3,811,141 research outputs found
Extensions of a result of Elekes and R\'onyai
Many problems in combinatorial geometry can be formulated in terms of curves
or surfaces containing many points of a cartesian product. In 2000, Elekes and
R\'onyai proved that if the graph of a polynomial contains points of an
cartesian product in , then the polynomial
has the form or . They used this to
prove a conjecture of Purdy which states that given two lines in
and points on each line, if the number of distinct distances between pairs
of points, one on each line, is at most , then the lines are parallel or
orthogonal. We extend the Elekes-R\'onyai Theorem to a less symmetric cartesian
product. We also extend the Elekes-R\'onyai Theorem to one dimension higher on
an cartesian product and an asymmetric cartesian
product. We give a proof of a variation of Purdy's conjecture with fewer points
on one of the lines. We finish with a lower bound for our main result in one
dimension higher with asymmetric cartesian product, showing that it is
near-optimal.Comment: 23 page
An O(n log n)-Time Algorithm for the Restricted Scaffold Assignment
The assignment problem takes as input two finite point sets S and T and
establishes a correspondence between points in S and points in T, such that
each point in S maps to exactly one point in T, and each point in T maps to at
least one point in S. In this paper we show that this problem has an O(n log
n)-time solution, provided that the points in S and T are restricted to lie on
a line (linear time, if S and T are presorted).Comment: 13 pages, 8 figure
O(N) models within the local potential approximation
Using Wegner-Houghton equation, within the Local Potential Approximation, we
study critical properties of O(N) vector models. Fixed Points, together with
their critical exponents and eigenoperators, are obtained for a large set of
values of N, including N=0 and N\to\infty. Polchinski equation is also treated.
The peculiarities of the large N limit, where a line of Fixed Points at d=2+2/n
is present, are studied in detail. A derivation of the equation is presented
together with its projection to zero modes.Comment: 27 pages, LaTeX with psfig, 7 PostScript figures. One reference
corrected and one added with respect to the journal versio
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