{"title":"Mathematical Formulation of Arithmetic Surface (3, 5) Over Q","authors":"","doi":"10.52783/jas.v9i1.1442","DOIUrl":"https://doi.org/10.52783/jas.v9i1.1442","url":null,"abstract":"","PeriodicalId":41066,"journal":{"name":"Journal of Algebraic Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70996155","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
It is often the case that a progenitor, P = m* n : N, factored by a subgroup generated by one or more relators, M = {iriwi, , 7TfcWfc), gives a finite group F, particularly, a classical group, simple group, or a sporadic group. In such instances, the presentation of the factor group, G = P/M = {x, y, t), is also a symmetric presentation of the finite group F. Symmetric presentations of groups allow us to represent, and ma nipulate, group elements in a manner that is typically more convenient than conventional techniques; in this sense, symmetric presentations are particularly useful in the study of large finite groups. In this thesis, we first construct, by manual double coset enumeration, the groups A5, S5, Sq, S7, and S7 x 3 as finite homomorphic images of the progenitors 2* 3 : S3, 2* 4 : A4, 2* 5 : A5, 3* 5 : S5, and 3* 5 : S5, respectively. We also demonstrate that their respective symmetric presentations enable us to represent, and manipulate, their group elements in a convenient (symmetric) fashion as well as to obtain, in most cases, useful permutation representations for their group elements. We devote the majority of our efforts to the construction, and manipulation, of M12: 2, or Aut(Mi2), the outer automorphism group of the Mathieu group M12. In particular, we construct, by the technique of manual double coset enumeration over S4, the group Aut(Mi2) as a finite homomorphic image of the progenitor 3* 4 : S4. By way of this construction, we show that Aut(Afi2) is isomorphic to 3* 4 : S4 factored by two relations and we conclude that the symmetric presentation (x, y,t | x4 = y2 = (yx)3 = t3 = [t,y] = [i33,?/] = (yxt)10 = ((x2y)2t)5 = e) defines the group Aut(Mi2). Finally, we demonstrate that this symmetric presentation enables us to express and manipulate every element of Aut(Mi2) either as a symmetric representation of the form 7rw, where tt is a permutation of S4 on 4 letters and w is a word of concatenated generators of length at most eight, or as a permutation representation on 7920 letters.
{"title":"Symmetric Presentations of Finite Groups","authors":"J. A. Roche","doi":"10.52783/jas.v9i1.1444","DOIUrl":"https://doi.org/10.52783/jas.v9i1.1444","url":null,"abstract":"It is often the case that a progenitor, P = m* n : N, factored by a subgroup generated by one or more relators, M = {iriwi, , 7TfcWfc), gives a finite group F, particularly, a classical group, simple group, or a sporadic group. In such instances, the presentation of the factor group, G = P/M = {x, y, t), is also a symmetric presentation of the finite group F. Symmetric presentations of groups allow us to represent, and ma nipulate, group elements in a manner that is typically more convenient than conventional techniques; in this sense, symmetric presentations are particularly useful in the study of large finite groups. In this thesis, we first construct, by manual double coset enumeration, the groups A5, S5, Sq, S7, and S7 x 3 as finite homomorphic images of the progenitors 2* 3 : S3, 2* 4 : A4, 2* 5 : A5, 3* 5 : S5, and 3* 5 : S5, respectively. We also demonstrate that their respective symmetric presentations enable us to represent, and manipulate, their group elements in a convenient (symmetric) fashion as well as to obtain, in most cases, useful permutation representations for their group elements. We devote the majority of our efforts to the construction, and manipulation, of M12: 2, or Aut(Mi2), the outer automorphism group of the Mathieu group M12. In particular, we construct, by the technique of manual double coset enumeration over S4, the group Aut(Mi2) as a finite homomorphic image of the progenitor 3* 4 : S4. By way of this construction, we show that Aut(Afi2) is isomorphic to 3* 4 : S4 factored by two relations and we conclude that the symmetric presentation (x, y,t | x4 = y2 = (yx)3 = t3 = [t,y] = [i33,?/] = (yxt)10 = ((x2y)2t)5 = e) defines the group Aut(Mi2). Finally, we demonstrate that this symmetric presentation enables us to express and manipulate every element of Aut(Mi2) either as a symmetric representation of the form 7rw, where tt is a permutation of S4 on 4 letters and w is a word of concatenated generators of length at most eight, or as a permutation representation on 7920 letters.","PeriodicalId":41066,"journal":{"name":"Journal of Algebraic Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70996423","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Guassian Lattice Reduction Algorithm in Two-Dimensions","authors":"","doi":"10.52783/jas.v9i1.1445","DOIUrl":"https://doi.org/10.52783/jas.v9i1.1445","url":null,"abstract":"","PeriodicalId":41066,"journal":{"name":"Journal of Algebraic Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70996491","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Entrepreneurial Skills Requirement in a Leading Economy Like India: An Empirical Study","authors":"","doi":"10.52783/jas.v9i1.1449","DOIUrl":"https://doi.org/10.52783/jas.v9i1.1449","url":null,"abstract":"","PeriodicalId":41066,"journal":{"name":"Journal of Algebraic Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70996583","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We exhibit the analog of the entropy map for multivariate Gaussian distributions on local fields. As in the real case, the image of this map lies in the supermodular cone and it determines the distribution of the valuation vector. In general, this map can be defined for non-archimedian valued fields whose valuation group is an additive subgroup of the real line, and it remains supermodular. We also explicitly compute the image of this map in dimension 3.
{"title":"The Gaussian entropy map in valued fields","authors":"Yassine El Maazouz","doi":"10.2140/astat.2022.13.1","DOIUrl":"https://doi.org/10.2140/astat.2022.13.1","url":null,"abstract":"We exhibit the analog of the entropy map for multivariate Gaussian distributions on local fields. As in the real case, the image of this map lies in the supermodular cone and it determines the distribution of the valuation vector. In general, this map can be defined for non-archimedian valued fields whose valuation group is an additive subgroup of the real line, and it remains supermodular. We also explicitly compute the image of this map in dimension 3.","PeriodicalId":41066,"journal":{"name":"Journal of Algebraic Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78882195","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-02-05DOI: 10.2140/astat.2021.12.213
Benjamin Hollering, S. Sullivant
Discrete max-linear Bayesian networks are directed graphical models specified by the same recursive structural equations as max-linear models but with discrete innovations. When all of the random variables in the model are binary, these models are isomorphic to the conjunctive Bayesian network (CBN) models of Beerenwinkel, Eriksson, and Sturmfels. Many of the techniques used to study CBN models can be extended to discrete max-linear models and similar results can be obtained. In particular, we extend the fact that CBN models are toric varieties after linear change of coordinates to all discrete max-linear models.
{"title":"Discrete max-linear Bayesian networks","authors":"Benjamin Hollering, S. Sullivant","doi":"10.2140/astat.2021.12.213","DOIUrl":"https://doi.org/10.2140/astat.2021.12.213","url":null,"abstract":"Discrete max-linear Bayesian networks are directed graphical models specified by the same recursive structural equations as max-linear models but with discrete innovations. When all of the random variables in the model are binary, these models are isomorphic to the conjunctive Bayesian network (CBN) models of Beerenwinkel, Eriksson, and Sturmfels. Many of the techniques used to study CBN models can be extended to discrete max-linear models and similar results can be obtained. In particular, we extend the fact that CBN models are toric varieties after linear change of coordinates to all discrete max-linear models.","PeriodicalId":41066,"journal":{"name":"Journal of Algebraic Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79424954","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-28DOI: 10.2140/astat.2020.11.213
L. Burigana, Michele Vicovaro
{"title":"Compatibility of distributions in probabilistic models: an algebraic frame and some characterizations","authors":"L. Burigana, Michele Vicovaro","doi":"10.2140/astat.2020.11.213","DOIUrl":"https://doi.org/10.2140/astat.2020.11.213","url":null,"abstract":"","PeriodicalId":41066,"journal":{"name":"Journal of Algebraic Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90695466","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-14DOI: 10.2140/astat.2021.12.187
Carlos Am'endola, Kathlén Kohn, Philipp Reichenbach, A. Seigal
We establish connections between invariant theory and maximum likelihood estimation for discrete statistical models. We show that norm minimization over a torus orbit is equivalent to maximum likelihood estimation in log-linear models. We use notions of stability under a torus action to characterize the existence of the maximum likelihood estimate, and discuss connections to scaling algorithms.
{"title":"Toric invariant theory for maximum likelihood estimation in log-linear models","authors":"Carlos Am'endola, Kathlén Kohn, Philipp Reichenbach, A. Seigal","doi":"10.2140/astat.2021.12.187","DOIUrl":"https://doi.org/10.2140/astat.2021.12.187","url":null,"abstract":"We establish connections between invariant theory and maximum likelihood estimation for discrete statistical models. We show that norm minimization over a torus orbit is equivalent to maximum likelihood estimation in log-linear models. We use notions of stability under a torus action to characterize the existence of the maximum likelihood estimate, and discuss connections to scaling algorithms.","PeriodicalId":41066,"journal":{"name":"Journal of Algebraic Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73006647","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-09DOI: 10.2140/astat.2021.12.167
B. Sturmfels, Simon Telen
We relate scattering amplitudes in particle physics to maximum likelihood estimation for discrete models in algebraic statistics. The scattering potential plays the role of the log-likelihood function, and its critical points are solutions to rational function equations. We study the ML degree of low-rank tensor models in statistics, and we revisit physical theories proposed by Arkani-Hamed, Cachazo and their collaborators. Recent advances in numerical algebraic geometry are employed to compute and certify critical points. We also discuss positive models and how to compute their string amplitudes.
{"title":"Likelihood equations and scattering amplitudes","authors":"B. Sturmfels, Simon Telen","doi":"10.2140/astat.2021.12.167","DOIUrl":"https://doi.org/10.2140/astat.2021.12.167","url":null,"abstract":"We relate scattering amplitudes in particle physics to maximum likelihood estimation for discrete models in algebraic statistics. The scattering potential plays the role of the log-likelihood function, and its critical points are solutions to rational function equations. We study the ML degree of low-rank tensor models in statistics, and we revisit physical theories proposed by Arkani-Hamed, Cachazo and their collaborators. Recent advances in numerical algebraic geometry are employed to compute and certify critical points. We also discuss positive models and how to compute their string amplitudes.","PeriodicalId":41066,"journal":{"name":"Journal of Algebraic Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85448473","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Editorial: A new beginning","authors":"Thomas W. Kahle, Sonja Petrović","doi":"10.2140/ASTAT.2020.11.1","DOIUrl":"https://doi.org/10.2140/ASTAT.2020.11.1","url":null,"abstract":"","PeriodicalId":41066,"journal":{"name":"Journal of Algebraic Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80376649","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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