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Numerical semigroups in a problem about cost-effective transport

  • Aureliano M. Robles-Pérez ORCID logo EMAIL logo und José Carlos Rosales ORCID logo
Veröffentlicht/Copyright: 14. Juni 2016
Forum Mathematicum
Aus der Zeitschrift Forum Mathematicum Band 29 Heft 2

Abstract

Let ℕ be the set of nonnegative integers. A problem about how to transport profitably an organized group of persons leads us to study the set T formed by the integers n such that the system of inequalities, with nonnegative integer coefficients,

a1⁢x1+⋯+ap⁢xp<n<b1⁢x1+⋯+bp⁢xp

has at least one solution in ℕp. We will see that T∪{0} is a numerical semigroup. Moreover, we will show that a numerical semigroup S can be obtained in this way if and only if {a+b-1,a+b+1}⊆S, for all a,b∈S∖{0}. In addition, we will demonstrate that such numerical semigroups form a Frobenius variety and we will study this variety. Finally, we show an algorithmic process in order to compute T.

Keywords: Diophantine inequalities; submonoids; numerical semigroups; non-homogeneous patterns; Frobenius varieties; trees; profitable transport
MSC 2010: 20M14; 11D07; 11D75

Communicated by Manfred Droste

Funding source: Ministerio de Economía y Competitividad

Award Identifier / Grant number: MTM2014-55367-P

Funding statement: Both authors are supported by the project MTM2014-55367-P, which is funded by Ministerio de Economía y Competitividad and Fondo Europeo de Desarrollo Regional FEDER, and by the Junta de Andalucía Grant Number FQM-343. The second author is also partially supported by Junta de Andalucía/Feder Grant Number FQM-5849.

Acknowledgements

The authors would like to thank the referee for providing constructive comments and help in improving the contents of this paper.

References

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Received: 2015-6-26
Revised: 2016-3-31
Published Online: 2016-6-14
Published in Print: 2017-3-1

© 2017 by De Gruyter

Artikel in diesem Heft

  1. Frontmatter
  2. A note on the Hilali conjecture
  3. Finite groups have more conjugacy classes
  4. The gauge action, DG Lie algebras and identities for Bernoulli numbers
  5. Unitary embeddings of finite loop spaces
  6. Topological complexity of planar polygon spaces with small genetic code
  7. Numerical semigroups in a problem about cost-effective transport
  8. On the smallest simultaneous power nonresidue modulo a prime
  9. Decompositions of the higher order polars of plane branches
  10. A formula for the expected volume of the Wiener sausage with constant drift
  11. The slice spectral sequence for the C4 analog of real K-theory
  12. Incidences between points and generalized spheres over finite fields and related problems
  13. Mean values of the Witten L-function for SU(2)
  14. A direct computation of the cohomology of the braces operad
  15. Abelian varieties over finite fields as basic abelian varieties
https://doi.org/10.1515/forum-2015-0123
Schlagwörter für diesen Artikel
Diophantine inequalities; submonoids; numerical semigroups; non-homogeneous patterns; Frobenius varieties; trees; profitable transport

Artikel in diesem Heft

  1. Frontmatter
  2. A note on the Hilali conjecture
  3. Finite groups have more conjugacy classes
  4. The gauge action, DG Lie algebras and identities for Bernoulli numbers
  5. Unitary embeddings of finite loop spaces
  6. Topological complexity of planar polygon spaces with small genetic code
  7. Numerical semigroups in a problem about cost-effective transport
  8. On the smallest simultaneous power nonresidue modulo a prime
  9. Decompositions of the higher order polars of plane branches
  10. A formula for the expected volume of the Wiener sausage with constant drift
  11. The slice spectral sequence for the C4 analog of real K-theory
  12. Incidences between points and generalized spheres over finite fields and related problems
  13. Mean values of the Witten L-function for SU(2)
  14. A direct computation of the cohomology of the braces operad
  15. Abelian varieties over finite fields as basic abelian varieties
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