Pub Date : 2022-04-12DOI: 10.2140/tunis.2023.5.171
Frédéric Charve
In this article we study the lifespan and asymptotics (in the large rotation and stratification regime) for the Primitive system for highly ill-prepared initial data in critical spaces. Compared to our previous works, we simplified the proof and made it adaptable to the Rotating fluids system with highly ill-prepared initial data decomposed as a sum of 2D horizontal part and a very large 3D part. We also provide explicit convergence rates.
{"title":"Asymptotics for the rotating fluids and primitive systems with large ill-prepared initial data in critical spaces","authors":"Frédéric Charve","doi":"10.2140/tunis.2023.5.171","DOIUrl":"https://doi.org/10.2140/tunis.2023.5.171","url":null,"abstract":"In this article we study the lifespan and asymptotics (in the large rotation and stratification regime) for the Primitive system for highly ill-prepared initial data in critical spaces. Compared to our previous works, we simplified the proof and made it adaptable to the Rotating fluids system with highly ill-prepared initial data decomposed as a sum of 2D horizontal part and a very large 3D part. We also provide explicit convergence rates.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42178182","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 : 2022-04-05DOI: 10.2140/tunis.2023.5.327
M. Feller
We show that the 2-Segal spaces (also called decomposition spaces) of Dyckerhoff-Kapranov and G'alvez-Kock-Tonks have a natural analogue within simplicial sets, which we call quasi-2-Segal sets, and that the two ideas enjoy a similar relationship as the one Segal spaces have with quasi-categories. In particular, we construct a model structure on the category of simplicial sets whose fibrant objects are the quasi-2-Segal sets which is Quillen equivalent to a model structure for complete 2-Segal spaces (where our notion of completeness comes from one of the equivalent characterizations of completeness for Segal spaces). We also prove a path space criterion, which says that a simplicial set is a quasi-2-Segal set if and only if its path spaces (also called d'ecalage) are quasi-categories, as well as an edgewise subdivision criterion.
{"title":"Quasi-2-Segal sets","authors":"M. Feller","doi":"10.2140/tunis.2023.5.327","DOIUrl":"https://doi.org/10.2140/tunis.2023.5.327","url":null,"abstract":"We show that the 2-Segal spaces (also called decomposition spaces) of Dyckerhoff-Kapranov and G'alvez-Kock-Tonks have a natural analogue within simplicial sets, which we call quasi-2-Segal sets, and that the two ideas enjoy a similar relationship as the one Segal spaces have with quasi-categories. In particular, we construct a model structure on the category of simplicial sets whose fibrant objects are the quasi-2-Segal sets which is Quillen equivalent to a model structure for complete 2-Segal spaces (where our notion of completeness comes from one of the equivalent characterizations of completeness for Segal spaces). We also prove a path space criterion, which says that a simplicial set is a quasi-2-Segal set if and only if its path spaces (also called d'ecalage) are quasi-categories, as well as an edgewise subdivision criterion.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":"58 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68571915","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 : 2022-03-30DOI: 10.2140/tunis.2022.4.183
L. Caporaso, J. Harris, B. Mazur
In [2] it is asserted that, assuming the truth of the Strong Lang Conjecture (Conjecture 1 below), a very strong form of boundedness holds: for every g ≥ 2 there is a finite bound N(g)—not depending on K!—such that for any number field K there are only finitely many isomorphism classes of curves of genus g defined over K with more than N(g) K-rational points. The issue is, in that statement do we mean finitely many isomorphism classes over K, or over the algebraic closure K? The paper asserts the statement in the stronger form—up to isomorphism over K—but the proof establishes only the weaker statement that there are finitely many curves with more than N(g) points up to isomorphism over K.
{"title":"Uniformity of rational points: an up-date and corrections","authors":"L. Caporaso, J. Harris, B. Mazur","doi":"10.2140/tunis.2022.4.183","DOIUrl":"https://doi.org/10.2140/tunis.2022.4.183","url":null,"abstract":"In [2] it is asserted that, assuming the truth of the Strong Lang Conjecture (Conjecture 1 below), a very strong form of boundedness holds: for every g ≥ 2 there is a finite bound N(g)—not depending on K!—such that for any number field K there are only finitely many isomorphism classes of curves of genus g defined over K with more than N(g) K-rational points. The issue is, in that statement do we mean finitely many isomorphism classes over K, or over the algebraic closure K? The paper asserts the statement in the stronger form—up to isomorphism over K—but the proof establishes only the weaker statement that there are finitely many curves with more than N(g) points up to isomorphism over K.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68572262","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}