Pub Date : 2023-11-23DOI: 10.4153/s0008439523000905
Boaz Tsaban
I provide simplified proofs for each of the following fundamental theorems regarding selection principles:
(1) 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.
(2) The Scheepers Diagram Last Theorem, due to Peng, completing all provable implications in the diagram.
(3) The Menger Game Theorem, due to Telgársky, determining when Bob has a winning strategy in the game version of Menger’s covering property.
(4) A lower bound on the additivity of Rothberger’s covering property, due to Carlson.
The simplified proofs lead to several new results.
我为以下有关选择原则的基本定理逐一提供了简化证明：(1)准正收敛定理，作者和兹德姆斯基提出，断言空间上连续函数空间的某一重要性质实际上被该空间的伯勒尔图像所保留。 (2)谢珀斯图最后定理，彭提出，完成了图中所有可证明的含义。(3) Telgársky 提出的门格尔博弈定理（Menger Game Theorem），确定了在门格尔覆盖性质的博弈版本中，鲍勃何时有获胜策略。 (4) 卡尔森提出的罗斯伯格覆盖性质可加性的下限。
{"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}
Pub Date : 2023-11-15DOI: 10.4153/s0008439523000887
Michael R. Pilla
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 $mathbb {C}^N$ 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.
{"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}
Pub Date : 2020-09-11DOI: 10.4153/S0008439520000776
Daniel López Garcia
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 $p,$ we can compute the orbit modulo p. We conjecture that the orbit in homology with coefficients in $mathbb {Z}$ can be determined by these orbits with coefficients in $mathbb {Z}_p$ .
{"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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