Pub Date : 1961-12-01DOI: 10.1017/S095018430000330X
F. Arscott
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.
当非对角线元素的每个叉乘(A ic i-1)为正时，对于这种形式的矩阵的潜根，我们已经知道了相当多的信息。1形式的多项式序列f r(λ)= | L r−λ我| r = 1, 2,…n,然后发现,很容易推断出(I)的0 n(λ)和f n_1(λ)interlace-that,连续两个零多项式的谎言一个零的其他(ii)的0 n的值(λ)f n *(λ)交替积极和消极,(iii)的0 f n(λ)——即所有的潜在根源L n——真正的不同。
{"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}
Pub Date : 1961-12-01DOI: 10.1017/S0950184300003293
N. Slater
{"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}
Pub Date : 1961-12-01DOI: 10.1017/S0950184300003323
C. Kilmister
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).
{"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}
Pub Date : 1960-12-01DOI: 10.1017/S0950184300003232
B. Josephson
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)O such that if | br — ar | <<52, then Abr)-Kar) | 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
通过(2)，F(x)= F '是a.c。因此30o使得如果{[ar, br)}是一个非重叠区间的Ja有限集，t (b， -ar) O使得如果| br-ar | K(br-ar)。现在如果/交流(ar, br),然后由(1 ), ,)-/(«,) | = 我“f ^ T I / '我= f f (a) | {br)。I Jar Jar emms - h J
{"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}
Pub Date : 1960-12-01DOI: 10.1017/S0950184300003220
G. Stokes
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
{"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}
Pub Date : 1960-12-01DOI: 10.1017/S0950184300003268
D. H. Parsons
{"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}
Pub Date : 1960-12-01DOI: 10.1017/S095018430000327X
L. Chambers
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
让m为n (x)的m为0(不需要不可或缺)。然后Relton (1)， p. 59，从nu梅里卡尔的考虑中得到了反映
{"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}
Pub Date : 1959-11-01DOI: 10.1017/S0950184300003190
D. H. Parsons
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 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)
{"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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