Im Jahr 2002 erschreckte die “Entdeckung”von Manindra Agrawal, Neeraj Kayal und Nitin Saxena die Welt der Theoretischen Informatik: Die Entscheidung, ob eine Zahl eine Primzahl ist oder nicht, ist in polynomieller Zeit zu finden. Bisher war man davon ausgegangen, dass ein Algorithmus die Entscheidung zwar in polynomieller Zeit treffen kann, aber u.U. sehr lange dafür braucht. Randomisierte Algorithmen sind zwar schneller, haben aber eine gewisse Fehlerquote beim Ergebnis. Der deterministische Algorithmus von Agrawal, Kayal und Saxena kann die Lösung in polynomieller Zeit finden, ohne auf bisher unbewiesene mathematische Theoreme zurückgreifen zu müssen.
{"title":"PRIMES in P","authors":"H. Szillat","doi":"10.1090/mbk/121/90","DOIUrl":"https://doi.org/10.1090/mbk/121/90","url":null,"abstract":"Im Jahr 2002 erschreckte die “Entdeckung”von Manindra Agrawal, Neeraj Kayal und Nitin Saxena die Welt der Theoretischen Informatik: Die Entscheidung, ob eine Zahl eine Primzahl ist oder nicht, ist in polynomieller Zeit zu finden. Bisher war man davon ausgegangen, dass ein Algorithmus die Entscheidung zwar in polynomieller Zeit treffen kann, aber u.U. sehr lange dafür braucht. Randomisierte Algorithmen sind zwar schneller, haben aber eine gewisse Fehlerquote beim Ergebnis. Der deterministische Algorithmus von Agrawal, Kayal und Saxena kann die Lösung in polynomieller Zeit finden, ohne auf bisher unbewiesene mathematische Theoreme zurückgreifen zu müssen.","PeriodicalId":423691,"journal":{"name":"100 Years of Math Milestones","volume":"28 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":"122074666","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}
Example 1.3. Let p be a prime number. For any 0 6= a ∈ Q, we can write a = p b c where m, b, c ∈ Z, (bc, p) = 1. Define |a|p = 1 pm and |0|p = 0, then |·|p is a nonarchimedean valuation on Q. Note that for different primes p and q, |·|p and |·|q are not equivalent. For z ∈ C, define |z|∞ = |z| (the usual absolute value). Then |·|∞ is an archimedean valuation on C(thus is not equivalent to |·|p for any p).
{"title":"Ostrowski’s theorem","authors":"G. Gim","doi":"10.1090/mbk/121/04","DOIUrl":"https://doi.org/10.1090/mbk/121/04","url":null,"abstract":"Example 1.3. Let p be a prime number. For any 0 6= a ∈ Q, we can write a = p b c where m, b, c ∈ Z, (bc, p) = 1. Define |a|p = 1 pm and |0|p = 0, then |·|p is a nonarchimedean valuation on Q. Note that for different primes p and q, |·|p and |·|q are not equivalent. For z ∈ C, define |z|∞ = |z| (the usual absolute value). Then |·|∞ is an archimedean valuation on C(thus is not equivalent to |·|p for any p).","PeriodicalId":423691,"journal":{"name":"100 Years of Math Milestones","volume":"81 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":"114283132","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":"A mathematician’s apology","authors":"F. FopSegre'","doi":"10.1090/mbk/121/28","DOIUrl":"https://doi.org/10.1090/mbk/121/28","url":null,"abstract":"","PeriodicalId":423691,"journal":{"name":"100 Years of Math Milestones","volume":"15 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":"123871334","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}
Enigma was an electro-mechanical machine that was used before and during the World War II by Germany to encrypt and decrypt secret messages. Invented by Arthur Scherbius in 1918 and meant initially as a commercial product for the enterprise community, Enigma turned out to be more successful with the German military forces [2]. Enigma had evolved throughout several years, gaining better cryptographic strength, but also being broken time and again.
{"title":"Breaking Enigma","authors":"D. Gabbasov","doi":"10.1090/mbk/121/31","DOIUrl":"https://doi.org/10.1090/mbk/121/31","url":null,"abstract":"Enigma was an electro-mechanical machine that was used before and during the World War II by Germany to encrypt and decrypt secret messages. Invented by Arthur Scherbius in 1918 and meant initially as a commercial product for the enterprise community, Enigma turned out to be more successful with the German military forces [2]. Enigma had evolved throughout several years, gaining better cryptographic strength, but also being broken time and again.","PeriodicalId":423691,"journal":{"name":"100 Years of Math Milestones","volume":"51 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":"121662595","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}
The notion of category stems from countability. The subsets of metric spaces are divided into two categories: first category and second category. Subsets of the first category can be thought of as small, and subsets of category two could be thought of as large, since it is usual that asset of the first category is a subset of some second category set; the verse inclusion never holds. Recall that a metric space is defined as a set with a distance function. Because this is the sole requirement on the set, the notion of category is versatile, and can be applied to various metric spaces, as is observed in Euclidian spaces, function spaces and sequence spaces. However, the Baire category theorem is used as a method of proving existence [1].
{"title":"Baire category theorem","authors":"Alana Liteanu","doi":"10.1090/mbk/121/87","DOIUrl":"https://doi.org/10.1090/mbk/121/87","url":null,"abstract":"The notion of category stems from countability. The subsets of metric spaces are divided into two categories: first category and second category. Subsets of the first category can be thought of as small, and subsets of category two could be thought of as large, since it is usual that asset of the first category is a subset of some second category set; the verse inclusion never holds. Recall that a metric space is defined as a set with a distance function. Because this is the sole requirement on the set, the notion of category is versatile, and can be applied to various metric spaces, as is observed in Euclidian spaces, function spaces and sequence spaces. However, the Baire category theorem is used as a method of proving existence [1].","PeriodicalId":423691,"journal":{"name":"100 Years of Math Milestones","volume":"7 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":"130654745","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}
The Poincaré conjecture is one of the few mathematical results that has managed to catch the interest of the mainstream media. Acknowledged as one of the most important open questions in mathematics, and endowed by the Clay Mathematics Institute with a reward of $1,000,000 for the first correct solution, it had bugged mathematicians for over a century. Then, in 2003, the reclusive Russian mathematician Grigory Perelman posted a series of online papers claiming to have solved the problem. He embarked on a tour of the USA, explaining the main ideas contained in his papers they seemed sound but lacked some detail.
{"title":"Poincaré conjecture","authors":"D. O'Shea","doi":"10.1090/mbk/121/91","DOIUrl":"https://doi.org/10.1090/mbk/121/91","url":null,"abstract":"The Poincaré conjecture is one of the few mathematical results that has managed to catch the interest of the mainstream media. Acknowledged as one of the most important open questions in mathematics, and endowed by the Clay Mathematics Institute with a reward of $1,000,000 for the first correct solution, it had bugged mathematicians for over a century. Then, in 2003, the reclusive Russian mathematician Grigory Perelman posted a series of online papers claiming to have solved the problem. He embarked on a tour of the USA, explaining the main ideas contained in his papers they seemed sound but lacked some detail.","PeriodicalId":423691,"journal":{"name":"100 Years of Math Milestones","volume":"9 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":"133928414","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 3𝑥+1 problem","authors":"G. Venturini","doi":"10.1090/mbk/121/20","DOIUrl":"https://doi.org/10.1090/mbk/121/20","url":null,"abstract":"","PeriodicalId":423691,"journal":{"name":"100 Years of Math Milestones","volume":"52 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":"131704837","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":"Class number one problem","authors":"Yuning Zhang","doi":"10.1090/mbk/121/54","DOIUrl":"https://doi.org/10.1090/mbk/121/54","url":null,"abstract":"","PeriodicalId":423691,"journal":{"name":"100 Years of Math Milestones","volume":"10 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":"126582057","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}
We de ne an L-function which uni es the L-function of a Galois representation and the L-function of an automorphic representation. It satis es analytic continuation, functional equation and Riemann Hypothesis. A nondeterministic Turing Machine is used. 2010 Mathematics Subject Classi cation 11R39
{"title":"The Langlands program","authors":"Hideto Ishihara","doi":"10.1090/mbk/121/55","DOIUrl":"https://doi.org/10.1090/mbk/121/55","url":null,"abstract":"We de ne an L-function which uni es the L-function of a Galois representation and the L-function of an automorphic representation. It satis es analytic continuation, functional equation and Riemann Hypothesis. A nondeterministic Turing Machine is used. 2010 Mathematics Subject Classi cation 11R39","PeriodicalId":423691,"journal":{"name":"100 Years of Math Milestones","volume":"21 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":"125273259","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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