{"title":"Latent Roots of Tri-Diagonal Matrices","authors":"F. Arscott","doi":"10.1017/S095018430000330X","DOIUrl":"https://doi.org/10.1017/S095018430000330X","url":null,"abstract":"A considerable amount is known about the latent roots of matrices of the form in the case when each cross-product of non-diagonal elements, a i c i-1 , is positive. One forms the sequence of polynomials f r (λ) = |L r −λI| for r = 1, 2, … n , and observes that then it is easy to deduce that (i) the zeros of f n (λ) and f n_1 (λ) interlace—that is, between two consecutive zeros of either polynomial lies precisely one zero of the other (ii) at the zeros of f n (λ) the values of f n-x (λ) are alternately positive and negative, (iii) all the zeros of f n (λ) — i.e. all the latent roots of L n —are real and different.","PeriodicalId":417997,"journal":{"name":"Edinburgh Mathematical Notes","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1961-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124015463","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":"Inertia Invariants of a Set of Particles","authors":"N. Slater","doi":"10.1017/S0950184300003293","DOIUrl":"https://doi.org/10.1017/S0950184300003293","url":null,"abstract":"","PeriodicalId":417997,"journal":{"name":"Edinburgh Mathematical Notes","volume":"70 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1961-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130999019","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":"The Existence of Integrals of Dynamical Systems Linear in the Velocities","authors":"C. Kilmister","doi":"10.1017/S0950184300003323","DOIUrl":"https://doi.org/10.1017/S0950184300003323","url":null,"abstract":"A dynamical system means here a system specified by generalised coordinates q α (α = 1, 2, …, n) and a Lagrangian L which is a quadratic polynomial in the generalised velocities, say (with a summation convention).","PeriodicalId":417997,"journal":{"name":"Edinburgh Mathematical Notes","volume":"2009 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1961-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125628055","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":"An Alternative Proof of a Theorem on the Lebesgue Integral","authors":"B. Josephson","doi":"10.1017/S0950184300003232","DOIUrl":"https://doi.org/10.1017/S0950184300003232","url":null,"abstract":"By (2), F(x)= f' is a.c. Hence 3<5,>O such that if {[ar, br)} is a Ja finite set of non-overlapping intervals and T.(b, — ar)<Su then ZF(br)-F(ar) <ie (4) By uniform continuity, 3<52>O such that if | br — ar | <<52, then Abr)-Kar) | <E (5) Now take 8 = min^ , S2, I, e/SK), and choose intervals [ar, br] to satisfy (3). The sum Z f(b^-f{ar) | may be divided into three parts, by putting fibr)-f(ar) into S, if/isa.c. in [ar, br], Z2 if/is not a.c. in [ar, b,] and f(br)-f(ar) ^K(br-ar), S3 if/is not a.c. in [ar, br] and f(br)-f(ar) >K(br-ar). Now if/is a.c. in [ar, br], then by (1), ,)-/(«,) | = I f\"f ^ T I / ' I = I F{br)F(a,) |. I Jar Jar E.M.S.—H J","PeriodicalId":417997,"journal":{"name":"Edinburgh Mathematical Notes","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1960-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129838596","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":"Linkages for the Trisection of an Angle and Duplication of the Cube","authors":"G. Stokes","doi":"10.1017/S0950184300003220","DOIUrl":"https://doi.org/10.1017/S0950184300003220","url":null,"abstract":"In this note some linkage systems for trisecting an angle and for finding the cube root of a number are described. The models are easily made and are of considerable pedagogic value","PeriodicalId":417997,"journal":{"name":"Edinburgh Mathematical Notes","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1960-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124486663","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":"Linear Ordinary Differential Equations with Constant Coefficients: Identification of Boole's Integral with that of Cauchy","authors":"D. H. Parsons","doi":"10.1017/S0950184300003268","DOIUrl":"https://doi.org/10.1017/S0950184300003268","url":null,"abstract":"","PeriodicalId":417997,"journal":{"name":"Edinburgh Mathematical Notes","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1960-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133077963","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 Properties of the Zeros of Bessel Functions","authors":"L. Chambers","doi":"10.1017/S095018430000327X","DOIUrl":"https://doi.org/10.1017/S095018430000327X","url":null,"abstract":"Let j nm be the m th positive zero of J n ( x ) ( n not necessarily integral). Then Relton (1), p. 59, has conjectured from numerical considerations that","PeriodicalId":417997,"journal":{"name":"Edinburgh Mathematical Notes","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1960-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128658387","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":"Linear Partial Differential Equations with Constant Coefficients: an Elementary Proof of an Existence Theorem","authors":"D. H. Parsons","doi":"10.1017/S0950184300003190","DOIUrl":"https://doi.org/10.1017/S0950184300003190","url":null,"abstract":"and any solution of R(DV ...,Dm)z = 0 (3) is also a solution of (1). The converse proposition, that every integral of (1) is the sum of an integral of (2) and an integral of (3), was postulated by Hadamard (1), in the case of m = 2, for linear equations with constant or variable coefficients, provided only that the two operators Q, R are commutative. This result was established by Cerf (2) and by Janet (3), who extended it to a very general case which certainly includes that under consideration here. The proof of the general theorem is not simple, however ; and in the case mentioned below (§3), in which the equation is fully reducible, most textbooks are content to assume the result without proof. We shall now give a purely elementary proof of this converse theorem, in the case when one of the factors of P, R say, is a power of a linear expression in Dv ..., Dm, which is not a factor of Q. We shall make the hypothesis that any partial differential equation of the form T(DV ...,Dm)z=4>{xl,...,xn) admits at least one integral, provided only that <f> satisfies sufficient conditions of continuity, and that the symbolic polynomial T is not identically zero. By suitable labelling, we may ensure that the linear factor of P contains Dv Thus let the equation considered be {(Dx-a2D2-...-amDm-bYQ(Dv ..., Dm)}z = 0, (4)","PeriodicalId":417997,"journal":{"name":"Edinburgh Mathematical Notes","volume":"61 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1959-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134186245","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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