{"title":"The GAGA principle","authors":"","doi":"10.1090/mbk/121/44","DOIUrl":"https://doi.org/10.1090/mbk/121/44","url":null,"abstract":"","PeriodicalId":423691,"journal":{"name":"100 Years of Math Milestones","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116965995","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}
An abelian group is finitely generated if there exist finitely many elements a1, a2, . . . , ak such that any element of G can be expressed as a sum c1a1 + c2a2 + . . .+ ckak, where the ci are integers and multiplication denotes repeated addition. Note that this representation need not be unique, so any finite group is also finitely generated. A subgroup of an abelian group G is a set H ⊆ G which is itself a group under the same operation. For any a ∈ G, a+H = {a+ h : h ∈ H} is a coset of H. a is called a representative of the coset a + H. If b ∈ a + H, then b − a ∈ H. Any two cosets of H are either equal or disjoint. The index of H in G, denoted [G : H], is the number of disjoint cosets of H. For a ∈ G, the order of a is the minimum positive integer k such that ka is the identity, or ∞ if there is no such k.
{"title":"Mordell’s theorem","authors":"Jonah Ostroff","doi":"10.1090/mbk/121/09","DOIUrl":"https://doi.org/10.1090/mbk/121/09","url":null,"abstract":"An abelian group is finitely generated if there exist finitely many elements a1, a2, . . . , ak such that any element of G can be expressed as a sum c1a1 + c2a2 + . . .+ ckak, where the ci are integers and multiplication denotes repeated addition. Note that this representation need not be unique, so any finite group is also finitely generated. A subgroup of an abelian group G is a set H ⊆ G which is itself a group under the same operation. For any a ∈ G, a+H = {a+ h : h ∈ H} is a coset of H. a is called a representative of the coset a + H. If b ∈ a + H, then b − a ∈ H. Any two cosets of H are either equal or disjoint. The index of H in G, denoted [G : H], is the number of disjoint cosets of H. For a ∈ G, the order of a is the minimum positive integer k such that ka is the identity, or ∞ if there is no such k.","PeriodicalId":423691,"journal":{"name":"100 Years of Math Milestones","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127068533","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":"Ackermann’s function","authors":"","doi":"10.1090/mbk/121/14","DOIUrl":"https://doi.org/10.1090/mbk/121/14","url":null,"abstract":"","PeriodicalId":423691,"journal":{"name":"100 Years of Math Milestones","volume":"60 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114191598","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":"The Gale–Shapely algorithm and the stable\u0000 marriage problem","authors":"","doi":"10.1090/mbk/121/50","DOIUrl":"https://doi.org/10.1090/mbk/121/50","url":null,"abstract":"","PeriodicalId":423691,"journal":{"name":"100 Years of Math Milestones","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116298993","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":"Tennenbaum’s proof of the irrationality of\u0000 √2","authors":"","doi":"10.1090/mbk/121/39","DOIUrl":"https://doi.org/10.1090/mbk/121/39","url":null,"abstract":"","PeriodicalId":423691,"journal":{"name":"100 Years of Math Milestones","volume":"62 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131928292","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":"The Metropolis algorithm","authors":"","doi":"10.1090/mbk/121/41","DOIUrl":"https://doi.org/10.1090/mbk/121/41","url":null,"abstract":"","PeriodicalId":423691,"journal":{"name":"100 Years of Math Milestones","volume":"56 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128259554","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":"William Stein developed Sage","authors":"","doi":"10.1090/mbk/121/93","DOIUrl":"https://doi.org/10.1090/mbk/121/93","url":null,"abstract":"","PeriodicalId":423691,"journal":{"name":"100 Years of Math Milestones","volume":"883 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126389509","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 the following paper, I will give a brief introduction to the theory of Diophantine sets as well as the theory of computability. I will then present the Matiyasevich-Robinson-Davis-Putnam (MRDP) theorem, which is immediately comprehensible given just a cursory understanding of the mathematical basics, and give some details of its proof. Finally, I will present some further work in the area of Diophantine computability and various applications or corollaries of the celebrated MRDP theorem.
{"title":"Hilbert’s tenth problem","authors":"Andrew J. Misner","doi":"10.1090/mbk/121/58","DOIUrl":"https://doi.org/10.1090/mbk/121/58","url":null,"abstract":"In the following paper, I will give a brief introduction to the theory of Diophantine sets as well as the theory of computability. I will then present the Matiyasevich-Robinson-Davis-Putnam (MRDP) theorem, which is immediately comprehensible given just a cursory understanding of the mathematical basics, and give some details of its proof. Finally, I will present some further work in the area of Diophantine computability and various applications or corollaries of the celebrated MRDP theorem.","PeriodicalId":423691,"journal":{"name":"100 Years of Math Milestones","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123854286","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":"Roth’s theorem","authors":"Jacques Verstraëte","doi":"10.1090/mbk/121/43","DOIUrl":"https://doi.org/10.1090/mbk/121/43","url":null,"abstract":"","PeriodicalId":423691,"journal":{"name":"100 Years of Math Milestones","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131834258","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 : 2019-06-12DOI: 10.1007/3-540-29462-7_13
Yu. V. Nesterenko
{"title":"Hilbert’s seventh problem","authors":"Yu. V. Nesterenko","doi":"10.1007/3-540-29462-7_13","DOIUrl":"https://doi.org/10.1007/3-540-29462-7_13","url":null,"abstract":"","PeriodicalId":423691,"journal":{"name":"100 Years of Math Milestones","volume":"67 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132131021","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：