. With the help of two types of results, one for real-valued functions, the other one for vector-valued functions, we show how the classical mean value theorems (in an equality form) and the concept of convexity (for functions and for sets) are closely related.
{"title":"Mean value theorems and convexity: An example of cross-fertilization of two mathematical items","authors":"J. Hiriart-Urruty","doi":"10.4171/em/406","DOIUrl":"https://doi.org/10.4171/em/406","url":null,"abstract":". With the help of two types of results, one for real-valued functions, the other one for vector-valued functions, we show how the classical mean value theorems (in an equality form) and the concept of convexity (for functions and for sets) are closely related.","PeriodicalId":41994,"journal":{"name":"Elemente der Mathematik","volume":"19 4","pages":""},"PeriodicalIF":0.1,"publicationDate":"2020-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/em/406","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41306176","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 following is a simplification of the proof of the infinitude of primes using continued fractions given by Barnes [1]. Assume that there are only finitely many prime numbers, namely 2, p1 = 3, . . . , pn. Let q = p1 · · · pn be the product of all odd primes; then q + 1 is not divisible by any odd prime, hence must be a power of 2. Since q +1 ≡ 2 mod 4, we must have q +1 = 2 and therefore q = 1: contradiction. Since no odd prime p ≡ 3 mod 4 can divide q + 1, the proof actually shows that there are infinitely many primes p ≡ 1 mod 4.
下面是对Barnes[1]给出的用连分式证明质数无穷的简化。假设只有有限个素数，即2,p1 = 3，…pn。设q = p1···pn为所有奇素数之积;那么q + 1不能被任何奇素数整除，因此它一定是2的幂。因为q +1≡2 mod 4，我们必须有q +1 = 2，因此q = 1:矛盾。因为没有奇数素数p≡3 mod 4能除q + 1，所以这个证明实际上表明有无穷多个素数p≡1 mod 4。
{"title":"A simple proof of the infinitude of primes","authors":"F. Lemmermeyer","doi":"10.4171/em/407","DOIUrl":"https://doi.org/10.4171/em/407","url":null,"abstract":"The following is a simplification of the proof of the infinitude of primes using continued fractions given by Barnes [1]. Assume that there are only finitely many prime numbers, namely 2, p1 = 3, . . . , pn. Let q = p1 · · · pn be the product of all odd primes; then q + 1 is not divisible by any odd prime, hence must be a power of 2. Since q +1 ≡ 2 mod 4, we must have q +1 = 2 and therefore q = 1: contradiction. Since no odd prime p ≡ 3 mod 4 can divide q + 1, the proof actually shows that there are infinitely many primes p ≡ 1 mod 4.","PeriodicalId":41994,"journal":{"name":"Elemente der Mathematik","volume":"75 1","pages":"80-80"},"PeriodicalIF":0.1,"publicationDate":"2020-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/em/407","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42271635","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":"Drittes Hilbertsches Problem und Dehn-Invariante – Eine Elementarisierung mit Kugeldreiecken","authors":"Max Leppmeier","doi":"10.4171/em/405","DOIUrl":"https://doi.org/10.4171/em/405","url":null,"abstract":"","PeriodicalId":41994,"journal":{"name":"Elemente der Mathematik","volume":"75 1","pages":"58-68"},"PeriodicalIF":0.1,"publicationDate":"2020-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/em/405","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45522483","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":"Mittelwerte als Minima","authors":"A. Schreiber","doi":"10.4171/em/400","DOIUrl":"https://doi.org/10.4171/em/400","url":null,"abstract":"","PeriodicalId":41994,"journal":{"name":"Elemente der Mathematik","volume":"75 1","pages":"23-31"},"PeriodicalIF":0.1,"publicationDate":"2020-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/em/400","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44246598","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":"Logical equivalence of the fundamental theorems on operators between Banach spaces","authors":"Friederike Liebaug, K. Spindler","doi":"10.4171/em/399","DOIUrl":"https://doi.org/10.4171/em/399","url":null,"abstract":"","PeriodicalId":41994,"journal":{"name":"Elemente der Mathematik","volume":"75 1","pages":"15-22"},"PeriodicalIF":0.1,"publicationDate":"2020-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/em/399","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46818943","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":"Zigzags with Bügi, Bernoulli, Euler and the Seidel–Entringer–Arnol’d triangle","authors":"P. Henry, G. Wanner","doi":"10.4171/em/393","DOIUrl":"https://doi.org/10.4171/em/393","url":null,"abstract":"","PeriodicalId":41994,"journal":{"name":"Elemente der Mathematik","volume":" ","pages":""},"PeriodicalIF":0.1,"publicationDate":"2019-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/em/393","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46246095","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 series representation for $pi$","authors":"H. Alzer","doi":"10.4171/em/395","DOIUrl":"https://doi.org/10.4171/em/395","url":null,"abstract":"","PeriodicalId":41994,"journal":{"name":"Elemente der Mathematik","volume":" ","pages":""},"PeriodicalIF":0.1,"publicationDate":"2019-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/em/395","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44976916","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 this article we prove a property characterizing the focal points of hyperbolas. 1 Chords through a point The property which we discuss here relates to the tangents of a hyperbola at the end points of a chord and their intersections with the asymptotes of the hyperbola. It is formulated by the following lemma.
{"title":"A characterization of the focals of hyperbolas","authors":"Paris Pamfilos","doi":"10.4171/EM/390","DOIUrl":"https://doi.org/10.4171/EM/390","url":null,"abstract":"In this article we prove a property characterizing the focal points of hyperbolas. 1 Chords through a point The property which we discuss here relates to the tangents of a hyperbola at the end points of a chord and their intersections with the asymptotes of the hyperbola. It is formulated by the following lemma.","PeriodicalId":41994,"journal":{"name":"Elemente der Mathematik","volume":" ","pages":""},"PeriodicalIF":0.1,"publicationDate":"2019-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/EM/390","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46887510","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":"Tupel aus $n$ natürlichen Zahlen, für die alle Summen verschieden sind, und ein Maßkonzentrations-Phänomen","authors":"E. Behrends","doi":"10.4171/EM/389","DOIUrl":"https://doi.org/10.4171/EM/389","url":null,"abstract":"","PeriodicalId":41994,"journal":{"name":"Elemente der Mathematik","volume":" ","pages":""},"PeriodicalIF":0.1,"publicationDate":"2019-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/EM/389","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46920894","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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