On the Multiplicity of a Proportionally Modular Numerical Semigroup
A proportionally modular numerical semigroup is the set Sa,b,c of nonnegative integer solutions to a Diophantine inequality of the form ax mod b≤cx, where a,b, and c are positive integers. A formula for the multiplicity of Sa,b,c, that is, mSa,b,c=kb/a for some positive integer k, is given by A. Mos...
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Hindawi Limited
2021-01-01
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| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2021/3982297 |
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doaj-a2f65dda39c64215a23b458480ab33ed2021-09-20T00:29:01ZengHindawi LimitedDiscrete Dynamics in Nature and Society1607-887X2021-01-01202110.1155/2021/3982297On the Multiplicity of a Proportionally Modular Numerical SemigroupZe Gu0School of Mathematics and StatisticsA proportionally modular numerical semigroup is the set Sa,b,c of nonnegative integer solutions to a Diophantine inequality of the form ax mod b≤cx, where a,b, and c are positive integers. A formula for the multiplicity of Sa,b,c, that is, mSa,b,c=kb/a for some positive integer k, is given by A. Moscariello. In this paper, we give a new proof of the formula and determine a better bound for k. Furthermore, we obtain k=1 for various cases and a formula for the number of the triples a,b,c such that k≠1 when the number a−c is fixed.http://dx.doi.org/10.1155/2021/3982297 |
| collection |
DOAJ |
| language |
English |
| format |
Article |
| sources |
DOAJ |
| author |
Ze Gu |
| spellingShingle |
Ze Gu On the Multiplicity of a Proportionally Modular Numerical Semigroup Discrete Dynamics in Nature and Society |
| author_facet |
Ze Gu |
| author_sort |
Ze Gu |
| title |
On the Multiplicity of a Proportionally Modular Numerical Semigroup |
| title_short |
On the Multiplicity of a Proportionally Modular Numerical Semigroup |
| title_full |
On the Multiplicity of a Proportionally Modular Numerical Semigroup |
| title_fullStr |
On the Multiplicity of a Proportionally Modular Numerical Semigroup |
| title_full_unstemmed |
On the Multiplicity of a Proportionally Modular Numerical Semigroup |
| title_sort |
on the multiplicity of a proportionally modular numerical semigroup |
| publisher |
Hindawi Limited |
| series |
Discrete Dynamics in Nature and Society |
| issn |
1607-887X |
| publishDate |
2021-01-01 |
| description |
A proportionally modular numerical semigroup is the set Sa,b,c of nonnegative integer solutions to a Diophantine inequality of the form ax mod b≤cx, where a,b, and c are positive integers. A formula for the multiplicity of Sa,b,c, that is, mSa,b,c=kb/a for some positive integer k, is given by A. Moscariello. In this paper, we give a new proof of the formula and determine a better bound for k. Furthermore, we obtain k=1 for various cases and a formula for the number of the triples a,b,c such that k≠1 when the number a−c is fixed. |
| url |
http://dx.doi.org/10.1155/2021/3982297 |
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AT zegu onthemultiplicityofaproportionallymodularnumericalsemigroup |
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