. In categorical data analysis, the 2 × 2 contingency tables are commonly used to assess the association between groups and responses, this is achieved by using some measures of association, such as the contingency coe ffi cient, odds ratio, risk relative, etc. In a Bayesian approach, the risk ratio is modeled according to a Beta-Binomial model, which has exact posterior distribution, due to the conjugacy property of the model. In this work, we provide the exact posterior distribution of the relative risk for the non-conjugate Kumaraswamy–Binomial model. The results are based on special functions and we give exact expressions for the posterior density, moments, and cumulative distribution. An example illustrates the theory.
{"title":"Exact Posterior distribution of risk ratio in the Kumaraswamy–Binomial model","authors":"J. A. A. Andrade, P. N. Rathie","doi":"10.5802/crmath.469","DOIUrl":"https://doi.org/10.5802/crmath.469","url":null,"abstract":". In categorical data analysis, the 2 × 2 contingency tables are commonly used to assess the association between groups and responses, this is achieved by using some measures of association, such as the contingency coe ffi cient, odds ratio, risk relative, etc. In a Bayesian approach, the risk ratio is modeled according to a Beta-Binomial model, which has exact posterior distribution, due to the conjugacy property of the model. In this work, we provide the exact posterior distribution of the relative risk for the non-conjugate Kumaraswamy–Binomial model. The results are based on special functions and we give exact expressions for the posterior density, moments, and cumulative distribution. An example illustrates the theory.","PeriodicalId":395483,"journal":{"name":"Comptes Rendus. Mathématique","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129129885","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}
In this paper, we introduce and study a set relative to singularities of plurisubharmonic functions. We prove that this set is countable under the condition h > 0 on B {0}. Mathematical subject classification (2010). 32U05, 32U15, 32U40, 32W20. Manuscript received 28 October 2022, revised 11 December 2022, accepted 18 December 2022.
{"title":"A note on the weighted log canonical threshold","authors":"Nguyen Van Phu, Nguyen Van Phua","doi":"10.5802/crmath.456","DOIUrl":"https://doi.org/10.5802/crmath.456","url":null,"abstract":"In this paper, we introduce and study a set relative to singularities of plurisubharmonic functions. We prove that this set is countable under the condition h > 0 on B {0}. Mathematical subject classification (2010). 32U05, 32U15, 32U40, 32W20. Manuscript received 28 October 2022, revised 11 December 2022, accepted 18 December 2022.","PeriodicalId":395483,"journal":{"name":"Comptes Rendus. Mathématique","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133663021","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":"Rational points on a certain genus 2 curve","authors":"Xuan Tho Nguyen","doi":"10.5802/crmath.471","DOIUrl":"https://doi.org/10.5802/crmath.471","url":null,"abstract":"","PeriodicalId":395483,"journal":{"name":"Comptes Rendus. Mathématique","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133116183","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}
. Itisexpectedthatthereexistlinebundlesonaquasi-a ffi nenon-singularsurfacewhichdonotadmit a flat connection. However, to the best of our knowledge there is no known example of such a line bundle. In this article we give several explicit examples of line bundles on certain non-singular, quasi-a ffi ne surfaces that cannot be equipped with a flat connection. Résumé. On s’attend à ce qu’il existe des fibrés en droites sur une surface non-singulière quasi-a ffi ne qui n’admettent pas de connexion plate mais, à notre connaissance, aucun exemple d’un tel fibré n’est connu. Dans cet article, nous en donnons plusieurs exemples explicites.
{"title":"Examples of non-flat bundles of rank one","authors":"Ananyo Dan, Agust'in Romano-Vel'azquez","doi":"10.5802/crmath.459","DOIUrl":"https://doi.org/10.5802/crmath.459","url":null,"abstract":". Itisexpectedthatthereexistlinebundlesonaquasi-a ffi nenon-singularsurfacewhichdonotadmit a flat connection. However, to the best of our knowledge there is no known example of such a line bundle. In this article we give several explicit examples of line bundles on certain non-singular, quasi-a ffi ne surfaces that cannot be equipped with a flat connection. Résumé. On s’attend à ce qu’il existe des fibrés en droites sur une surface non-singulière quasi-a ffi ne qui n’admettent pas de connexion plate mais, à notre connaissance, aucun exemple d’un tel fibré n’est connu. Dans cet article, nous en donnons plusieurs exemples explicites.","PeriodicalId":395483,"journal":{"name":"Comptes Rendus. Mathématique","volume":"144 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122541227","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}
. A well-known result of Fulton–Lazarsfeld ensures the connectedness of degeneracy loci under an ampleness condition. We extend it to positive characteristic, along with the variants for degeneracy loci of symmetric and alternating maps of even rank, due to Tu in characteristic zero. The proof uses the explicit determination of the top étale cohomology group of an algebraic variety, a result communicated by Esnault.
{"title":"The connectedness of degeneracy loci in positive characteristic","authors":"Rémi Lodh","doi":"10.5802/crmath.448","DOIUrl":"https://doi.org/10.5802/crmath.448","url":null,"abstract":". A well-known result of Fulton–Lazarsfeld ensures the connectedness of degeneracy loci under an ampleness condition. We extend it to positive characteristic, along with the variants for degeneracy loci of symmetric and alternating maps of even rank, due to Tu in characteristic zero. The proof uses the explicit determination of the top étale cohomology group of an algebraic variety, a result communicated by Esnault.","PeriodicalId":395483,"journal":{"name":"Comptes Rendus. Mathématique","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121162510","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 start from the contractive functional equation proposed in [4], where it was shown that the polynomial solution of functional equation can be used to initialize a Neural Network structure, with a controlled accuracy. We propose a novel algorithm, where the functional equation is solved with a converging iterative algorithm which can be realized as a Machine Learning training method iteratively with respect to the number of layers. The proof of convergence is performed with respect to the L ∞ norm. Numerical tests illustrate the theory and show that stochastic gradient descent methods can be used with good accuracy for this problem.
{"title":"A convergent Deep Learning algorithm for approximation of polynomials","authors":"B. Després","doi":"10.5802/crmath.462","DOIUrl":"https://doi.org/10.5802/crmath.462","url":null,"abstract":". We start from the contractive functional equation proposed in [4], where it was shown that the polynomial solution of functional equation can be used to initialize a Neural Network structure, with a controlled accuracy. We propose a novel algorithm, where the functional equation is solved with a converging iterative algorithm which can be realized as a Machine Learning training method iteratively with respect to the number of layers. The proof of convergence is performed with respect to the L ∞ norm. Numerical tests illustrate the theory and show that stochastic gradient descent methods can be used with good accuracy for this problem.","PeriodicalId":395483,"journal":{"name":"Comptes Rendus. Mathématique","volume":"69 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133282219","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}
. In this note, we check that a complex projective algebraic variety has (at most) countably many real forms. We more generally prove it when the field of reals is replaced with a field that has only countably many finite extensions up to isomorphism. The verification consists in gathering known results about automorphism groups and Galois cohomology. This contrasts with the recent discovery by A. Bot of an a ffi ne real variety with uncountably many real forms.
{"title":"Projective varieties have countably many real forms","authors":"Timothée L. Labinet","doi":"10.5802/crmath.441","DOIUrl":"https://doi.org/10.5802/crmath.441","url":null,"abstract":". In this note, we check that a complex projective algebraic variety has (at most) countably many real forms. We more generally prove it when the field of reals is replaced with a field that has only countably many finite extensions up to isomorphism. The verification consists in gathering known results about automorphism groups and Galois cohomology. This contrasts with the recent discovery by A. Bot of an a ffi ne real variety with uncountably many real forms.","PeriodicalId":395483,"journal":{"name":"Comptes Rendus. Mathématique","volume":"188 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139358384","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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