{"title":"Mouse, and if","authors":"W. Briggs","doi":"10.1007/978-1-4842-5187-4_4","DOIUrl":"https://doi.org/10.1007/978-1-4842-5187-4_4","url":null,"abstract":"","PeriodicalId":382565,"journal":{"name":"C++20 for Lazy Programmers","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125924071","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":"Loops, Input, and char","authors":"Will Briggs","doi":"10.1007/978-1-4842-5187-4_5","DOIUrl":"https://doi.org/10.1007/978-1-4842-5187-4_5","url":null,"abstract":"","PeriodicalId":382565,"journal":{"name":"C++20 for Lazy Programmers","volume":"794 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126184051","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":"Functions","authors":"Will Briggs","doi":"10.1142/9789813274587_0002","DOIUrl":"https://doi.org/10.1142/9789813274587_0002","url":null,"abstract":". We prove that every continuous function on a separable infinite-dimensional Hilbert space X can be uniformly approximated by C ∞ smooth functions with no critical points . This kind of result can be regarded as a sort of very strong approximate version of the Morse-Sard theorem. Some consequences of the main theorem are as follows. Every two disjoint closed subsets of X can be separated by a one-codimensional smooth manifold which is a level set of a smooth function with no critical points; this fact may be viewed as a nonlinear analogue of the geometrical version of the Hahn-Banach theorem. In particular, every closed set in X can be uniformly approximated by open sets whose boundaries are C ∞ smooth one-codimensional submanifolds of X . Finally, since every Hilbert manifold is diffeomorphic to an open subset of the Hilbert space, all of these results still hold if one replaces the Hilbert space X with any smooth manifold M modelled on X .","PeriodicalId":382565,"journal":{"name":"C++20 for Lazy Programmers","volume":"136 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116382310","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":"C++20 for Lazy Programmers: Quick, Easy, and Fun C++ for Beginners","authors":"Will Briggs","doi":"10.1007/978-1-4842-6306-8","DOIUrl":"https://doi.org/10.1007/978-1-4842-6306-8","url":null,"abstract":"","PeriodicalId":382565,"journal":{"name":"C++20 for Lazy Programmers","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115327908","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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