Random numerical semigroups and sums of subsets of cyclic groups
We investigate properties of random numerical semigroups using a probabilistic model based on the Erdös-Rényi model for random graphs and propose a new probabilistic model. We provide a new and more elementary proof of a lower bound of the expected embedding dimension, genus, and Frobenius number of...
- Autores:
-
Morales Duarte, Santiago
- Tipo de recurso:
- Trabajo de grado de pregrado
- Fecha de publicación:
- 2023
- Institución:
- Universidad de los Andes
- Repositorio:
- Séneca: repositorio Uniandes
- Idioma:
- eng
- OAI Identifier:
- oai:repositorio.uniandes.edu.co:1992/73668
- Acceso en línea:
- https://hdl.handle.net/1992/73668
- Palabra clave:
- Numerical semigroups
Probabilistic methods
Matemáticas
- Rights
- openAccess
- License
- https://repositorio.uniandes.edu.co/static/pdf/aceptacion_uso_es.pdf
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Random numerical semigroups and sums of subsets of cyclic groups |
| title |
Random numerical semigroups and sums of subsets of cyclic groups |
| spellingShingle |
Random numerical semigroups and sums of subsets of cyclic groups Numerical semigroups Probabilistic methods Matemáticas |
| title_short |
Random numerical semigroups and sums of subsets of cyclic groups |
| title_full |
Random numerical semigroups and sums of subsets of cyclic groups |
| title_fullStr |
Random numerical semigroups and sums of subsets of cyclic groups |
| title_full_unstemmed |
Random numerical semigroups and sums of subsets of cyclic groups |
| title_sort |
Random numerical semigroups and sums of subsets of cyclic groups |
| dc.creator.fl_str_mv |
Morales Duarte, Santiago |
| dc.contributor.advisor.none.fl_str_mv |
Bogart, Tristram |
| dc.contributor.author.none.fl_str_mv |
Morales Duarte, Santiago |
| dc.contributor.jury.none.fl_str_mv |
Quiroz Salazar, Adolfo José |
| dc.subject.keyword.eng.fl_str_mv |
Numerical semigroups Probabilistic methods |
| topic |
Numerical semigroups Probabilistic methods Matemáticas |
| dc.subject.themes.spa.fl_str_mv |
Matemáticas |
| description |
We investigate properties of random numerical semigroups using a probabilistic model based on the Erdös-Rényi model for random graphs and propose a new probabilistic model. We provide a new and more elementary proof of a lower bound of the expected embedding dimension, genus, and Frobenius number of a random semigroup, and provide a tighter probabilistic upper bound. Our results derive from the application of the Probabilistic Method to the generation of random numerical semigroups and observations about sums of uniformly random subsets of cyclic groups. We include experiments that motivated our results. |
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2023 |
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2023-12-06 |
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2024-01-31T14:30:19Z |
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2024-01-31T14:30:19Z |
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Trabajo de grado - Pregrado |
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eng |
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eng |
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A. Assi, M. D’Anna, and P. A. Garcı́a-Sánchez, Numerical semigroups and applications. Springer Nature, 2020, vol. 3. J. De Loera, C. O’Neill, and D. Wilburne, “Random numerical semigroups and a simplicial complex of irreducible semigroups,” The Electronic Journal of Combinatorics, P4–37, 2018. C. O’Neill, Numsgps-sage, https://github.com/coneill-math/numsgps-sage, 2013. M. Delgado, P. Garcıa-Sánchez, and J. Morais, “Numericalsgps,” A GAP package for numerical semigroups. Available via http://www. gap-system. org, 2015. M. Delgado, “Intpic,” a GAP package for drawing integers, Available via http://www. fc.up.pt/cmup/mdelgado/software, 2013. S. Morales, Randnumsgps, https://github.com/smoralesduarte/randnumsgps, 2023. N. Alon and J. H. Spencer, The Probabilistic Method. John Wiley & Sons, 2016. J. Park and H. Pham, “A proof of the Kahn–Kalai conjecture,” Journal of the American Mathematical Society, 2023. J. C. Rosales, P. A. Garcı́a-Sánchez, et al., Numerical semigroups. Springer, 2009. J. Grime. “How to order 43 mcnuggets - numberphile,” Youtube. (2012), [Online]. Avail- able: https://www.youtube.com/watch?v=vNTSugyS038&ab_channel=Numberphile. J. L. Ramı́rez-Alfonsı́n, “Complexity of the Frobenius problem,” Combinatorica, vol. 16, pp. 143–147, 1996. I. Aliev, M. Henk, and A. Hinrichs, “Expected Frobenius numbers,” Journal of Combi- natorial Theory, Series A, vol. 118, no. 2, pp. 525–531, 2011. R. Apéry, “Sur les branches superlinéaires des courbes algébriques,” CR Acad. Sci. Paris, vol. 222, no. 1198, p. 2000, 1946. E. S. Selmer, “On the linear Diophantine problem of Frobenius,” 1977. H. S. Wilf, “A circle-of-lights algorithm for the “money-changing problem”,” The American Mathematical Monthly, vol. 85, no. 7, pp. 562–565, 1978. M. Delgado, “Conjecture of Wilf: A survey,” Numerical Semigroups: IMNS 2018, pp. 39–62, 2020. V. I. Arnold, “Weak asymptotics for the numbers of solutions of Diophantine problems,” Functional Analysis and Its Applications, vol. 33, no. 4, pp. 292–293, 1999. P. Erdös and R. Graham, “On a linear Diophantine problem of Frobenius,” Acta Arithmetica, vol. 1, no. 21, pp. 399–408, 1972. I. M. Aliev and P. M. Gruber, “An optimal lower bound for the Frobenius problem,” Journal of Number Theory, vol. 123, no. 1, pp. 71–79, 2007. V. I. Arnold, Arnold’s problems. Springer, 2004. R. P. Stanley, Combinatorics and commutative algebra. Springer Science & Business Media, 2007, vol. 41. G. H. Hardy and E. M. Wright, An introduction to the theory of numbers. Oxford university press, 1979. W. Feller, An introduction to probability theory and its applications. John Wiley & Sons, 1971, vol. 1. |
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Bogart, TristramMorales Duarte, SantiagoQuiroz Salazar, Adolfo José2024-01-31T14:30:19Z2024-01-31T14:30:19Z2023-12-06https://hdl.handle.net/1992/73668instname:Universidad de los Andesreponame:Repositorio Institucional Sénecarepourl:https://repositorio.uniandes.edu.co/We investigate properties of random numerical semigroups using a probabilistic model based on the Erdös-Rényi model for random graphs and propose a new probabilistic model. We provide a new and more elementary proof of a lower bound of the expected embedding dimension, genus, and Frobenius number of a random semigroup, and provide a tighter probabilistic upper bound. Our results derive from the application of the Probabilistic Method to the generation of random numerical semigroups and observations about sums of uniformly random subsets of cyclic groups. We include experiments that motivated our results.MatemáticoPregrado47 páginasapplication/pdfengUniversidad de los AndesMatemáticasFacultad de CienciasDepartamento de Matemáticashttps://repositorio.uniandes.edu.co/static/pdf/aceptacion_uso_es.pdfinfo:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Random numerical semigroups and sums of subsets of cyclic groupsTrabajo de grado - Pregradoinfo:eu-repo/semantics/bachelorThesisinfo:eu-repo/semantics/acceptedVersionhttp://purl.org/coar/resource_type/c_7a1fTexthttp://purl.org/redcol/resource_type/TPNumerical semigroupsProbabilistic methodsMatemáticasA. Assi, M. D’Anna, and P. A. Garcı́a-Sánchez, Numerical semigroups and applications. Springer Nature, 2020, vol. 3.J. De Loera, C. O’Neill, and D. Wilburne, “Random numerical semigroups and a simplicial complex of irreducible semigroups,” The Electronic Journal of Combinatorics, P4–37, 2018.C. O’Neill, Numsgps-sage, https://github.com/coneill-math/numsgps-sage, 2013.M. Delgado, P. Garcıa-Sánchez, and J. Morais, “Numericalsgps,” A GAP package for numerical semigroups. Available via http://www. gap-system. org, 2015.M. Delgado, “Intpic,” a GAP package for drawing integers, Available via http://www. fc.up.pt/cmup/mdelgado/software, 2013.S. Morales, Randnumsgps, https://github.com/smoralesduarte/randnumsgps, 2023.N. Alon and J. H. Spencer, The Probabilistic Method. John Wiley & Sons, 2016.J. Park and H. Pham, “A proof of the Kahn–Kalai conjecture,” Journal of the American Mathematical Society, 2023.J. C. Rosales, P. A. Garcı́a-Sánchez, et al., Numerical semigroups. Springer, 2009.J. Grime. “How to order 43 mcnuggets - numberphile,” Youtube. (2012), [Online]. Avail- able: https://www.youtube.com/watch?v=vNTSugyS038&ab_channel=Numberphile.J. L. Ramı́rez-Alfonsı́n, “Complexity of the Frobenius problem,” Combinatorica, vol. 16, pp. 143–147, 1996.I. Aliev, M. Henk, and A. Hinrichs, “Expected Frobenius numbers,” Journal of Combi- natorial Theory, Series A, vol. 118, no. 2, pp. 525–531, 2011.R. Apéry, “Sur les branches superlinéaires des courbes algébriques,” CR Acad. Sci. Paris, vol. 222, no. 1198, p. 2000, 1946.E. S. Selmer, “On the linear Diophantine problem of Frobenius,” 1977.H. S. Wilf, “A circle-of-lights algorithm for the “money-changing problem”,” The American Mathematical Monthly, vol. 85, no. 7, pp. 562–565, 1978.M. Delgado, “Conjecture of Wilf: A survey,” Numerical Semigroups: IMNS 2018, pp. 39–62, 2020.V. I. Arnold, “Weak asymptotics for the numbers of solutions of Diophantine problems,” Functional Analysis and Its Applications, vol. 33, no. 4, pp. 292–293, 1999.P. Erdös and R. Graham, “On a linear Diophantine problem of Frobenius,” Acta Arithmetica, vol. 1, no. 21, pp. 399–408, 1972.I. M. Aliev and P. M. Gruber, “An optimal lower bound for the Frobenius problem,” Journal of Number Theory, vol. 123, no. 1, pp. 71–79, 2007.V. I. Arnold, Arnold’s problems. Springer, 2004.R. P. Stanley, Combinatorics and commutative algebra. Springer Science & Business Media, 2007, vol. 41.G. H. Hardy and E. M. Wright, An introduction to the theory of numbers. Oxford university press, 1979.W. Feller, An introduction to probability theory and its applications. 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