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.
在1955年的一篇论文中，Ky Fan, Olga Taussky和John Todd提出了Wirtinger型不等式的离散类比，通过取极限，他们能够恢复连续不等式。我们将他们的技巧推广到积分中带有权函数的混合和高阶导数和不等式。我们还考虑了Muller和Redheffer的类似不等式，并利用这些不等式推导了有序数对的充分必要条件，使第一个数是某个周期函数的第k阶导数的平方范数，第二个数是同一周期函数的第m阶导数的平方范数。
{"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}
{"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}
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.
{"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}
{"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}
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
设Plt . iv (R)表示m由n个秩为In的矩阵组成的集合，其中e个元素来自特征为o的域。给定Plt . iv (R)表示5EPltnm的集合n，使得R5T = o。注意，5E . iv (R)当且仅当RE。V(5)，我们假设REfYtmn和5E的结果如下:Iv (r):引理1。当且仅当X = 0时，XR = 0。RYT = 0当且仅当Y = ZS，对于某个矩阵Z。明显的规定是X, Y, Z分别有m,n,n - in列，Y和Z有相同的行数。利用这些引理，我们证明:定理。如果R ' R2EPltmn ' StEff (R，)， S2Eff (R2)，则R,RJ是非奇异当且仅当SISi' LS非奇异，在这种情况下
{"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}
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.
{"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}
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.
设X是一个函数的空间，比如X c c (K)， K局部紧合Hausdorff，设X X*是X上的整数积分，设M* c X*是“简单”函数s的给定子空间，那么我们就想要得到它!!n leM~ for给定n, leM ';， cm *;M”，作为一种合适的“空间”，在空间X，空间X，空间X下，空间X，空间X，空间X下，空间X，空间X，空间X，空间X，空间X，空间X，空间X，空间X，空间X，空间X，空间X，空间X，空间X，空间X，空间X，空间X，空间X，空间X，空间X，空间X，空间X，空间X，空间X，空间X，空间X，空间X，空间X，空间X，空间X，空间X，空间X。
{"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}
{"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}
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.
估计均值和s ta nd ard d e viati我们g th e c罗年代两三个观测样本从一个正常popul ati的帐目分钟ati的年代lippage意味着由抽样investi封闭的年代的立筋y。李电子e“res ult年代,whi ch表明th at th ese的使用年代tatistics不建议noncont米娜ted sa mpl,出生的再保险e b y thi s螺栓y为续一分钟t.e d sa mpl s我们噢。关键词:Outli - e - s，含最小二样本，稳健估计，三选二。
{"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}
(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
{"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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：