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A Minimum Problem for Finite Sets of Real Numbers with Nonnegative Sum

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  • G. Chiaselotti
  • G. Marino
  • C. Nardi

Abstract

Let 𠑛 and 𠑟 be two integers such that 0 < 𠑟 ≤ 𠑛 ; we denote by 𠛾 ( 𠑛 , 𠑟 ) [ 𠜂 ( 𠑛 , 𠑟 ) ] the minimum [maximum] number of the nonnegative partial sums of a sum ∑ 𠑛 1 = 1 𠑎 𠑖 ≥ 0 , where 𠑎 1 , … , 𠑎 𠑛 are 𠑛 real numbers arbitrarily chosen in such a way that 𠑟 of them are nonnegative and the remaining 𠑛 − 𠑟 are negative. We study the following two problems: ( 𠑃 1 ) which are the values of 𠛾 ( 𠑛 , 𠑟 ) and 𠜂 ( 𠑛 , 𠑟 ) for each 𠑛 and 𠑟 , 0 < 𠑟 ≤ 𠑛 ? ( 𠑃 2 ) if 𠑞 is an integer such that 𠛾 ( 𠑛 , 𠑟 ) ≤ 𠑞 ≤ 𠜂 ( 𠑛 , 𠑟 ) , can we find 𠑛 real numbers 𠑎 1 , … , 𠑎 𠑛 , such that 𠑟 of them are nonnegative and the remaining 𠑛 − 𠑟 are negative with ∑ 𠑛 1 = 1 𠑎 𠑖 ≥ 0 , such that the number of the nonnegative sums formed from these numbers is exactly 𠑞 ?

Suggested Citation

  • G. Chiaselotti & G. Marino & C. Nardi, 2012. "A Minimum Problem for Finite Sets of Real Numbers with Nonnegative Sum," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-15, May.
    • Handle: RePEc:hin:jnljam:847958
    DOI: 10.1155/2012/847958
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    Cited by:

    1. Chiaselotti, G. & Gentile, T. & Oliverio, P.A., 2014. "Parallel and sequential dynamics of two discrete models of signed integer partitions," Applied Mathematics and Computation, Elsevier, vol. 232(C), pages 1249-1261.

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