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Dot product rearrangements

Author

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  • Paul Erdos
  • Gary Weiss

Abstract

Let a = ( a n ) , x = ( x n ) denote nonnegative sequences; x = ( x π ( n ) ) denotes the rearranged sequence determined by the permutation π , a ⋅ x denotes the dot product ∑ a n x n ; and S ( a , x ) denotes { a ⋅ x π : π is a permuation of the positive integers } . We examine S ( a , x ) as a subset of the nonnegative real line in certain special circumstances. The main result is that if a n ↑ ∞ , then S ( a , x ) = [ a ⋅ x , ∞ ] for every x n ↓ ≠ 0 if and only if a n + 1 / a n is uniformly bounded.

Suggested Citation

  • Paul Erdos & Gary Weiss, 1983. "Dot product rearrangements," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 6, pages 1-10, January.
  • Handle: RePEc:hin:jijmms:962681
    DOI: 10.1155/S0161171283000368
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