{"title":"Ratio tests for the convergence of integrals","authors":"W. Ferrar","doi":"10.1017/S0950184300002548","DOIUrl":"https://doi.org/10.1017/S0950184300002548","url":null,"abstract":"1. Ratio tests for the convergence or divergence of infinite series of positive terms are well known; they are used in and out of season. On the other hand, ratio tests for infinite integrals are never used. What is the reason for this disparity between series and integrals?","PeriodicalId":417997,"journal":{"name":"Edinburgh Mathematical Notes","volume":"59 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133082163","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 remark about canonical forms","authors":"H. Perfect","doi":"10.1017/S0950184300000276","DOIUrl":"https://doi.org/10.1017/S0950184300000276","url":null,"abstract":"A comparison of the rational and classical canonical forms of a square matrix reveals that for a nilpotent matrix the two are identical. In this note I describe how we may utilise this fact in solving the problem of reducing a given matrix to classical canonical form. I believe that the point which I try to make in what follows is one which is not always explicitly remarked upon in the literature, and it has therefore seemed to me to be worth while to stress it here.","PeriodicalId":417997,"journal":{"name":"Edinburgh Mathematical Notes","volume":"107 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132313542","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 further note on differentials","authors":"E. Phillips","doi":"10.1017/S095018430000255X","DOIUrl":"https://doi.org/10.1017/S095018430000255X","url":null,"abstract":"~{f (x).x}^-kf(x) when x>X, (7) ax then, as a little calculation shows, f(x)^Ax-~, (8) where A is a positive constant. No one would prefer (7) to (8) as a criterion of convergence and (8), like (6), is a well-known test for the convergence of infinite integrals. The next test, in the usual order, is given by taking <f>(x)=x log x in Theorems 1 and 2. That the test is useless may be seen from the fact (mildly interesting in its proof) that","PeriodicalId":417997,"journal":{"name":"Edinburgh Mathematical Notes","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117003636","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 proof of the “Theorem of the Means.”","authors":"C. Walsh","doi":"10.1017/S0950184300000045","DOIUrl":"https://doi.org/10.1017/S0950184300000045","url":null,"abstract":"","PeriodicalId":417997,"journal":{"name":"Edinburgh Mathematical Notes","volume":"90 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122540194","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":"Some Parameters of Sampling Distributions Simply Obtained","authors":"L. M. Brown","doi":"10.1017/S0950184300000094","DOIUrl":"https://doi.org/10.1017/S0950184300000094","url":null,"abstract":"In the theory of statistics a set of quantities a 1 , a 2 , …, a v is considered, and called a distribution. The moments of this distribution about its origin are defined by the equations .","PeriodicalId":417997,"journal":{"name":"Edinburgh Mathematical Notes","volume":"197 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115090151","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":"On the sum of the r -th powers of the first n integers","authors":"A. Waterson","doi":"10.1017/S0950184300002974","DOIUrl":"https://doi.org/10.1017/S0950184300002974","url":null,"abstract":"In this note an explicit expression is obtained for the sum of the r-th powers of the first n integers. The result is equivalent to the well-known result in terms of Bernoulli numbers and the equivalence is not difficult to establish. However, the method given here is elementary and self-contained and provides an excellent exercise on. the manipulation of determinants. Throughout the note, the following notation will be used: S r s l ' + 2 ' + . . . + n, n f=n(n-l)(n-2)...(n-r+","PeriodicalId":417997,"journal":{"name":"Edinburgh Mathematical Notes","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116078515","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 model of a hyperboloid of one sheet and its asymptotic cone","authors":"A. G. Walker","doi":"10.1017/S0950184300000215","DOIUrl":"https://doi.org/10.1017/S0950184300000215","url":null,"abstract":"In this article is described the construction of a thread model of a hyperboloid of one sheet ( H ) and its asymptotic cone ( C ). It ia simple to make, requiring only cardboard and thread, and can be made collapsible and of pocket size if desired. The model consists of two hinged pieces of cardboard (intersecting planes π and ) on which are drawn circles S H , respectively in which the planes meet H , and the concentric circles S C , respectively in which the planes meet C . A number of generators of the same system on H are now represented by threads joining S H and , and the corresponding parallel generators of C are represented by threads joining S C and . In order to ensure that these generators are well spaced, those of C are taken at equal eccentric angles apart in a principal elliptic section. The main theorem used in the design is that if l a generator of C , then the tangent plane to C at points of l meets H in two generators both of which are parallel to l","PeriodicalId":417997,"journal":{"name":"Edinburgh Mathematical Notes","volume":"144 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132184600","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 plane quartic curve with twelve undulations","authors":"W. L. Edge","doi":"10.1017/S0950184300000197","DOIUrl":"https://doi.org/10.1017/S0950184300000197","url":null,"abstract":"where x, y, z are homogeneous coordinates in a plane, was encountered by Ciani [Palermo Rendiconli, Vol. 13, 1899] in his search for plane quartic curves that were invariant under harmonic inversions. If x, y, z undergo any permutation the ternary quartic form on the left of (1) is not altered; nor is it altered if any, or all, of x, y, z be multiplied by — 1. There thus arises an octahedral group 0 of ternary collineations for which every curve of the pencil is invariant. Since (1) may also be written","PeriodicalId":417997,"journal":{"name":"Edinburgh Mathematical Notes","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131012186","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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