1,184 research outputs found
The Complexity of Order Type Isomorphism
The order type of a point set in maps each -tuple of points to
its orientation (e.g., clockwise or counterclockwise in ). Two point sets
and have the same order type if there exists a mapping from to
for which every -tuple of and the
corresponding tuple in have the same
orientation. In this paper we investigate the complexity of determining whether
two point sets have the same order type. We provide an algorithm for
this task, thereby improving upon the algorithm
of Goodman and Pollack (1983). The algorithm uses only order type queries and
also works for abstract order types (or acyclic oriented matroids). Our
algorithm is optimal, both in the abstract setting and for realizable points
sets if the algorithm only uses order type queries.Comment: Preliminary version of paper to appear at ACM-SIAM Symposium on
Discrete Algorithms (SODA14
An Algebra of Pieces of Space -- Hermann Grassmann to Gian Carlo Rota
We sketch the outlines of Gian Carlo Rota's interaction with the ideas that
Hermann Grassmann developed in his Ausdehnungslehre of 1844 and 1862, as
adapted and explained by Giuseppe Peano in 1888. This leads us past what Rota
variously called 'Grassmann-Cayley algebra', or 'Peano spaces', to the Whitney
algebra of a matroid, and finally to a resolution of the question "What,
really, was Grassmann's regressive product?". This final question is the
subject of ongoing joint work with Andrea Brini, Francesco Regonati, and
William Schmitt.
The present paper was presented at the conference "The Digital Footprint of
Gian-Carlo Rota: Marbles, Boxes and Philosophy" in Milano on 17 Feb 2009. It
will appear in proceedings of that conference, to be published by Springer
Verlag.Comment: 28 page
Constructing packings in Grassmannian manifolds via alternating projection
This paper describes a numerical method for finding good packings in
Grassmannian manifolds equipped with various metrics. This investigation also
encompasses packing in projective spaces. In each case, producing a good
packing is equivalent to constructing a matrix that has certain structural and
spectral properties. By alternately enforcing the structural condition and then
the spectral condition, it is often possible to reach a matrix that satisfies
both. One may then extract a packing from this matrix.
This approach is both powerful and versatile. In cases where experiments have
been performed, the alternating projection method yields packings that compete
with the best packings recorded. It also extends to problems that have not been
studied numerically. For example, it can be used to produce packings of
subspaces in real and complex Grassmannian spaces equipped with the
Fubini--Study distance; these packings are valuable in wireless communications.
One can prove that some of the novel configurations constructed by the
algorithm have packing diameters that are nearly optimal.Comment: 41 pages, 7 tables, 4 figure
R.A.Fisher, design theory, and the Indian connection
Design Theory, a branch of mathematics, was born out of the experimental
statistics research of the population geneticist R. A. Fisher and of Indian
mathematical statisticians in the 1930s. The field combines elements of
combinatorics, finite projective geometries, Latin squares, and a variety of
further mathematical structures, brought together in surprising ways. This
essay will present these structures and ideas as well as how the field came
together, in itself an interesting story.Comment: 11 pages, 3 figure
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