2,206,646 research outputs found
Approximate Minimum Diameter
We study the minimum diameter problem for a set of inexact points. By
inexact, we mean that the precise location of the points is not known. Instead,
the location of each point is restricted to a contineus region (\impre model)
or a finite set of points (\indec model). Given a set of inexact points in
one of \impre or \indec models, we wish to provide a lower-bound on the
diameter of the real points.
In the first part of the paper, we focus on \indec model. We present an
time
approximation algorithm of factor for finding minimum diameter
of a set of points in dimensions. This improves the previously proposed
algorithms for this problem substantially.
Next, we consider the problem in \impre model. In -dimensional space, we
propose a polynomial time -approximation algorithm. In addition, for
, we define the notion of -separability and use our algorithm for
\indec model to obtain -approximation algorithm for a set of
-separable regions in time
Diameter Perfect Lee Codes
Lee codes have been intensively studied for more than 40 years. Interest in
these codes has been triggered by the Golomb-Welch conjecture on the existence
of the perfect error-correcting Lee codes. In this paper we deal with the
existence and enumeration of diameter perfect Lee codes. As main results we
determine all for which there exists a linear diameter-4 perfect Lee code
of word length over and prove that for each there are
uncountable many diameter-4 perfect Lee codes of word length over This
is in a strict contrast with perfect error-correcting Lee codes of word length
over \ as there is a unique such code for and its is
conjectured that this is always the case when is a prime. We produce
diameter perfect Lee codes by an algebraic construction that is based on a
group homomorphism. This will allow us to design an efficient algorithm for
their decoding. We hope that this construction will turn out to be useful far
beyond the scope of this paper
The monic integer transfinite diameter
We study the problem of finding nonconstant monic integer polynomials,
normalized by their degree, with small supremum on an interval I. The monic
integer transfinite diameter t_M(I) is defined as the infimum of all such
supremums. We show that if I has length 1 then t_M(I) = 1/2.
We make three general conjectures relating to the value of t_M(I) for
intervals I of length less that 4. We also conjecture a value for t_M([0, b])
where 0 < b < 1. We give some partial results, as well as computational
evidence, to support these conjectures.
We define two functions that measure properties of the lengths of intervals I
with t_M(I) on either side of t. Upper and lower bounds are given for these
functions.
We also consider the problem of determining t_M(I) when I is a Farey
interval. We prove that a conjecture of Borwein, Pinner and Pritsker concerning
this value is true for an infinite family of Farey intervals.Comment: 32 pages, 5 figure
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