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Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences最新文献

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A note on the time dependence of the effective axis and angle of a rotation 关于有效轴和旋转角度的时间依赖性的说明
Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences Pub Date : 1971-07-01 DOI: 10.6028/JRES.075B.002
H. Gelman
{"title":"A note on the time dependence of the effective axis and angle of a rotation","authors":"H. Gelman","doi":"10.6028/JRES.075B.002","DOIUrl":"https://doi.org/10.6028/JRES.075B.002","url":null,"abstract":"The time dependent rotation of one orthogonal coordinate system with respect to a fixed one is considered in the parametrization based on the effective axis and angle of the rotation, a parametri zation which has recently been used to discuss the irreducible representations of the rotation group. The method of the intrinsic vector is used to derive the equations of motion fur the instantaneous effective axis and angle. A new representation of the angular velocity is obtained in a rotating or· thogonal coordinate system generated by a unit vector along the effective axis, and a new geometrical interpretation of the effective angle is given.","PeriodicalId":166823,"journal":{"name":"Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences","volume":"58 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1971-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130915351","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: 10
SOME ELEMENTARY FORMULAS IN "MATRIX CALCULUS" AND THEIR APPLICATIONS “矩阵演算”中的几个基本公式及其应用
Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences Pub Date : 1971-07-01 DOI: 10.6028/JRES.075B.001
J. Fong
{"title":"SOME ELEMENTARY FORMULAS IN \"MATRIX CALCULUS\" AND THEIR APPLICATIONS","authors":"J. Fong","doi":"10.6028/JRES.075B.001","DOIUrl":"https://doi.org/10.6028/JRES.075B.001","url":null,"abstract":"A COLLECTION OF ELEMENTARY FORMULAS FOR CALCULATING THE GRADIENTS OF SCALAR- AND MATRIX-VALUED FUNCTIONS OF ONE MATRIX ARGUMENT IS PRESENTED. USING SOME OF THE WELL-KNOWN PROPERTIES OF THE OPERATOR \"TRACE\" ON SQUARE MATRICES, ALTERNATIVE DEFINITIONS OF GRADIENTS AND SIMPLE EXAMPLES OF CALCULATING THEM USING THE PRODUCT RULE AND THE CHAIN RULE FOR DIFFERENTIATION ARE TREATED IN AN EXPOSITORY FASHION IN BOTH COMPONENT AND MATRIX NOTATIONS WITH EMPHASIS ON THE LATTER. TWO EXAMPLES IN CONTINUUM MECHANICS ARE PRESENTED TO ILLUSTRATE THE APPLICATON OF THE SO-CALLED \"MATRIX CALCULUS\" OF DIFFERENTIABLE FUNCTIONS. /JR-MS/","PeriodicalId":166823,"journal":{"name":"Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences","volume":"57 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1971-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121485889","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
Some Theorems on Tensor Composite Graphs 张量复合图的几个定理
Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences Pub Date : 1970-10-01 DOI: 10.6028/JRES.074B.021
M. Capobianco
{"title":"Some Theorems on Tensor Composite Graphs","authors":"M. Capobianco","doi":"10.6028/JRES.074B.021","DOIUrl":"https://doi.org/10.6028/JRES.074B.021","url":null,"abstract":"","PeriodicalId":166823,"journal":{"name":"Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences","volume":"56 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1970-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129286804","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
Error estimates for the solution of linear algebraic systems 线性代数系统解的误差估计
Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences Pub Date : 1970-10-01 DOI: 10.6028/JRES.074B.024
Brother Kenneth E. Fitzgerald
{"title":"Error estimates for the solution of linear algebraic systems","authors":"Brother Kenneth E. Fitzgerald","doi":"10.6028/JRES.074B.024","DOIUrl":"https://doi.org/10.6028/JRES.074B.024","url":null,"abstract":"In thi s paper bounds for th e e rror of a computed inverse of a matrix a re developed. These are th e n app lied to th e solution of a s in gle sys tem. Methods for improving th e app rox imate inverse are th e n discussed with so me observa tions on th e dange rs involved in the ir pract ical use on a co mpute r a nd so me sa feguard s a re indica ted. So me compute r p rogra ms for matrix invers ion are th en evaluated by means of th e bound s deve loped.","PeriodicalId":166823,"journal":{"name":"Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1970-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127096924","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
A property of triangle groups 三角形群的一个性质
Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences Pub Date : 1970-10-01 DOI: 10.6028/JRES.074B.020
J. Lehner
{"title":"A property of triangle groups","authors":"J. Lehner","doi":"10.6028/JRES.074B.020","DOIUrl":"https://doi.org/10.6028/JRES.074B.020","url":null,"abstract":"The F .groups a re the g roups possessing fai t hful rep rese nt at io ns by Fuchsian groups of the firs t kind ; the ir presenta tions a re known eXI)licit ly. Amo ng the F ·groups a re the well · know n triangle groups C = {x, ylx\"=Y\"= {xy}'·= l}. H p , q, r are dis tin ct pri me intege rs, eve ry prope r norm a l su bgroup of fin it e inde x in C has no e le me nts of fini te order. In t hi s pape r it is proved that a mong the F ·groups onl y the triangle groups with di s tinct prime p , q, r have thi s prope rt y.","PeriodicalId":166823,"journal":{"name":"Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1970-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132485865","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
The Diophantine Approximation of Roots of Positive Integers 正整数根的丢番图近似
Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences Pub Date : 1970-10-01 DOI: 10.6028/JRES.074B.022
C. Osgood
{"title":"The Diophantine Approximation of Roots of Positive Integers","authors":"C. Osgood","doi":"10.6028/JRES.074B.022","DOIUrl":"https://doi.org/10.6028/JRES.074B.022","url":null,"abstract":"Rece nt ly Schinzel [4] I and Dave nport [1] have each obtain ed a res u lt of the followin g so rt : Let a be a real algeb raic numbe r of degree r ~ 2. Le t s be a positive real numbe r large r than s( r), whe re for Daven port r > s( r ) = tr+ 0(1 ) > t r while for Schinzel s( r) = 3(r/2) 1/2 Th e n th ere ex is ts an e ffectively co mputable positive integer qo(a , s) s uc h th at , with a t mos t one exce p· tion, every pair of relatively prim e integers p and q with q ~ qo (a) sa tis fi es th e in equ ality","PeriodicalId":166823,"journal":{"name":"Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences","volume":"404 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1970-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116493920","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: 3
The minimum number of problems to cover all subproblems 覆盖所有子问题的最小问题数
Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences Pub Date : 1970-10-01 DOI: 10.6028/jres.074b.023
H. J. Greenberg
{"title":"The minimum number of problems to cover all subproblems","authors":"H. J. Greenberg","doi":"10.6028/jres.074b.023","DOIUrl":"https://doi.org/10.6028/jres.074b.023","url":null,"abstract":"Th e fo ll ow ing proble m is motiva te d and the n so lved, using the theo ry of sys tems of di s tinct re presentatives . Le t 111 = {I . 2 . . , III}. a nd for each sequ e nce u of di s tinc t me mbers of 111, le t (u ) be th e assoc ia te d subse t. S uppose give n a mathe mati ca l probl e m 1' (5) for e ac h s ubset 5 of 111, and a n a lgo rithm A whi c h whe n a ppli ed to u so lves not on ly 1'( (u» b ut a lso all 1' ( (0») where T is an initi a l segme nt of u. What is the s malles t numbe r of a pplicat ions of A need ed to solve the e ntire e nse mble of prob le ms { p (S):S C ilf }?","PeriodicalId":166823,"journal":{"name":"Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences","volume":"171 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1970-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121829500","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
On the singular values of a product of matrices 关于矩阵乘积的奇异值
Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences Pub Date : 1970-10-01 DOI: 10.6028/JRES.074B.025
W. Watkins
{"title":"On the singular values of a product of matrices","authors":"W. Watkins","doi":"10.6028/JRES.074B.025","DOIUrl":"https://doi.org/10.6028/JRES.074B.025","url":null,"abstract":"The purpose of this note is to give necessary and sufficient conditions for the singular values of a product of matrices to be equal to certain products of their singular values. We then analyze the case of equa]jty in a matrix inequality of Os trows ki . Th e s ingular values of an n·square co mplex matrix X are the positive square roots of the eigen· values of X*X, where X* is the conjugate tran spose of X. Denote the singular values of X by a t (X) , ... , an (X), arranged so that al (X) ~ ... ~ an(X) > 0 (all matrices are assumed to be nonsi ngular). Let A and B be n,·square complex matri ces and let A = UH, B = VK be the polar factorizations of A a nd B. In the factorization s U and V are unitary matrices and Hand K are positive·definit e hermitian matri ces. THEOREM 1: Let k be a positive integer less than n. Then","PeriodicalId":166823,"journal":{"name":"Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences","volume":"69 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1970-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121943669","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
Selecting nonlinear transformations for the evaluation of improper integrals 选择非线性变换求反常积分
Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences Pub Date : 1970-07-01 DOI: 10.6028/jres.074b.017
T. Atchison
{"title":"Selecting nonlinear transformations for the evaluation of improper integrals","authors":"T. Atchison","doi":"10.6028/jres.074b.017","DOIUrl":"https://doi.org/10.6028/jres.074b.017","url":null,"abstract":"Recent lit erature concerning the use of nonlinear transformations to evaluate numeri ca lly certain improper integral s of the first kind involves the determination of a transformation function g to im· prove the approxim ation. By approximating a given integrandIby an integrab le fun c tion II and then determining an associated g function for II , a nonlinear transformation may be constructe d which will yie ld an improved a pproximation of the improper integral of;:","PeriodicalId":166823,"journal":{"name":"Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences","volume":"56 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1970-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133434493","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
On the spheroidal functions 关于球面函数
Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences Pub Date : 1970-07-01 DOI: 10.6028/JRES.074B.018
D. Rhodes
{"title":"On the spheroidal functions","authors":"D. Rhodes","doi":"10.6028/JRES.074B.018","DOIUrl":"https://doi.org/10.6028/JRES.074B.018","url":null,"abstract":"","PeriodicalId":166823,"journal":{"name":"Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences","volume":"382 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1970-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116060637","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: 18
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