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Integer solutions to interval linear equations and unique measurement

Author: Peter Fishburn
Journal: Proc. Amer. Math. Soc. 129 (2001), 1595-1599
MSC (2000): Primary 05A99, 11D04, 91E45
Published electronically: November 15, 2000
MathSciNet review: 1814085
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Abstract | References | Similar Articles | Additional Information


Every system of $n$ linearly independent homogeneous linear equations in $n+1$ unknowns with coefficients in $\{1,0,-1\}$ has a unique (up to multiplication by $-1$) non-zero solution vector $d= (d_1, d_2, \ldots , d_{n+1} )$ in which the $d_j$'s are integers with no common divisor greater than 1. It is known that, for large $n$, $\vert \sum d_j \vert$ can be arbitrarily greater than $2^n$. We prove that if every equation, written as $\sum_A x_i - \sum_B x_i =0$, is such that $A$ and $B$ are intervals of contiguous indices, then $\vert\sum d_j \vert \le 2^n$. This confirms conjectures of the author and Fred Roberts that arose in the theory of unique finite measurement.

References [Enhancements On Off] (What's this?)

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Additional Information

Peter Fishburn
Affiliation: AT&T Laboratories, Room C227, 180 Park Avenue, Florham Park, New Jersey 07932

Keywords: Linear equations, integer solutions, measurement theory
Received by editor(s): September 14, 1999
Published electronically: November 15, 2000
Communicated by: Mark J. Ablowitz
Article copyright: © Copyright 2000 American Mathematical Society

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