arXiv: History and Overview最新文献_第3页

Polish mathematicians and mathematics in World War I. Part II. Russian Empire 第一次世界大战中的波兰数学家和数学。俄罗斯帝国

arXiv: History and Overview Pub Date : 2018-04-06 DOI: 10.4467/2543702XSHS.19.004.11010

Stanisław Domoradzki, Małgorzata Stawiska

引用次数: 3

Yet another proof of the quadratic reciprocity law 这是二次互易律的又一个证明

arXiv: History and Overview Pub Date : 2018-03-31 DOI: 10.4064/AA180321-10-7

A. Czogala, P. Koprowski

引用次数: 0

Mathematical Encounters. 数学的邂逅。

arXiv: History and Overview Pub Date : 2018-03-24 DOI: 10.4310/iccm.2018.v6.n2.a11

J. Sotomayor

引用次数: 5

Lorenzen’s Reshaping of Krull’s Fundamentalsatz for Integral Domains (1938–1953) Lorenzen对Krull的积分域基本原理的重塑(1938-1953)

arXiv: History and Overview Pub Date : 2018-03-08 DOI: 10.1007/978-3-030-65824-3_9

S. Neuwirth

引用次数: 6

Steady and Stable: Numerical Investigations of Nonlinear Partial Differential Equations 稳定与稳定:非线性偏微分方程的数值研究

arXiv: History and Overview Pub Date : 2018-02-24 DOI: 10.1007/978-3-319-66065-3_11

R. Harwood

引用次数: 1

On a numerical upper bound for the extended Goldbach conjecture 关于扩展哥德巴赫猜想的数值上界

arXiv: History and Overview Pub Date : 2017-12-29 DOI: 10.25911/5D9EFB1B88F6C

David Quarel

{"title":"On a numerical upper bound for the extended Goldbach conjecture","authors":"David Quarel","doi":"10.25911/5D9EFB1B88F6C","DOIUrl":"https://doi.org/10.25911/5D9EFB1B88F6C","url":null,"abstract":"The Goldbach conjecture states that every even number can be decomposed as the sum of two primes. Let $D(N)$ denote the number of such prime decompositions for an even $N$. It is known that $D(N)$ can be bounded above by $$ D(N) leq C^* Theta(N), quad Theta(N):= frac{N}{log^2 N}prod_{substack{p|N p>2}} left( 1 + frac{1}{p-2}right)prod_{p>2}left(1-frac{1}{(p-1)^2}right) $$ where $C^*$ denotes Chen's constant. It is conjectured that $C^*=2$. In 2004, Wu showed that $C^* leq 7.8209$. We attempted to replicate his work in computing Chen's constant, and in doing so we provide an improved approximation of the Buchstab function $omega(u)$, begin{align*} omega(u)=1/u, & quad (1leq uleq 2), (u omega(u))'=omega(u-1), & quad (ugeq 2). end{align*} based on work done by Cheer and Goldston. For each interval $[j,j+1]$, they expressed $omega(u)$ as a Taylor expansion about $u=j+1$. We expanded about the point $u=j+0.5$, so $omega(u)$ was never evaluated more than $0.5$ away from the center of the Taylor expansion, which gave much stronger error bounds. \u0000Issues arose while using this Taylor expansion to compute the required integrals for Chen's constant, so we proceeded with solving the above differential equation to obtain $omega(u)$, and then integrating the result. Although the values that were obtained undershot Wu's results, we pressed on and refined Wu's work by discretising his integrals with finer granularity. The improvements to Chen's constant were negligible (as predicted by Wu). This provides experimental evidence, but not a proof, that were Wu's integrals computed on smaller intervals in exact form, the improvement to Chen's constant would be negligible. Thus, any substantial improvement on Chen's constant likely requires a radically different method to what Wu provided.","PeriodicalId":429168,"journal":{"name":"arXiv: History and Overview","volume":"645 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120846644","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}

引用次数: 2

Cauchy, infinitesimals and ghosts of departed quantifiers 柯西，无限小和离去量词的幽灵

arXiv: History and Overview Pub Date : 2017-12-01 DOI: 10.15330/ms.47.2.115-144

J. Bair, Piotr Błaszczyk, R. Ely, V. Henry, V. Kanovei, Karin U. Katz, M. Katz, T. Kudryk, Semen Samsonovich Kutateladze, T. Mcgaffey, T. Mormann, D. Schaps, David Sherry

{"title":"Cauchy, infinitesimals and ghosts of departed quantifiers","authors":"J. Bair, Piotr Błaszczyk, R. Ely, V. Henry, V. Kanovei, Karin U. Katz, M. Katz, T. Kudryk, Semen Samsonovich Kutateladze, T. Mcgaffey, T. Mormann, D. Schaps, David Sherry","doi":"10.15330/ms.47.2.115-144","DOIUrl":"https://doi.org/10.15330/ms.47.2.115-144","url":null,"abstract":"Procedures relying on infinitesimals in Leibniz, Euler and Cauchy have been interpreted in both a Weierstrassian and Robinson's frameworks. The latter provides closer proxies for the procedures of the classical masters. Thus, Leibniz's distinction between assignable and inassignable numbers finds a proxy in the distinction between standard and nonstandard numbers in Robinson's framework, while Leibniz's law of homogeneity with the implied notion of equality up to negligible terms finds a mathematical formalisation in terms of standard part. It is hard to provide parallel formalisations in a Weierstrassian framework but scholars since Ishiguro have engaged in a quest for ghosts of departed quantifiers to provide a Weierstrassian account for Leibniz's infinitesimals. Euler similarly had notions of equality up to negligible terms, of which he distinguished two types: geometric and arithmetic. Euler routinely used product decompositions into a specific infinite number of factors, and used the binomial formula with an infinite exponent. Such procedures have immediate hyperfinite analogues in Robinson's framework, while in a Weierstrassian framework they can only be reinterpreted by means of paraphrases departing significantly from Euler's own presentation. Cauchy gives lucid definitions of continuity in terms of infinitesimals that find ready formalisations in Robinson's framework but scholars working in a Weierstrassian framework bend over backwards either to claim that Cauchy was vague or to engage in a quest for ghosts of departed quantifiers in his work. Cauchy's procedures in the context of his 1853 sum theorem (for series of continuous functions) are more readily understood from the viewpoint of Robinson's framework, where one can exploit tools such as the pointwise definition of the concept of uniform convergence. \u0000Keywords: historiography; infinitesimal; Latin model; butterfly model","PeriodicalId":429168,"journal":{"name":"arXiv: History and Overview","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122333563","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}

引用次数: 11

Introducing an old calculating instrument in a new technologies environment: a praxeological analysis of students' tasks using different registers 在新技术环境中引入旧的计算工具:使用不同寄存器的学生任务的行动学分析

arXiv: History and Overview Pub Date : 2017-12-01 DOI: 10.26220/REV.2839

Caroline Poisard

{"title":"Introducing an old calculating instrument in a new technologies environment: a praxeological analysis of students' tasks using different registers","authors":"Caroline Poisard","doi":"10.26220/REV.2839","DOIUrl":"https://doi.org/10.26220/REV.2839","url":null,"abstract":"The Chinese abacus is the resource presented in this paper, to teach and learn number sense and place-value system at primary level. The Chinese abacus can be material, virtual (software) or drawn on a worksheet. We present three tasks and analyse them in term of techniques and relative knowledge. We show how these tasks can be solved by students in different registers (material, software, paper-and-pencil, fingers, oral) which is important for both students' understanding and teachers' activity. KEYWORDS Material and virtual resources, praxeology, task, technique, technology, register, number sense, place-value system, Chinese abacus. R{'E}SUM{'E} Le boulier chinois est la ressource pr{'e}sent{'e}e dans cet article, pour enseigner et apprendre la construction du nombre et le syst{`e}me de num{'e}ration d{'e}cimal {`a} l'{'e}cole. Le boulier chinois peut {^e}tre mat{'e}riel, virtuel (logiciel) ou dessin{'e} sur une feuille. Nous pr{'e}sentons trois t{^a}ches et les analysons en terme de techniques et connaissances sous-jacentes. Nous montrons comment ces t{^a}ches peuvent {^e}tre r{'e}solues par des {'e}l{`e}ves dans diff{'e}rents registres (mat{'e}riel, logiciel, papier-crayon, mains, oral) ce qui est important pour la compr{'e}hension des {'e}l{`e}ves et {'e}galement pour l'activit{'e} des professeurs.","PeriodicalId":429168,"journal":{"name":"arXiv: History and Overview","volume":"58 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114993969","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}

引用次数: 0

Measure Theory and Integration By and For the Learner 学习者的测量理论与整合

arXiv: History and Overview Pub Date : 2017-11-03 DOI: 10.16929/sbs/2016.0005

G. Lo, Aladji Babacar Niang

{"title":"Measure Theory and Integration By and For the Learner","authors":"G. Lo, Aladji Babacar Niang","doi":"10.16929/sbs/2016.0005","DOIUrl":"https://doi.org/10.16929/sbs/2016.0005","url":null,"abstract":"Measure Theory and Integration is exposed with the clear aim to help beginning learners to perfectly master its essence. In opposition of a delivery of the contents in an academic and vertical course, the knowledge is broken into exercises which are left to the learners for solutions. Hints are present at any corner to help readers to achieve the solutions. In that way, the knowledge is constructed by the readers by summarizing the results of one or a group of exercises. \u0000Each chapter is organized into Summary documents which contain the knowledge, Discovery documents which give the learner the opportunity to extract the knowledge himself through exercises and into Solution Documents which offer detailed answers for the exercises. Exceptionally, a few number of results (A key lemma related the justification of definition of the integral of a non-negative function, the Caratheodory's theorem and the Lebesgue-Stieljes measure on $mathbb{R}^d$) are presented in appendix documents and given for reading in small groups. \u0000The full theory is presented in the described way. We highly expect that any student who goes through the materials, alone or in a small group or under the supervision of an assistant will gain a very solid knowledge in the subject and by the way ensure a sound foundation for studying disciplines such as Probability Theory, Statistics, Functional Analysis, etc. \u0000The materials have been successfully used as such in normal real analysis classes several times.","PeriodicalId":429168,"journal":{"name":"arXiv: History and Overview","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128795216","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}

引用次数: 10

Physics in Riemann’s Mathematical Papers 黎曼数学论文中的物理学

arXiv: History and Overview Pub Date : 2017-11-02 DOI: 10.1007/978-3-319-60039-0_6

A. Papadopoulos

引用次数: 12