Pub Date : 1900-01-01DOI: 10.1017/S0950184300002548
W. Ferrar
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?
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Pub Date : 1900-01-01DOI: 10.1017/S0950184300000276
H. Perfect
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.
{"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}
Pub Date : 1900-01-01DOI: 10.1017/S0950184300000045
C. Walsh
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Pub Date : 1900-01-01DOI: 10.1017/S095018430000255X
E. Phillips
~{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 (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
{"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}
Pub Date : 1900-01-01DOI: 10.1017/S0950184300000094
L. M. Brown
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 .
{"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}
Pub Date : 1900-01-01DOI: 10.1017/S0950184300000343
H. W. Turnbull
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Pub Date : 1900-01-01DOI: 10.1017/S0950184300000197
W. L. Edge
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
其中x, y, z是平面上的齐次坐标，是Ciani [Palermo Rendiconli, Vol. 13, 1899]在寻找调和反转下不变的平面四次曲线时遇到的。如果x, y, z发生任何排列，则式(1)左侧的三元四次形式不改变;如果x, y, z中的任何一个或全部乘以- 1，它也不会改变。这样就产生了一个三元共线的八面体群0，其中铅笔的每条曲线都是不变的。因为(1)也可以写
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Pub Date : 1900-01-01DOI: 10.1017/S0950184300002974
A. Waterson
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+
本文给出了前n个整数的r次幂和的显式表达式。该结果与众所周知的伯努利数的结果是等价的，并且这种等价性不难建立。然而，这里给出的方法是基本的和独立的，并提供了一个很好的练习。行列式的操作。在整个笔记中，将使用以下符号:S r S l ' + 2 ' +…+ n, n f=n(n- 1)(n-2)…(n-r+
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