{"title":"The Arithmetic of Points on a Conic and Projectivities","authors":"H. Coxeter, G. Beck","doi":"10.3888/TMJ.21-2","DOIUrl":"https://doi.org/10.3888/TMJ.21-2","url":null,"abstract":"","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69962562","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":"Peak Response of Single-Degree-of-Freedom Systems to Swept-Frequency Excitation","authors":"Christopher H. Reed, A. Kabe","doi":"10.3888/tmj.21-1","DOIUrl":"https://doi.org/10.3888/tmj.21-1","url":null,"abstract":"","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69962503","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}
I. Falcão, Fernando Miranda, R. Severino, Joana Soares
Research at the Centre of Mathematics at the University of Minho was financed by Portuguese Funds through FCT - Fundacao para a Ciencia e a Tecnologia, within the Project UID/MAT/00013/2013. Research at the Economics Politics Research Unit was carried out within the funding with COMPETE reference number POCI-01-0145-FEDER-006683 (UID/ECO/03182/2013), with the FCT/MEC’s (Fundacao para a Ciencia e a Tecnologia, I.P.) financial support through national funding and by the European Regional Development Fund through the Operational Programme on “Competitiveness and Internationalization - COMPETE 2020” under the PT2020 Partnership Agreement.
Minho大学数学中心的研究由葡萄牙基金通过FCT-Fundacao para a Ciencia e a Tecnologia在UID/MAT/00113/2013项目中资助。经济政治研究部的研究是在COMPETE参考号为POCI-010145-FDER-006863（UID/CO/03182/2013）的资助范围内进行的，FCT/MEC（Fundacao para a Ciencia e a Tecnologia，I.P.）通过国家资金提供财政支持，欧洲区域发展基金通过PT2020伙伴关系协议下的“竞争力和国际化-竞争力2020”运营计划提供财政支持。
{"title":"Computational Aspects of Quaternionic Polynomials - Part I: Manipulating, Evaluating and Factoring","authors":"I. Falcão, Fernando Miranda, R. Severino, Joana Soares","doi":"10.3888/TMJ.20-4","DOIUrl":"https://doi.org/10.3888/TMJ.20-4","url":null,"abstract":"Research at the Centre of Mathematics at the University of Minho was financed by Portuguese Funds \u0000through FCT - Fundacao para a Ciencia e a Tecnologia, within the Project UID/MAT/00013/2013. Research \u0000at the Economics Politics Research Unit was carried out within the funding with COMPETE \u0000reference number POCI-01-0145-FEDER-006683 (UID/ECO/03182/2013), with the FCT/MEC’s \u0000(Fundacao para a Ciencia e a Tecnologia, I.P.) financial support through national funding and by the \u0000European Regional Development Fund through the Operational Programme on “Competitiveness and \u0000Internationalization - COMPETE 2020” under the PT2020 Partnership Agreement.","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42504197","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":"Improving the Kruskal–Katona Bounds for Complete Subgraphs of a Graph","authors":"R. Cowen","doi":"10.3888/TMJ.20-6","DOIUrl":"https://doi.org/10.3888/TMJ.20-6","url":null,"abstract":"","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69962375","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 modular group Γ is the set of all 2 × 2 matrices with integral elements and determinant 1. That is, Γ is the special linear group of 2 × 2 matrices over the integers, Γ = SL(2, Z). It forms a group under matrix multiplication. If M = a b c d ∈ Γ Then M defines a map f M (z) = az + b cz + d of the extended complex plane to itself. Here f M (−d/c) = ∞, and f M (∞) = a/c. Suppose that N = α β γ δ. Then by direct calculation we find that f N (f M (z)) = (αa + βc)z + (αb + βd) (γa + δc)z + (γb + δd) = f N M (z). While it might seem surprising that the composition of one such rational function with another would be connected with matrix multiplication, the mystery can be dispelled by considering how M and N transform the 2-dimensional vectors C 2. Suppose that M z 1 z 2 = w 1 w 2 , N w 1 w 2 = t 1 t 2. Then N M z 1 z 2 = t 1 t 2. Now suppose we consider these vectors in terms of projective geometry. Two vectors are then considered to be the same if their coordinates are proportional (i.e., the vectors are colinear). In other words, if c is a non-zero complex number, then z 1 z 2 and cz 1 cz 2 are considered to be the same. Thus a projective point z 1 : z 2 is determined by the ratio z = z 1 /z 2 if its coordinates, and the image w 1 : w 2 is determined by the ratio w = w 1 /w 2 of its coordinates. But then w = w 1 w 2 = az 1 + bz 2 cz 1 + dz 2 = az 1 /z 2 + b cz 1 /z 2 + d = az + b cz + d , so the map from z to w reflects a linear transformation in projective coordinates.
模群Γ是所有2 × 2矩阵的集合，其元素为整，行列式为1。即Γ是2 × 2矩阵在整数上的特殊线性群，Γ = SL(2, Z)，它在矩阵乘法下形成一个群。若M = a b c d∈Γ则M定义了扩展复平面到自身的映射f M (z) = az + b cz + d。这里f M (- d/c) =∞，f M(∞)= a/c。假设N = α β γ δ。然后通过直接计算，我们发现f N (f M (z)) = (αa + βc)z + (αb + βd) (γa + δc)z + (γb + δd) = f N M (z)。虽然一个这样的有理函数与另一个函数的组合与矩阵乘法有关似乎令人惊讶，但通过考虑M和N如何变换二维向量c2可以消除这个谜团。假设mz1z2 = w1w2, nw1w2 = t1 t2。那么nmz1z2 = t1 t1。现在假设我们用射影几何来考虑这些向量。如果两个向量的坐标成比例(即向量共线)，则认为它们是相同的。换句话说，如果c是一个非零的复数，那么z1z2和c1cz2被认为是相同的。因此，投影点z1: z2由其坐标的比值z = z1 / z2决定，图像w1: w2由其坐标的比值w = w1 / w2决定。然后w = w1w2 = az1 + bz2cz1 + dz2 = az1 / z2 + bcz1 / z2 + d = az + bcz1 / z2 + d，所以从z到w的映射反映了射影坐标中的线性变换。
{"title":"The Modular Group","authors":"P. Mccreary, Teri Jo Murphy, Christan Carter","doi":"10.3888/TMJ.20-3","DOIUrl":"https://doi.org/10.3888/TMJ.20-3","url":null,"abstract":"The modular group Γ is the set of all 2 × 2 matrices with integral elements and determinant 1. That is, Γ is the special linear group of 2 × 2 matrices over the integers, Γ = SL(2, Z). It forms a group under matrix multiplication. If M = a b c d ∈ Γ Then M defines a map f M (z) = az + b cz + d of the extended complex plane to itself. Here f M (−d/c) = ∞, and f M (∞) = a/c. Suppose that N = α β γ δ. Then by direct calculation we find that f N (f M (z)) = (αa + βc)z + (αb + βd) (γa + δc)z + (γb + δd) = f N M (z). While it might seem surprising that the composition of one such rational function with another would be connected with matrix multiplication, the mystery can be dispelled by considering how M and N transform the 2-dimensional vectors C 2. Suppose that M z 1 z 2 = w 1 w 2 , N w 1 w 2 = t 1 t 2. Then N M z 1 z 2 = t 1 t 2. Now suppose we consider these vectors in terms of projective geometry. Two vectors are then considered to be the same if their coordinates are proportional (i.e., the vectors are colinear). In other words, if c is a non-zero complex number, then z 1 z 2 and cz 1 cz 2 are considered to be the same. Thus a projective point z 1 : z 2 is determined by the ratio z = z 1 /z 2 if its coordinates, and the image w 1 : w 2 is determined by the ratio w = w 1 /w 2 of its coordinates. But then w = w 1 w 2 = az 1 + bz 2 cz 1 + dz 2 = az 1 /z 2 + b cz 1 /z 2 + d = az + b cz + d , so the map from z to w reflects a linear transformation in projective coordinates.","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"9 1","pages":"564-582"},"PeriodicalIF":0.0,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69962348","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 Wolfram Language Approach to Real Numerical Algebraic Plane Curves","authors":"Barry H. Dayton","doi":"10.3888/tmj.20-7","DOIUrl":"https://doi.org/10.3888/tmj.20-7","url":null,"abstract":"","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69962388","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":"Aspects of Input Shaping Control of Flexible Mechanical Systems","authors":"D. Adair, M. Jaeger","doi":"10.3888/TMJ.19-3","DOIUrl":"https://doi.org/10.3888/TMJ.19-3","url":null,"abstract":"","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69962731","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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