116,929 research outputs found
Crossing Patterns of Segments
AbstractIt is shown that for every c>0 there exists c′>0 satisfying the following condition. Let S be a system of n straight-line segments in the plane, which determine at least cn2 crossings. Then there are two disjoint at least c′n-element subsystems, S1, S2⊂S, such that every element of S1 crosses all elements of S2
Quasi-Parallel Segments and Characterization of Unique Bichromatic Matchings
Given n red and n blue points in general position in the plane, it is
well-known that there is a perfect matching formed by non-crossing line
segments. We characterize the bichromatic point sets which admit exactly one
non-crossing matching. We give several geometric descriptions of such sets, and
find an O(nlogn) algorithm that checks whether a given bichromatic set has this
property.Comment: 31 pages, 24 figure
Short and random: Modelling the effects of (proto-)neural elongations
To understand how neurons and nervous systems first evolved, we need an
account of the origins of neural elongations: Why did neural elongations (axons
and dendrites) first originate, such that they could become the central
component of both neurons and nervous systems? Two contrasting conceptual
accounts provide different answers to this question. Braitenberg's vehicles
provide the iconic illustration of the dominant input-output (IO) view. Here
the basic role of neural elongations is to connect sensors to effectors, both
situated at different positions within the body. For this function, neural
elongations are thought of as comparatively long and specific connections,
which require an articulated body involving substantial developmental processes
to build. Internal coordination (IC) models stress a different function for
early nervous systems. Here the coordination of activity across extended parts
of a multicellular body is held central, in particular for the contractions of
(muscle) tissue. An IC perspective allows the hypothesis that the earliest
proto-neural elongations could have been functional even when they were
initially simple short and random connections, as long as they enhanced the
patterning of contractile activity across a multicellular surface. The present
computational study provides a proof of concept that such short and random
neural elongations can play this role. While an excitable epithelium can
generate basic forms of patterning for small body-configurations, adding
elongations allows such patterning to scale up to larger bodies. This result
supports a new, more gradual evolutionary route towards the origins of the very
first full neurons and nervous systems.Comment: 12 pages, 5 figures, Keywords: early nervous systems, neural
elongations, nervous system evolution, computational modelling, internal
coordinatio
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