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Journal of Applied Mathematics
Volume 2012, Article ID 847958, 15 pages
http://dx.doi.org/10.1155/2012/847958
Research Article

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

1Dipartimento di Matematica, Universitá della Calabria, Via Pietro Bucci, Cubo 30B, 87036 Arcavacata di Rende, Italy
2Dipartimento di Matematica, Universitá della Calabria, Via Pietro Bucci, Cubo 30C, 87036 Arcavacata di Rende, Italy

Received 6 February 2012; Accepted 2 March 2012

Academic Editor: Yonghong Yao

Copyright © 2012 G. Chiaselotti et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let 𝑛 and π‘Ÿ be two integers such that 0 < π‘Ÿ ≀ 𝑛 ; we denote by 𝛾 ( 𝑛 , π‘Ÿ ) [ πœ‚ ( 𝑛 , π‘Ÿ ) ] the minimum [maximum] number of the nonnegative partial sums of a sum βˆ‘ 𝑛 1 = 1 π‘Ž 𝑖 β‰₯ 0 , where π‘Ž 1 , … , π‘Ž 𝑛 are 𝑛 real numbers arbitrarily chosen in such a way that π‘Ÿ of them are nonnegative and the remaining 𝑛 βˆ’ π‘Ÿ are negative. We study the following two problems: ( 𝑃 1 ) which are the values of 𝛾 ( 𝑛 , π‘Ÿ ) and πœ‚ ( 𝑛 , π‘Ÿ ) for each 𝑛 and π‘Ÿ , 0 < π‘Ÿ ≀ 𝑛 ? ( 𝑃 2 ) if π‘ž is an integer such that 𝛾 ( 𝑛 , π‘Ÿ ) ≀ π‘ž ≀ πœ‚ ( 𝑛 , π‘Ÿ ) , can we find 𝑛 real numbers π‘Ž 1 , … , π‘Ž 𝑛 , such that π‘Ÿ of them are nonnegative and the remaining 𝑛 βˆ’ π‘Ÿ are negative with βˆ‘ 𝑛 1 = 1 π‘Ž 𝑖 β‰₯ 0 , such that the number of the nonnegative sums formed from these numbers is exactly π‘ž ?