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On certain discrete inequalities and their continous analogues 若干离散不等式及其连续类似物
Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics Pub Date : 1966-07-01 DOI: 10.6028/JRES.070B.021
A. M. Pfeffer
{"title":"On certain discrete inequalities and their continous analogues","authors":"A. M. Pfeffer","doi":"10.6028/JRES.070B.021","DOIUrl":"https://doi.org/10.6028/JRES.070B.021","url":null,"abstract":"In a 1955 paper, Ky Fan, Olga Taussky, and John Todd presented discrete analogues of inequalities of Wirtinger type, and by taking limits they were able to recover the continuous inequalities. We generalize their techniques to mixed and higher derivatives and inequalities with weight functions in the integrals. We have also considered analogues of inequalities of Muller and Redheffer and have used these inequalities to derive a necessary and sufficient condition on ordered pairs of numbers so that the first number is the square norm of the k th derivative of some periodic function and the second number is the square norm of the m th derivative of the same periodic function.","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1966-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124358670","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}
引用次数: 20
Finding a rank-maximizing matrix block 寻找一个秩最大化矩阵块
Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics Pub Date : 1966-07-01 DOI: 10.6028/jres.070b.020
A. J. Goldman, M. Newman
{"title":"Finding a rank-maximizing matrix block","authors":"A. J. Goldman, M. Newman","doi":"10.6028/jres.070b.020","DOIUrl":"https://doi.org/10.6028/jres.070b.020","url":null,"abstract":"","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1966-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125064795","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}
引用次数: 0
Error bounds for asymptotic solutions of differential equations.I. Distinct eigenvalue case 微分方程渐近解的误差界。不同特征值情况
Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics Pub Date : 1966-07-01 DOI: 10.6028/JRES.070B.017
F. Stenger
{"title":"Error bounds for asymptotic solutions of differential equations.I. Distinct eigenvalue case","authors":"F. Stenger","doi":"10.6028/JRES.070B.017","DOIUrl":"https://doi.org/10.6028/JRES.070B.017","url":null,"abstract":"The method of Olver for bounding the error term in the asymptotic so lutions of a second-order equation having a n irregular singul arity at infinity is extended to the general system of n first-order equations in the case when the eigenvalu es of the lead coefficient matrix are distinct. Vector and norm bounds are given for the difference between an actual solution vector and a partial su m of a formal so lution vector. Two cases are distinguished geometrica ll y: In one it is possible to express the error vec tor by a s ingle Volterra vec tor integral equat.ion; in the other it is necessary to use a simultaneous pair of Volterra vector integral equations. Some ne w inequalities for integral equations are given in an append ix.","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1966-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132514530","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}
引用次数: 9
Error bounds for asymptotic solutions of differential equations. 2. The general case 微分方程渐近解的误差界。2. 一般情况下
Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics Pub Date : 1966-05-01 DOI: 10.6028/JRES.070B.018
F. Stenger
{"title":"Error bounds for asymptotic solutions of differential equations. 2. The general case","authors":"F. Stenger","doi":"10.6028/JRES.070B.018","DOIUrl":"https://doi.org/10.6028/JRES.070B.018","url":null,"abstract":"","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1966-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134240575","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}
引用次数: 5
Pairs of nonsingular matrices 非奇异矩阵对
Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics Pub Date : 1966-04-01 DOI: 10.6028/JRES.070B.014
K. Goldberg
{"title":"Pairs of nonsingular matrices","authors":"K. Goldberg","doi":"10.6028/JRES.070B.014","DOIUrl":"https://doi.org/10.6028/JRES.070B.014","url":null,"abstract":"Let Plt llln denote the se t of m by n matrices of rank In, with e ntries from a field of characteri sti c O. Given REPltmn let .IV (R) denote the set of 5EPltnm , n such that R5T = O. Note thal 5E .IV(R) if and only if RE.!V (5) , We assume as known the following results for REfYtmn and 5E.IV (R): LEMMA 1. XR = 0 if and only if X = O. LEMMA 2. RYT = 0 if and only if Y = ZS , for some matrix Z. The obvious stipulation s are that X, Y, and Z have m,n and n-In column s respec tively, and Y and Z have the same number of rows . Using these lemmas we prove: THEOREM. If R\" R2EPltmn ' StEff (R,) , S2Eff (R2) then R,RJ is nonsingular if and only if SISi' LS nonsingular, in which case","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1966-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130211047","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}
引用次数: 1
Treatment of outliers in samples of size three 大小为3的样本中异常值的处理
Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics Pub Date : 1966-04-01 DOI: 10.6028/JRES.070B.011
F. J. Anscombe, B. A. Barron
{"title":"Treatment of outliers in samples of size three","authors":"F. J. Anscombe, B. A. Barron","doi":"10.6028/JRES.070B.011","DOIUrl":"https://doi.org/10.6028/JRES.070B.011","url":null,"abstract":"A reading that is a long way from most of the others in a series of replicate determinations is called an outlier. A particular procedure for rejecting outliers, and a lso a particular procedure for modifying outliers , are considered for samples of size three, s upposed drawn from a common normal population except that one of the three readings may have an added bias . Numerical results are given illus· trating the effects of the procedures on es timation of the location paramete r. The calculations support a tentative general conclus ion that estimation by least squares shou ld us ually be tempered by suc· cessive application of both a rejec tion rule and a modification rule.","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"89 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1966-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132773868","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}
引用次数: 11
On Abstract Numerical Integrations 关于抽象数值积分
Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics Pub Date : 1966-04-01 DOI: 10.6028/JRES.070B.010
J. Sopka
{"title":"On Abstract Numerical Integrations","authors":"J. Sopka","doi":"10.6028/JRES.070B.010","DOIUrl":"https://doi.org/10.6028/JRES.070B.010","url":null,"abstract":"Let X be a s pace of func tions, say X c C(K), K locally co mpact Hausdorff, le t IEX* be an int~gral on X and le t M* C X* be a given subspace of \"simple\" functiona ls, then it is des ired to obta in!!n leM~ for given n, leM\";, C M*; M\";, bein g a su itable\" dim ens ion a l s llb s pace d e t e rmin ed so th a t: I I annihi lates a given finit e dim ensiona l subs pace X, C X. In thi s ge neral context , the abst ract ana lysis of nllm e ri ca l integration is de ve loped and ce rtain speci fi c app licatio ns are made.","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1966-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125301285","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}
引用次数: 0
Upper bounds for determinant of a row stochastic matrix 行随机矩阵行列式的上界
Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics Pub Date : 1966-04-01 DOI: 10.6028/JRES.070B.015
K. Goldberg
{"title":"Upper bounds for determinant of a row stochastic matrix","authors":"K. Goldberg","doi":"10.6028/JRES.070B.015","DOIUrl":"https://doi.org/10.6028/JRES.070B.015","url":null,"abstract":"","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1966-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121839200","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}
引用次数: 2
A Note on Contaminated Samples of Size Three 3号受污染样品的说明
Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics Pub Date : 1966-04-01 DOI: 10.6028/JRES.070B.012
T. A. Willke
{"title":"A Note on Contaminated Samples of Size Three","authors":"T. A. Willke","doi":"10.6028/JRES.070B.012","DOIUrl":"https://doi.org/10.6028/JRES.070B.012","url":null,"abstract":"Estimation of the mean a nd s ta nd ard d e viati on us in g th e c lo sest two of three observations in a sample fro m a normal popul ati on with conta min ati on by s lippage of the mean is investi gated by a sampling s tud y. Li ebl e in 's res ult s , whi ch indicated th a t the use of th ese s tatistics is not advisable for noncont a mina ted sa mples, a re born e out b y thi s stud y for cont a min a t.e d sa mples a s we ll. Key Word s : Outli e rs, cont a min a te d samples, robus t estimates, best two of three.","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1966-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125122125","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}
引用次数: 4
On a sequence of points of interest for numerical quadrature 关于数值求积分的兴趣点序列
Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics Pub Date : 1966-04-01 DOI: 10.6028/JRES.070B.009
S. Haber
{"title":"On a sequence of points of interest for numerical quadrature","authors":"S. Haber","doi":"10.6028/JRES.070B.009","DOIUrl":"https://doi.org/10.6028/JRES.070B.009","url":null,"abstract":"(where XI I), X2) •• • ,xIS) are the S coo rdinates of the point X i), we call R the \"error\" of the quadrature form ula (1). R depends on the integrand f and on the se t of points XI, ... ,XN; for R to be r ela ti vely small for som e wide class of functions th e se t of points s hould be, in so me se nse, we ll distributed in the cube Cs . An appropriate se nse of \" we ll di st ributed\" is that , for a regio n A in Cs , the numbe r of points of th e se t which li e in A s hould be approximately N times the volume of A. Restric ting attention to regions which are inte rval s with one vertex at th e origin-i.e. , which are of th e form","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1966-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122584347","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}
引用次数: 20
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