contextuality
provided by the presence of an underlying rock-paper-scissors cyclic dynamics
leads to a violation of Bell's inequality, which means it is of a non-classical
type. We work out an explicit quantum-mechanical representation in Hilbert
space for the lizard situation and show that it faithfully models a set of
experimental data collected on three throat-colored morphs of a specific lizard
species. Furthermore, we investigate the Hilbert space modeling, and show that
the states describing the lizard competitions contain entanglement for each one
of the considered confrontations of lizards with different competing
strategies, which renders it no longer possible to interpret these states of
the competing lizards as compositions of states of the individual lizards.Comment: 28 page
How Does Colour Experience Represent the World?
Many favor representationalism about color experience. To a first approximation, this view holds that experiencing is like believing. In particular, like believing, experiencing is a matter of representing the world to be a certain way. Once you view color experience along these lines, you face a big question: do our color experiences represent the world as it really is? For instance, suppose you see a tomato. Representationalists claim that having an experience with this sensory character is necessarily connected with representing a distinctive quality as pervading a round area out there in external space. Let us call it “sensible redness” to highlight the fact that the representation of this property is necessarily connected with the sensory character of the experience. Is this property, sensible redness, really co-instantiated with roundness out there in the space before you
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