219,858 research outputs found
On point sets with many unit distances in few directions
We study the problem of the maximum number of unit distances among n points in the plane under the additional restriction that we count only those unit distances that occur in a xed set of k directions taking the maximum over all sets of n points and all sets of k directions We prove that for xed k and suciently large n n k the extremal sets are essentially sections of lattices bounded by edges parallel to the k directions and of equal lengt
The number of unit distances is almost linear for most norms
We prove that there exists a norm in the plane under which no n-point set
determines more than O(n log n log log n) unit distances. Actually, most norms
have this property, in the sense that their complement is a meager set in the
metric space of all norms (with the metric given by the Hausdorff distance of
the unit balls)
Random Geometric Graphs and Isometries of Normed Spaces
Given a countable dense subset of a finite-dimensional normed space ,
and , we form a random graph on by joining, independently and with
probability , each pair of points at distance less than . We say that
is `Rado' if any two such random graphs are (almost surely) isomorphic.
Bonato and Janssen showed that in almost all are Rado. Our
main aim in this paper is to show that is the unique normed space
with this property: indeed, in every other space almost all sets are
non-Rado. We also determine which spaces admit some Rado set: this turns out to
be the spaces that have an direct summand. These results answer
questions of Bonato and Janssen.
A key role is played by the determination of which finite-dimensional normed
spaces have the property that every bijective step-isometry (meaning that the
integer part of distances is preserved) is in fact an isometry. This result may
be of independent interest
Incidence estimates for well spaced tubes
We prove analogues of the Szemer\'edi-Trotter theorem and other incidence
theorems using -tubes in place of straight lines, assuming that the
-tubes are well-spaced in a strong sense.Comment: 17 page
An error analysis of probabilistic fibre tracking methods: average curves optimization
Fibre tractography using diffusion tensor imaging is a promising method for estimating the pathways of white matter tracts in the human brain. The success of fibre tracking methods ultimately depends upon the accuracy of the fibre tracking algorithms and the quality of the data. Uncertainty and its representation have an important role to play in fibre tractography methods to infer useful information from real world noisy diffusion weighted data. Probabilistic fibre tracking approaches have received considerable interest recently for resolving orientational uncertainties. In this study, an average curves approach was used to investigate the impact of SNR and tensor field geometry on the accuracy of three different types of probabilistic tracking algorithms. The accuracy was assessed using simulated data and a range of tract geometries. The average curves representations were employed to represent the optimal fibre path of probabilistic tracking curves. The results are compared with streamline tracking on both simulated and in vivo data
On the computation of zone and double zone diagrams
Classical objects in computational geometry are defined by explicit
relations. Several years ago the pioneering works of T. Asano, J. Matousek and
T. Tokuyama introduced "implicit computational geometry", in which the
geometric objects are defined by implicit relations involving sets. An
important member in this family is called "a zone diagram". The implicit nature
of zone diagrams implies, as already observed in the original works, that their
computation is a challenging task. In a continuous setting this task has been
addressed (briefly) only by these authors in the Euclidean plane with point
sites. We discuss the possibility to compute zone diagrams in a wide class of
spaces and also shed new light on their computation in the original setting.
The class of spaces, which is introduced here, includes, in particular,
Euclidean spheres and finite dimensional strictly convex normed spaces. Sites
of a general form are allowed and it is shown that a generalization of the
iterative method suggested by Asano, Matousek and Tokuyama converges to a
double zone diagram, another implicit geometric object whose existence is known
in general. Occasionally a zone diagram can be obtained from this procedure.
The actual (approximate) computation of the iterations is based on a simple
algorithm which enables the approximate computation of Voronoi diagrams in a
general setting. Our analysis also yields a few byproducts of independent
interest, such as certain topological properties of Voronoi cells (e.g., that
in the considered setting their boundaries cannot be "fat").Comment: Very slight improvements (mainly correction of a few typos); add DOI;
Ref [51] points to a freely available computer application which implements
the algorithms; to appear in Discrete & Computational Geometry (available
online
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