Pub Date : 2016-02-25DOI: 10.1007/978-1-4614-4684-2_9
F. Furinghetti, José Manuel Matos, M. Menghini
{"title":"From Mathematics and Education, to Mathematics Education","authors":"F. Furinghetti, José Manuel Matos, M. Menghini","doi":"10.1007/978-1-4614-4684-2_9","DOIUrl":"https://doi.org/10.1007/978-1-4614-4684-2_9","url":null,"abstract":"","PeriodicalId":429168,"journal":{"name":"arXiv: History and Overview","volume":"55 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129287635","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 present, discuss and generalize an elegant geometrical proof of the law of cosines, due to Al Cuoco.
我们提出、讨论并推广了Al Cuoco关于余弦定理的一个优雅的几何证明。
{"title":"Un problema da discutere: una rappresentazione geometrica del teorema del coseno","authors":"C. Bernardi","doi":"10.1400/232399","DOIUrl":"https://doi.org/10.1400/232399","url":null,"abstract":"We present, discuss and generalize an elegant geometrical proof of the law of cosines, due to Al Cuoco.","PeriodicalId":429168,"journal":{"name":"arXiv: History and Overview","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124394876","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 survey some major contributions to Riemann's moduli space and Teichm{"u}ller space. Our report has a historical character, but the stress is on the chain of mathematical ideas. We start with the introduction of Riemann surfaces, and we end with the discovery of some of the basic structures of Riemann's moduli space and Teichm{"u}ller space. We point out several facts which seem to be unknown to many algebraic geometers and analysts working in the theory. The period we are interested in starts with Riemann, in 1851, and ends in the early 1960s, when Ahlfors and Bers confirmed that Teichm{"u}ller's results were correct.This paper was written for the book "Lipman Bers, a life in Mathematics," edited by Linda Keen , Irwin Kra and Rubi Rodriguez (Amercian Mathematical Society, 2015). It is dedicated to the memory of Lipman Bers who was above all a complex analyst and spent a large part of his life and energy working on the analytic structure of Teichm{"u}ller space. His work on analysis is nevertheless inseparable from geometry and topology. In this survey, we highlight the relations and the logical dependence between this work and the works of Riemann, Poincar{'e}, Klein, Brouwer, Siegel, Teichm{"u}ller, Weil, Grothendieck and others. We explain the motivation behind the ideas. In doing so, we point out several facts which seem to be unknown to many Teichm{"u}ller theorists.
本文综述了黎曼模空间和泰奇姆空间的一些重要贡献。我们的报告具有历史性质，但重点是数学思想的链条。我们从黎曼曲面的介绍开始，并以黎曼模空间和泰希姆勒空间的一些基本结构的发现结束。我们指出了几个事实，这似乎是许多代数几何学者和分析在理论工作不知道。我们感兴趣的时期从1851年黎曼开始，到20世纪60年代初结束，当时阿尔福斯和贝尔斯证实了泰希姆勒的结果是正确的。本文是为Linda Keen, Irwin Kra和Rubi Rodriguez编辑的《Lipman Bers, a life in Mathematics》(美国数学学会，2015)一书撰写的。它是为了纪念李普曼·伯斯，他首先是一个复杂的分析家，他的大部分生命和精力都花在了研究泰希姆空间的分析结构上。然而，他在分析方面的工作与几何学和拓扑学密不可分。在这篇综述中，我们强调了这部作品与黎曼、庞加莱、克莱因、布劳维尔、西格尔、泰希姆勒、韦尔、格罗滕迪克等人的作品之间的关系和逻辑依赖。我们解释这些想法背后的动机。在这样做的过程中，我们指出了许多泰希姆勒理论家似乎不知道的几个事实。
{"title":"On the early history of moduli and Teichm{ü}ller spaces","authors":"N. A'campo, L. Ji, A. Papadopoulos","doi":"10.1090/mbk/093/13","DOIUrl":"https://doi.org/10.1090/mbk/093/13","url":null,"abstract":"We survey some major contributions to Riemann's moduli space and Teichm{\"u}ller space. Our report has a historical character, but the stress is on the chain of mathematical ideas. We start with the introduction of Riemann surfaces, and we end with the discovery of some of the basic structures of Riemann's moduli space and Teichm{\"u}ller space. We point out several facts which seem to be unknown to many algebraic geometers and analysts working in the theory. The period we are interested in starts with Riemann, in 1851, and ends in the early 1960s, when Ahlfors and Bers confirmed that Teichm{\"u}ller's results were correct.This paper was written for the book \"Lipman Bers, a life in Mathematics,\" edited by Linda Keen , Irwin Kra and Rubi Rodriguez (Amercian Mathematical Society, 2015). It is dedicated to the memory of Lipman Bers who was above all a complex analyst and spent a large part of his life and energy working on the analytic structure of Teichm{\"u}ller space. His work on analysis is nevertheless inseparable from geometry and topology. In this survey, we highlight the relations and the logical dependence between this work and the works of Riemann, Poincar{'e}, Klein, Brouwer, Siegel, Teichm{\"u}ller, Weil, Grothendieck and others. We explain the motivation behind the ideas. In doing so, we point out several facts which seem to be unknown to many Teichm{\"u}ller theorists.","PeriodicalId":429168,"journal":{"name":"arXiv: History and Overview","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126414069","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}
This paper deals with the celebrated Euclidean theorem about isosceles triangles, comparing different proofs.
本文讨论了著名的关于等腰三角形的欧几里得定理，比较了不同的证明。
{"title":"Gli angoli alla base di un triangolo isoscele","authors":"C. Bernardi","doi":"10.1400/238418","DOIUrl":"https://doi.org/10.1400/238418","url":null,"abstract":"This paper deals with the celebrated Euclidean theorem about isosceles triangles, comparing different proofs.","PeriodicalId":429168,"journal":{"name":"arXiv: History and Overview","volume":"65 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123898071","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}
A three-polar, cf. T. Gregor, J. Haluv{s}ka, Lexicographical ordering and field operations in the complex plane. Stud. Mat. 41(2014), 123--133., $HSV-RGB$ Colour space $triangle$ was introduced and studied. It was equipped with operations of addition, subtraction, multiplication, and (partially) division. Achromatic Grey Hues form an ideal $mathfrak{S}$. Factorizing $triangle$ by the ideal $mathfrak{S}$, we obtain a field $triangle | mathfrak{S}$. An element (i.e an individual Colour) in $triangle | mathfrak{S}$ is a triplet of three triangular coefficients. The set of all triangular coefficients is a subset of a semi-field of parabolic-complex functions. For the parabolic-complex number set, cf.~A. A. Harkin--J. B. Harkin, Geometry of general complex numbers. Mathematics magazine, 77(2004), 118--129.
{"title":"On fields inspired with the polar HSV -- RGB theory of Colour","authors":"J. Haluska","doi":"10.18778/7969-663-5.06","DOIUrl":"https://doi.org/10.18778/7969-663-5.06","url":null,"abstract":"A three-polar, cf. T. Gregor, J. Haluv{s}ka, Lexicographical ordering and field operations in the complex plane. Stud. Mat. 41(2014), 123--133., $HSV-RGB$ Colour space $triangle$ was introduced and studied. It was equipped with operations of addition, subtraction, multiplication, and (partially) division. Achromatic Grey Hues form an ideal $mathfrak{S}$. Factorizing $triangle$ by the ideal $mathfrak{S}$, we obtain a field $triangle | mathfrak{S}$. An element (i.e an individual Colour) in $triangle | mathfrak{S}$ is a triplet of three triangular coefficients. The set of all triangular coefficients is a subset of a semi-field of parabolic-complex functions. For the parabolic-complex number set, cf.~A. A. Harkin--J. B. Harkin, Geometry of general complex numbers. Mathematics magazine, 77(2004), 118--129.","PeriodicalId":429168,"journal":{"name":"arXiv: History and Overview","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128402574","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 : 2015-11-25DOI: 10.4310/ICCM.2015.V3.N1.A10
Fang-fang Li, Yong Zhang
When studying the history of mathematical symbols, one finds that the development of mathematical symbols in China is a significant piece of Chinese history; however, between the beginning of mathematics and modern day mathematics in China, there exists a long blank period. Let us focus on the development of Chinese mathematical symbols, and find out the significance of their origin, evolution, rise and fall within Chinese mathematics.
{"title":"On Mathematical Symbols in China","authors":"Fang-fang Li, Yong Zhang","doi":"10.4310/ICCM.2015.V3.N1.A10","DOIUrl":"https://doi.org/10.4310/ICCM.2015.V3.N1.A10","url":null,"abstract":"When studying the history of mathematical symbols, one finds that the development of mathematical symbols in China is a significant piece of Chinese history; however, between the beginning of mathematics and modern day mathematics in China, there exists a long blank period. Let us focus on the development of Chinese mathematical symbols, and find out the significance of their origin, evolution, rise and fall within Chinese mathematics.","PeriodicalId":429168,"journal":{"name":"arXiv: History and Overview","volume":"122 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134604227","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 : 2015-10-24DOI: 10.1007/978-3-319-54469-4_14
J. Jost
{"title":"Object oriented models vs. data analysis -- is this the right alternative?","authors":"J. Jost","doi":"10.1007/978-3-319-54469-4_14","DOIUrl":"https://doi.org/10.1007/978-3-319-54469-4_14","url":null,"abstract":"","PeriodicalId":429168,"journal":{"name":"arXiv: History and Overview","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128444173","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 : 2015-10-18DOI: 10.13140/RG.2.1.3906.3763
H. Gouin
In his treatise addressed to Dositheus of Pelusium, Archimedes of Syracuse obtained the result of which he was the most proud: a sphere has two-thirds the volume of its circumscribing cylinder. At his request a sculpted sphere and cylinder were placed on his tomb near Syracuse. Usually, it is admitted that to find this formula, Archimedes used a half polygon inscribed in a semicircle; then he performed rotations of these two figures to obtain a set of trunks in a sphere. This set of trunks allowed him to determine the volume. In our opinion, Archimedes was so clever that he found the proof with shorter demonstration. Archimedes did not need to know π to prove the result and the Pythagorean theorem was probably the key to the proof.
{"title":"Archimedes' famous-theorem","authors":"H. Gouin","doi":"10.13140/RG.2.1.3906.3763","DOIUrl":"https://doi.org/10.13140/RG.2.1.3906.3763","url":null,"abstract":"In his treatise addressed to Dositheus of Pelusium, Archimedes of Syracuse obtained the result of which he was the most proud: a sphere has two-thirds the volume of its circumscribing cylinder. At his request a sculpted sphere and cylinder were placed on his tomb near Syracuse. Usually, it is admitted that to find this formula, Archimedes used a half polygon inscribed in a semicircle; then he performed rotations of these two figures to obtain a set of trunks in a sphere. This set of trunks allowed him to determine the volume. In our opinion, Archimedes was so clever that he found the proof with shorter demonstration. Archimedes did not need to know π to prove the result and the Pythagorean theorem was probably the key to the proof.","PeriodicalId":429168,"journal":{"name":"arXiv: History and Overview","volume":"170 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115967112","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 Mathematical Thought and Its Objects, Charles Parsons argues that our knowledge of the iterability of functions on the natural numbers and of the validity of complete induction is not intuitive knowledge; Brouwer disagrees on both counts. I will compare Parsons' argument with Brouwer's and defend the latter. I will not argue that Parsons is wrong once his own conception of intuition is granted, as I do not think that that is the case. But I will try to make two points: (1) Using elements from Husserl and from Brouwer, Brouwer's claims can be justified in more detail than he has done; (2) There are certain elements in Parsons' discussion that, when developed further, would lead to Brouwer's notion thus analysed, or at least something relevantly similar to it. (This version contains a postscript of May, 2015.)
{"title":"Intuition, iteration, induction","authors":"M. Atten","doi":"10.31235/osf.io/d36hp","DOIUrl":"https://doi.org/10.31235/osf.io/d36hp","url":null,"abstract":"In Mathematical Thought and Its Objects, Charles Parsons argues that our knowledge of the iterability of functions on the natural numbers and of the validity of complete induction is not intuitive knowledge; Brouwer disagrees on both counts. I will compare Parsons' argument with Brouwer's and defend the latter. I will not argue that Parsons is wrong once his own conception of intuition is granted, as I do not think that that is the case. But I will try to make two points: (1) Using elements from Husserl and from Brouwer, Brouwer's claims can be justified in more detail than he has done; (2) There are certain elements in Parsons' discussion that, when developed further, would lead to Brouwer's notion thus analysed, or at least something relevantly similar to it. (This version contains a postscript of May, 2015.)","PeriodicalId":429168,"journal":{"name":"arXiv: History and Overview","volume":"96 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114840681","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 : 2015-09-02DOI: 10.4310/ICCM.2016.V4.N2.A4
J. Milne
The statement of the Riemann hypothesis makes sense for all global fields, not just the rational numbers. For function fields, it has a natural restatement in terms of the associated curve. Weil's work on the Riemann hypothesis for curves over finite fields led him to state his famous "Weil conjectures", which drove much of the progress in algebraic and arithmetic geometry in the following decades. In this article, I describe Weil's work and some of the ensuing progress: Weil cohomology (etale, crystalline); Grothendieck's standard conjectures; motives; Deligne's proof; Hasse-Weil zeta functions and Langlands functoriality.
{"title":"The Riemann Hypothesis over Finite Fields: From Weil to the Present Day","authors":"J. Milne","doi":"10.4310/ICCM.2016.V4.N2.A4","DOIUrl":"https://doi.org/10.4310/ICCM.2016.V4.N2.A4","url":null,"abstract":"The statement of the Riemann hypothesis makes sense for all global fields, not just the rational numbers. For function fields, it has a natural restatement in terms of the associated curve. Weil's work on the Riemann hypothesis for curves over finite fields led him to state his famous \"Weil conjectures\", which drove much of the progress in algebraic and arithmetic geometry in the following decades. \u0000In this article, I describe Weil's work and some of the ensuing progress: Weil cohomology (etale, crystalline); Grothendieck's standard conjectures; motives; Deligne's proof; Hasse-Weil zeta functions and Langlands functoriality.","PeriodicalId":429168,"journal":{"name":"arXiv: History and Overview","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116887383","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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