Multidimensional integer trigonometry

Q3 Mathematics Communications in Mathematics Pub Date : 2023-02-06 DOI:10.46298/cm.10919
J. Blackman, James Dolan, O. Karpenkov
{"title":"Multidimensional integer trigonometry","authors":"J. Blackman, James Dolan, O. Karpenkov","doi":"10.46298/cm.10919","DOIUrl":null,"url":null,"abstract":"This paper is dedicated to providing an introduction into multidimensional\ninteger trigonometry. We start with an exposition of integer trigonometry in\ntwo dimensions, which was introduced in 2008, and use this to generalise these\ninteger trigonometric functions to arbitrary dimension. We then move on to\nstudy the basic properties of integer trigonometric functions. We find integer\ntrigonometric relations for transpose and adjacent simplicial cones, and for\nthe cones which generate the same simplices. Additionally, we discuss the\nrelationship between integer trigonometry, the Euclidean algorithm, and\ncontinued fractions. Finally, we use adjacent and transpose cones to introduce\na notion of best approximations of simplicial cones. In two dimensions, this\nnotion of best approximation coincides with the classical notion of the best\napproximations of real numbers.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2023-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications in Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.46298/cm.10919","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Mathematics","Score":null,"Total":0}
引用次数: 2

Abstract

This paper is dedicated to providing an introduction into multidimensional integer trigonometry. We start with an exposition of integer trigonometry in two dimensions, which was introduced in 2008, and use this to generalise these integer trigonometric functions to arbitrary dimension. We then move on to study the basic properties of integer trigonometric functions. We find integer trigonometric relations for transpose and adjacent simplicial cones, and for the cones which generate the same simplices. Additionally, we discuss the relationship between integer trigonometry, the Euclidean algorithm, and continued fractions. Finally, we use adjacent and transpose cones to introduce a notion of best approximations of simplicial cones. In two dimensions, this notion of best approximation coincides with the classical notion of the best approximations of real numbers.
查看原文
分享 分享
微信好友 朋友圈 QQ好友 复制链接
本刊更多论文
多维整数三角学
本文致力于介绍多维积分三角法。我们从2008年引入的二维整数三角函数的阐述开始,并用它将这些整数三角函数推广到任意维度。然后我们继续研究整数三角函数的基本性质。我们找到了转置和相邻单形锥以及生成相同单形的锥的整数三角关系。此外,我们还讨论了整数三角法、欧几里得算法和连续分数之间的关系。最后,我们使用相邻锥和转置锥引入了单锥最佳逼近的概念。在二维中,这种最佳近似的概念与实数的最佳近似的经典概念一致。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 去求助
来源期刊
Communications in Mathematics
Communications in Mathematics Mathematics-Mathematics (all)
CiteScore
1.00
自引率
0.00%
发文量
26
审稿时长
45 weeks
期刊介绍: Communications in Mathematics publishes research and survey papers in all areas of pure and applied mathematics. To be acceptable for publication, the paper must be significant, original and correct. High quality review papers of interest to a wide range of scientists in mathematics and its applications are equally welcome.
期刊最新文献
Sharp Restriction Theory Weak polynomial identities of small degree for the Weyl algebra A complete invariant for doodles on a 2-sphere Lie pairs Non-associative algebraic structures: classification and structure
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
现在去查看 取消
×
提示
确定
0
微信
客服QQ
Book学术公众号 扫码关注我们
反馈
×
意见反馈
请填写您的意见或建议
请填写您的手机或邮箱
已复制链接
已复制链接
快去分享给好友吧!
我知道了
×
扫码分享
扫码分享
Book学术官方微信
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术
文献互助 智能选刊 最新文献 互助须知 联系我们:info@booksci.cn
Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。
Copyright © 2023 Book学术 All rights reserved.
ghs 京公网安备 11010802042870号 京ICP备2023020795号-1