Pub Date : 2016-09-09DOI: 10.6084/M9.FIGSHARE.3814761.V1
T. Sochi
The present text is a collection of notes about differential geometry prepared to some extent as part of tutorials about topics and applications related to tensor calculus. They can be regarded as continuation to the previous notes on tensor calculus as they are based on the materials and conventions given in those documents. They can be used as a reference for a first course on the subject or as part of a course on tensor calculus.
{"title":"Principles of Differential Geometry","authors":"T. Sochi","doi":"10.6084/M9.FIGSHARE.3814761.V1","DOIUrl":"https://doi.org/10.6084/M9.FIGSHARE.3814761.V1","url":null,"abstract":"The present text is a collection of notes about differential geometry prepared to some extent as part of tutorials about topics and applications related to tensor calculus. They can be regarded as continuation to the previous notes on tensor calculus as they are based on the materials and conventions given in those documents. They can be used as a reference for a first course on the subject or as part of a course on tensor calculus.","PeriodicalId":429168,"journal":{"name":"arXiv: History and Overview","volume":"43 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"113971757","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 translation of "Verallgemeinerung des Sylow'schen Satzes" by F. G. Frobenius, Sitzungsberichte K. Preuss. Akad. Wiss. Berlin, 1895 (II).
{"title":"A translation of \"Verallgemeinerung des Sylow'schen Satzes\" by F. G. Frobenius","authors":"R. Andreev","doi":"10.3931/e-rara-18880","DOIUrl":"https://doi.org/10.3931/e-rara-18880","url":null,"abstract":"A translation of \"Verallgemeinerung des Sylow'schen Satzes\" by F. G. Frobenius, Sitzungsberichte K. Preuss. Akad. Wiss. Berlin, 1895 (II).","PeriodicalId":429168,"journal":{"name":"arXiv: History and Overview","volume":"102 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124278825","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 : 2016-08-13DOI: 10.1007/978-3-319-44950-0_25
T. Holm
{"title":"Transforming Post-Secondary Education in Mathematics","authors":"T. Holm","doi":"10.1007/978-3-319-44950-0_25","DOIUrl":"https://doi.org/10.1007/978-3-319-44950-0_25","url":null,"abstract":"","PeriodicalId":429168,"journal":{"name":"arXiv: History and Overview","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131217984","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 discuss variations of the zero-sum game where Bob selects two distinct numbers, and Alice learns one of them to make a guess which of the numbers is the larger.
我们讨论了零和博弈的变体，鲍勃选择了两个不同的数字，爱丽丝学习其中一个来猜测哪个数字更大。
{"title":"Guess the Larger Number","authors":"A. Gnedin","doi":"10.14708/MA.V44I1.1205","DOIUrl":"https://doi.org/10.14708/MA.V44I1.1205","url":null,"abstract":"We discuss variations of the zero-sum game where Bob selects two distinct numbers, and Alice learns one of them to make a guess which of the numbers is the larger.","PeriodicalId":429168,"journal":{"name":"arXiv: History and Overview","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130169336","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 : 2016-05-02DOI: 10.1007/978-3-319-44418-5_19
R. O. Wells
{"title":"The Geometry of Manifolds and the Perception of Space","authors":"R. O. Wells","doi":"10.1007/978-3-319-44418-5_19","DOIUrl":"https://doi.org/10.1007/978-3-319-44418-5_19","url":null,"abstract":"","PeriodicalId":429168,"journal":{"name":"arXiv: History and Overview","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116334589","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 paper we consider a few Calculus optimization problems in which we notice peculiar patterns. In each of these cases there is a geometric explanation for the pattern showing that it is not just a coincidence.
{"title":"It is not a Coincidence! On Curious Patterns in Calculus Optimization Problems","authors":"Maria Nogin","doi":"10.17654/ME016030319","DOIUrl":"https://doi.org/10.17654/ME016030319","url":null,"abstract":"In this paper we consider a few Calculus optimization problems in which we notice peculiar patterns. In each of these cases there is a geometric explanation for the pattern showing that it is not just a coincidence.","PeriodicalId":429168,"journal":{"name":"arXiv: History and Overview","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123878877","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 study of the mathematics and geometry of ancient civilizations is a task which seems to be very difficult or even impossible to fulfil, if few written documents, or none at all, had survived from the past. However, besides the direct information that we can have from written documents, we can gain some indirect evidence on mathematics and geometry also from the analysis of the decorations we find on artifacts. Here, for instance, we will show that ancient Egyptians were able of making geometric constructions using compass and straightedge, quite before the Greek Oenopides of Chois, who lived around 450 BC, had declared some of their basic principles. In fact, a wood panel covered by an overlapping circles grid pattern, found in the tomb of Kha, an architect who served three kings of 18th Dynasty (1400-1350 BC), evidences that some simple constructions with compass and straightedge were used in ancient Egypt about nine centuries before Oenopides' time.
研究古代文明的数学和几何似乎是一项非常困难甚至不可能完成的任务，如果从过去流传下来的书面文件很少，或者根本没有。然而，除了我们可以从书面文件中获得的直接信息外，我们还可以从分析我们在人工制品上发现的装饰中获得一些关于数学和几何的间接证据。例如，在这里，我们将展示古埃及人能够使用指南针和直尺制作几何结构，这远远早于生活在公元前450年左右的希腊人奥诺匹德斯(Oenopides of Chois)宣布他们的一些基本原则。事实上，在为18王朝(公元前1400-1350年)的三位国王服务的建筑师Kha的坟墓中发现的一块覆盖着重叠圆圈网格图案的木板，证明了在奥诺匹德斯时代之前大约9个世纪的古埃及就已经使用了一些简单的指南针和直尺建筑。
{"title":"Overlapping Circles Grid Drawn with Compass and Straightedge on an Egyptian Artifact of 14th Century BC","authors":"Amelia Carolina Sparavigna, M. M. Baldi","doi":"10.2139/ssrn.2750125","DOIUrl":"https://doi.org/10.2139/ssrn.2750125","url":null,"abstract":"The study of the mathematics and geometry of ancient civilizations is a task which seems to be very difficult or even impossible to fulfil, if few written documents, or none at all, had survived from the past. However, besides the direct information that we can have from written documents, we can gain some indirect evidence on mathematics and geometry also from the analysis of the decorations we find on artifacts. Here, for instance, we will show that ancient Egyptians were able of making geometric constructions using compass and straightedge, quite before the Greek Oenopides of Chois, who lived around 450 BC, had declared some of their basic principles. In fact, a wood panel covered by an overlapping circles grid pattern, found in the tomb of Kha, an architect who served three kings of 18th Dynasty (1400-1350 BC), evidences that some simple constructions with compass and straightedge were used in ancient Egypt about nine centuries before Oenopides' time.","PeriodicalId":429168,"journal":{"name":"arXiv: History and Overview","volume":"145 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115191407","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 report on a workshop for grade eleven high school students, which took place in the framework of a university open day. During the workshop the participants first discovered the key properties of the intuitive concept of distance from real life examples. After this preparation, the formal definition of a metric space was introduced and discussed in small groups by means of problem-oriented exercise sessions.
{"title":"An open day in the metric space","authors":"Sven-Ake Wegner, Katrin Rolka","doi":"10.1093/TEAMAT/HRW025","DOIUrl":"https://doi.org/10.1093/TEAMAT/HRW025","url":null,"abstract":"We report on a workshop for grade eleven high school students, which took place in the framework of a university open day. During the workshop the participants first discovered the key properties of the intuitive concept of distance from real life examples. After this preparation, the formal definition of a metric space was introduced and discussed in small groups by means of problem-oriented exercise sessions.","PeriodicalId":429168,"journal":{"name":"arXiv: History and Overview","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121866483","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 : 2016-02-29DOI: 10.1007/978-3-319-17187-6_32
M. Menghini
{"title":"From Practical Geometry to the Laboratory Method: The Search for an Alternative to Euclid in the History of Teaching Geometry","authors":"M. Menghini","doi":"10.1007/978-3-319-17187-6_32","DOIUrl":"https://doi.org/10.1007/978-3-319-17187-6_32","url":null,"abstract":"","PeriodicalId":429168,"journal":{"name":"arXiv: History and Overview","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132314125","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 : 2016-02-26DOI: 10.23943/princeton/9780691171920.003.0002
T. Khovanova
Kasha-eating dragons introduce advanced mathematics. The goal of this paper is twofold: to entertain people who know advanced mathematics and inspire people who don’t. Suppose a four-armed dragon is sitting on every face of a cube. Each dragon has a bowl of kasha in front of him. Dragons are very greedy, so instead of eating their own kasha, they try to steal kasha from their neighbors. Every minute every dragon extends four arms to the four neighboring faces on the cube and tries to get the kasha from the bowls there. As four arms are fighting for every bowl of kasha, each arm manages to steal one-fourth of what is in the bowl. Thus each dragon steals one-fourth of the kasha of each of his neighbors, while at the same time all of his own kasha is stolen. Given the initial amounts of kasha in every bowl, what is the asymptotic behavior of the amounts of kasha? Why do these dragons eat kasha? Kasha (buckwheat porridge) is very healthy. But for mathematicians, kasha represents a continuous entity. You can view the amount of kasha in a bowl as a real number. Another common food that works for this purpose is soup, but liquid soup is difficult to steal with your bare hands. We do not want to s soup spilled all over our cube, do we? If kasha seems too exotic, you can imagine less exotic and less healthy mashed potatoes. How does this relate to advanced mathematics? For starters, it relates to linear algebra [3]. We can consider the amounts of kasha as six real numbers, as there are six bowls, one on each of the six faces of the cube. We can view this six-tuple that represents kasha at each moment as a vector in a six-dimensional vector space of possible amounts of kasha. To be able to view the amounts of kasha as a vector, we need to make a leap of faith and assume that negative amounts of kasha are possible. I just hope that if my readers have enough imagination to envision six four-armed dragons on the faces of the cube, then they can also imagine negative kasha. The bowl with 2 pounds of kasha means that if you put two pounds of kasha into this bowl, it becomes empty. For those who wonder why dragons would fight for negative kasha, this is how mathematics works. We make unrealistic assumptions, solve the problem, and then hope that the solution translates to reality anyway.
{"title":"Dragons and Kasha","authors":"T. Khovanova","doi":"10.23943/princeton/9780691171920.003.0002","DOIUrl":"https://doi.org/10.23943/princeton/9780691171920.003.0002","url":null,"abstract":"Kasha-eating dragons introduce advanced mathematics. The goal of this paper is twofold: to entertain people who know advanced mathematics and inspire people who don’t. Suppose a four-armed dragon is sitting on every face of a cube. Each dragon has a bowl of kasha in front of him. Dragons are very greedy, so instead of eating their own kasha, they try to steal kasha from their neighbors. Every minute every dragon extends four arms to the four neighboring faces on the cube and tries to get the kasha from the bowls there. As four arms are fighting for every bowl of kasha, each arm manages to steal one-fourth of what is in the bowl. Thus each dragon steals one-fourth of the kasha of each of his neighbors, while at the same time all of his own kasha is stolen. Given the initial amounts of kasha in every bowl, what is the asymptotic behavior of the amounts of kasha? Why do these dragons eat kasha? Kasha (buckwheat porridge) is very healthy. But for mathematicians, kasha represents a continuous entity. You can view the amount of kasha in a bowl as a real number. Another common food that works for this purpose is soup, but liquid soup is difficult to steal with your bare hands. We do not want to s soup spilled all over our cube, do we? If kasha seems too exotic, you can imagine less exotic and less healthy mashed potatoes. How does this relate to advanced mathematics? For starters, it relates to linear algebra [3]. We can consider the amounts of kasha as six real numbers, as there are six bowls, one on each of the six faces of the cube. We can view this six-tuple that represents kasha at each moment as a vector in a six-dimensional vector space of possible amounts of kasha. To be able to view the amounts of kasha as a vector, we need to make a leap of faith and assume that negative amounts of kasha are possible. I just hope that if my readers have enough imagination to envision six four-armed dragons on the faces of the cube, then they can also imagine negative kasha. The bowl with 2 pounds of kasha means that if you put two pounds of kasha into this bowl, it becomes empty. For those who wonder why dragons would fight for negative kasha, this is how mathematics works. We make unrealistic assumptions, solve the problem, and then hope that the solution translates to reality anyway.","PeriodicalId":429168,"journal":{"name":"arXiv: History and Overview","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126163182","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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