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A note on multiplicative property of the Smith normal form 关于史密斯范式的乘法性质的注记
Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences Pub Date : 1972-07-01 DOI: 10.6028/JRES.076B.016
M. Marcus, E. E. Underwood
{"title":"A note on multiplicative property of the Smith normal form","authors":"M. Marcus, E. E. Underwood","doi":"10.6028/JRES.076B.016","DOIUrl":"https://doi.org/10.6028/JRES.076B.016","url":null,"abstract":"In [2; p. 33]1 the following interesting result appears: If A and Bare n-square matrices over a principal ideal domain Rand g.c.d. (det (A), det (B»= 1 then S(AB)= S(A)S(B) where -SeA) is the Smith normal form of A. The purpose of this note is to present a simple proof of the result that uses elementary properties of compound matrices. LEMMA. Let Q=diag (q., ... , qn), P=diag (p., ... , Pn), qllqj' Pllpj,j=l, ... , nand g.c.d. (Pb qJ)= 1, i, j= 1, . .. , n. Let U be an n-square matrix with the property that g.c.d. (Ull' U2!, _ .. , Unl)= g.c.d. (Ull' U12, ... , Utn)= 1. Then the g.c.d. of all the entries in QUP is Plqt. PROOF. Obviously PlqlIQUP, i.e., Ptql divides every entry of QUP. Write QUP= PlqtD. Suppose that plD where p is a prime. It is simple to see that the first row and column of Dare respectively","PeriodicalId":166823,"journal":{"name":"Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences","volume":"170 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1972-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121342929","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
Distance coordinates with respect to a triangle of reference 相对于参考三角形的距离坐标
Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences Pub Date : 1972-07-01 DOI: 10.6028/JRES.076B.010
K. Goldberg
{"title":"Distance coordinates with respect to a triangle of reference","authors":"K. Goldberg","doi":"10.6028/JRES.076B.010","DOIUrl":"https://doi.org/10.6028/JRES.076B.010","url":null,"abstract":"With respect to a triangle of reference A IA 2A3 • each point P in the plane of the triangle . has unique area coordinates: P = (b l • b, . b3 ) with 6 1 + 'b2 + b3 = 1. Distance coordinates are introduced such that P= (d l • d 2 • d3 ], with d,. the di stance from P to A k • It is shown that there is an' explicit function [(XI. X,. X3) such thatf(d;. d~. d5) = 0 is necessary and sufficient for P = [d l • d2 • d3 ], each d k nonnegative. The partial derivatives fdxI. X2. X3) = a[(x l . X2. X 3 ) laxk are such that b,. = fdd;. d;' d~) for each k. Other results relating the bk and the dj,' are given. The use of f(x\" x •• X3) in solving geometric problems is shown.","PeriodicalId":166823,"journal":{"name":"Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1972-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132336822","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
Normal matrices with entries from an arbitrary field of characteristic not-equal 2 条目来自特征不等于2的任意域的正规矩阵
Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences Pub Date : 1972-07-01 DOI: 10.6028/JRES.076B.009
M. Pearl, A. Penn
{"title":"Normal matrices with entries from an arbitrary field of characteristic not-equal 2","authors":"M. Pearl, A. Penn","doi":"10.6028/JRES.076B.009","DOIUrl":"https://doi.org/10.6028/JRES.076B.009","url":null,"abstract":"","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":"1972-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130929421","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 TABLE OF INTEGRALS OF THE ERROR FUNCTION. II, ADDITIONS AND CORRECTIONS 误差函数的积分表。二、补充和更正
Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences Pub Date : 1971-12-01 DOI: 10.6028/JRES.075B.003
M. Geller, E. W. Ng
{"title":"A TABLE OF INTEGRALS OF THE ERROR FUNCTION. II, ADDITIONS AND CORRECTIONS","authors":"M. Geller, E. W. Ng","doi":"10.6028/JRES.075B.003","DOIUrl":"https://doi.org/10.6028/JRES.075B.003","url":null,"abstract":"Integrals of products of error functions with other functions are presented, taking into account a combination of the error function with powers, a combination of the error function with exponentials and powers, a combination of the error function with exponentials of more complicated arguments, definite integrals from Laplace transforms, and a combination of the error function with trigonometric functions. Other integrals considered include a combination of the error function with logarithms and powers, a combination of two error functions, and a combination of the error function with other special functions.","PeriodicalId":166823,"journal":{"name":"Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences","volume":"29 10","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1971-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132359983","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: 23
Linear transformations on matrices 矩阵上的线性变换
Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences Pub Date : 1971-07-01 DOI: 10.6028/JRES.075B.008
M. Marcus
{"title":"Linear transformations on matrices","authors":"M. Marcus","doi":"10.6028/JRES.075B.008","DOIUrl":"https://doi.org/10.6028/JRES.075B.008","url":null,"abstract":"","PeriodicalId":166823,"journal":{"name":"Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences","volume":"109 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":"115118887","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: 66
An ergodic source which is not a function of any finite Markov process 不是任何有限马尔可夫过程函数的遍历源
Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences Pub Date : 1971-07-01 DOI: 10.6028/JRES.075B.005
R. Traub
{"title":"An ergodic source which is not a function of any finite Markov process","authors":"R. Traub","doi":"10.6028/JRES.075B.005","DOIUrl":"https://doi.org/10.6028/JRES.075B.005","url":null,"abstract":"","PeriodicalId":166823,"journal":{"name":"Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences","volume":"96 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":"123990651","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
On entire functions of exponential type 关于指数型的整个函数
Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences Pub Date : 1971-07-01 DOI: 10.6028/JRES.075B.004
S. Shah, W. Sisarcick
{"title":"On entire functions of exponential type","authors":"S. Shah, W. Sisarcick","doi":"10.6028/JRES.075B.004","DOIUrl":"https://doi.org/10.6028/JRES.075B.004","url":null,"abstract":"Le t J be a n en tire fun c ti o n a nd le t p. \"\" 1 and 1(1 , r)= {f:~ 1 f 11)(re iO) I \"dO} 1/\". for a U s uffic ie ntly la rge r, the n J is of ex pone nti a l type not exceeding. {2 log (l-t. ~) + 1 + log (2N) !} .. If thi s co ndition is re place d by re lated co nditi ons, th e n a lso is of expo ne nti a l t ype. An e ntire fun c tion f(z) is said to be of bounde d ind ex if and only if th ere exi sts a non-negative integer N (ind e pe nde nt of z) s uch thatO\"'j \",N j!-k! (1.1) for all k and all z, and the smallest s uc h integer N is calle d the index off(z) ([1], [4] , [5]).1 It is known that a function of bounde d inde x N is of ex ponenti al type not exceedin g N+ 1 [6] but that a function of expon e ntial type need not be of bounde d inde x. In fac t any e ntire fun c tion havin g ze ros of arbitrarily large multipli city is not of bound e d index and th ere exist fun c tion s with simple zeros and of exponential type whic h are not of bounded index [8]. In a recent paper [2] Fred Gross considers interesting variations of condition (1.1) and proves the following THEOREM A: Let f be entire and C a positive constant. If there exists a positive integer N such that for k=O, 1,. . , N, f satisfies one ofthefollowing,for all z with l z I sufficiently large:","PeriodicalId":166823,"journal":{"name":"Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences","volume":"60 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":"128578260","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: 15
Eigenvalues of sums of Hermitian matrices .III. 厄米矩阵和的特征值。iii。
Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences Pub Date : 1971-07-01 DOI: 10.6028/JRES.075B.007
R. C. Thompson, L. F. Garbanati
{"title":"Eigenvalues of sums of Hermitian matrices .III.","authors":"R. C. Thompson, L. F. Garbanati","doi":"10.6028/JRES.075B.007","DOIUrl":"https://doi.org/10.6028/JRES.075B.007","url":null,"abstract":"","PeriodicalId":166823,"journal":{"name":"Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences","volume":"36 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":"133513278","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: 14
A numerical solution of integral equation of the first kind applied to slit correction in small-angle x-ray scattering 第一类积分方程的数值解应用于小角x射线散射缝缝校正
Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences Pub Date : 1971-07-01 DOI: 10.6028/JRES.075B.009
J. Mazur
{"title":"A numerical solution of integral equation of the first kind applied to slit correction in small-angle x-ray scattering","authors":"J. Mazur","doi":"10.6028/JRES.075B.009","DOIUrl":"https://doi.org/10.6028/JRES.075B.009","url":null,"abstract":"","PeriodicalId":166823,"journal":{"name":"Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences","volume":"37 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":"122684104","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 new type of boundary value coupling for second order Sturm-Liouville systems 一类新的二阶Sturm-Liouville系统边值耦合
Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences Pub Date : 1971-07-01 DOI: 10.6028/jres.075b.006
John E. Geist
{"title":"A new type of boundary value coupling for second order Sturm-Liouville systems","authors":"John E. Geist","doi":"10.6028/jres.075b.006","DOIUrl":"https://doi.org/10.6028/jres.075b.006","url":null,"abstract":"A natural genera lization of the familiar second order Sturm-Liouville syste m is presented. T his gene rali za tion co ns is ts of considering a number of differe ntia l equ ations defined on difTerent inte rvals, instead of ju st one equation on one inte rva l. The self-adjoint characte r of the diffe rential equations is re ta ined in th e gene rali za tion, but th e boundary conditions a re re laxed conside ra bl y. The mos t gene ral boundary cond itions which can be accommod ated by thi s so rt of gene ralization of S turm-Liouvi ll e theory a re di scussed. Th e exis te nce of e igenva lues is proved , and a gene ra lized orthogonalit y and a weak e igenfun ction expansion theorem are de rived.","PeriodicalId":166823,"journal":{"name":"Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences","volume":"12 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":"114054842","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
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