{"title":"Selection principles and proofs from the Book","authors":"Boaz Tsaban","doi":"10.4153/s0008439523000905","DOIUrl":"https://doi.org/10.4153/s0008439523000905","url":null,"abstract":"<p>I provide simplified proofs for each of the following fundamental theorems regarding selection principles: </p><ol><li><p><span>(1)</span> The Quasinormal Convergence Theorem, due to the author and Zdomskyy, asserting that a certain, important property of the space of continuous functions on a space is actually preserved by Borel images of that space.</p></li><li><p><span>(2)</span> The Scheepers Diagram Last Theorem, due to Peng, completing all provable implications in the diagram.</p></li><li><p><span>(3)</span> The Menger Game Theorem, due to Telgársky, determining when Bob has a winning strategy in the game version of Menger’s covering property.</p></li><li><p><span>(4)</span> A lower bound on the additivity of Rothberger’s covering property, due to Carlson.</p></li></ol><p></p><p>The simplified proofs lead to several new results.</p>","PeriodicalId":501184,"journal":{"name":"Canadian Mathematical Bulletin","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138566896","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":"Linear fractional self-maps of the unit ball","authors":"Michael R. Pilla","doi":"10.4153/s0008439523000887","DOIUrl":"https://doi.org/10.4153/s0008439523000887","url":null,"abstract":"<p>Determining the range of complex maps plays a fundamental role in the study of several complex variables and operator theory. In particular, one is often interested in determining when a given holomorphic function is a self-map of the unit ball. In this paper, we discuss a class of maps in <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231128072208832-0343:S0008439523000887:S0008439523000887_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$mathbb {C}^N$</span></span></img></span></span> that generalize linear fractional maps. We then proceed to determine precisely when such a map is a self-map of the unit ball. In particular, we take a novel approach, obtaining numerous new results about this class of maps along the way.</p>","PeriodicalId":501184,"journal":{"name":"Canadian Mathematical Bulletin","volume":"1217 30","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138510713","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":"Homology supported in Lagrangian submanifolds in mirror quintic threefolds","authors":"Daniel López Garcia","doi":"10.4153/S0008439520000776","DOIUrl":"https://doi.org/10.4153/S0008439520000776","url":null,"abstract":"Abstract In this note, we study homology classes in the mirror quintic Calabi–Yau threefold that can be realized by special Lagrangian submanifolds. We have used Picard–Lefschetz theory to establish the monodromy action and to study the orbit of Lagrangian vanishing cycles. For many prime numbers \u0000$p,$\u0000 we can compute the orbit modulo p. We conjecture that the orbit in homology with coefficients in \u0000$mathbb {Z}$\u0000 can be determined by these orbits with coefficients in \u0000$mathbb {Z}_p$\u0000 .","PeriodicalId":501184,"journal":{"name":"Canadian Mathematical Bulletin","volume":"29 49","pages":"709 - 724"},"PeriodicalIF":0.0,"publicationDate":"2020-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141204975","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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