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Comments on the discrete matrix model of population dynamics 关于种群动力学离散矩阵模型的几点评述
Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences Pub Date : 1974-04-01 DOI: 10.6028/JRES.078B.012
Charles R. Johnson, R. Freese
{"title":"Comments on the discrete matrix model of population dynamics","authors":"Charles R. Johnson, R. Freese","doi":"10.6028/JRES.078B.012","DOIUrl":"https://doi.org/10.6028/JRES.078B.012","url":null,"abstract":"This paper exa mines several aspects of the discrete matrix model of population trans ition. Certain appropriate applications of matrix theory and exploita tion of the s pec ifi c form of the model s hould serve to enhance its already well-developed s tatus. The aspects dealt wit h in clude (1) a simplification of the Perron-Frobe nius theory; (2) row and co lumn sum bounds on maximal e igenvalues; (3) relations between osciUations in a population and the remaining e ige nvalues; (4) implications of stab ility for th e transition matrix; and (5) relation s between characte ri s ti c quantities of a sta bl e population.","PeriodicalId":166823,"journal":{"name":"Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1974-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130604445","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
A property of equivalence 等价的一个性质
Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences Pub Date : 1974-04-01 DOI: 10.6028/JRES.078B.011
M. Newman
{"title":"A property of equivalence","authors":"M. Newman","doi":"10.6028/JRES.078B.011","DOIUrl":"https://doi.org/10.6028/JRES.078B.011","url":null,"abstract":"Let R be a prin cipal id eal rin g. W e write A E B, if A and B a re matrices over R wh ich are equivale nt (see [1] for a co mple te di scussion of thi s topic). The Kronecke r product of any two matrices A and B will be de noted by A @ B. The follow in g res ult was s ugges ted by a re mark made by W. D. Wallis in hi s s urvey paper [2]: THEOREM: Suppose that K, A, Bare nonsinguLar matrices over R such that K @ A E K @ B. Then AEB. It is not actu ally necessary to assume that A and Bare nonsin gular , bu t doin g so simpljfies the exposi ti on. We firs t prove th e followin g: LEMMA: Suppose that the sets","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":"1974-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114612687","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 conjecture on a matrix group with 2 generators 具有两个生成元的矩阵群上的一个猜想
Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences Pub Date : 1974-04-01 DOI: 10.6028/JRES.078B.010
M. Newman
{"title":"A conjecture on a matrix group with 2 generators","authors":"M. Newman","doi":"10.6028/JRES.078B.010","DOIUrl":"https://doi.org/10.6028/JRES.078B.010","url":null,"abstract":"","PeriodicalId":166823,"journal":{"name":"Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences","volume":"45 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1974-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116166745","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
A new proof of pick's theorem 匹克定理的新证明
Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences Pub Date : 1974-04-01 DOI: 10.6028/jres.078b.014
S. Minsker
{"title":"A new proof of pick's theorem","authors":"S. Minsker","doi":"10.6028/jres.078b.014","DOIUrl":"https://doi.org/10.6028/jres.078b.014","url":null,"abstract":"f3 real. (See, for instance, [1], p. 224, ex. 4.) We offer the following proof whic h, although it also uses Bieberbach's result, is considerably different. PROOF: Let cP (z) J~ . Then cP sends the open unit disk into itself. Let cP 1= cP and inductively 1 a? define cPn=cP OcPlI t. If cPll(Z)=An,1 z+AIl,z Z2+ . .. ,it is clear that A\"'=M' A I ,2= M' AII ,I = A\", AIl I,t, and A II ,2 = AI ,I A nI,2 + A I,2 A;' _1.1' It follows that An,1 = A;'., and A II ,2 =A I ,zA :' ,-; 1 (l +A 1 , 1 +ALI + . . +A :',-;1). Now cPli/A 11 , 1 E!F for each n, so Bieberbach's theorem implies that IA\" ,2/A\" ,11 :s;: 2, or","PeriodicalId":166823,"journal":{"name":"Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1974-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115869752","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
Bessel functions I and J of complex argument and integer order 贝塞尔函数I和J的复数参数和整数阶
Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences Pub Date : 1973-07-01 DOI: 10.6028/JRES.077B.012
David J. Sookne
{"title":"Bessel functions I and J of complex argument and integer order","authors":"David J. Sookne","doi":"10.6028/JRES.077B.012","DOIUrl":"https://doi.org/10.6028/JRES.077B.012","url":null,"abstract":"","PeriodicalId":166823,"journal":{"name":"Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1973-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123921668","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
Certification of an algorithm for Bessel functions of complex argument 复参数贝塞尔函数算法的证明
Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences Pub Date : 1973-07-01 DOI: 10.6028/JRES.077B.013
David J. Sookne
{"title":"Certification of an algorithm for Bessel functions of complex argument","authors":"David J. Sookne","doi":"10.6028/JRES.077B.013","DOIUrl":"https://doi.org/10.6028/JRES.077B.013","url":null,"abstract":"using multiprecis ion ari thm etic [2).2 Th e 60·bit mantissa of each part of the test value was sub· tracted from the correspo ndin g manti ssa of the check value , and the res ult expressed as a multiple, m , say, of the last bit. For most e rrors mentioned in this certification . 1 m 1 ~ 7. Bit comparison was used in preference to calc ulation of relative error in order to simplify co mputa tion s. An error of m bits in the mantissa corresponds to a relative error between 171 .2 60 and 111 • 259• This tes t is too stri ct , however, near a zero of the real or im agin ary part of th e fun ction be ing tested. For co mpl ex values of z, it is more reali stic to compare the bit error with the greater of the real and im aginary parts , since the relative error in a complex number ~+ i'Y/ is (o~+ iO'Y/) /I ~+ i'Y/ I, o~ and o'Y/ de notin g th e respective errors in the real and imaginary parts. For arguments in th e neighborhoods of zeroes of Re JII (Z), 1m J,, (z), Re l ,,(z) , or 1m J,,(z), an assess me nt of absolute errors was made to 60 bin ary places by right·shifting the mantissa of both test and c heck values before subtraction. Thi s was done whenever the order n was less than 1 Z 1 and the modulus of the 1 check value less than 2\"' These cases are distinguished by asterisks in the computer printout;","PeriodicalId":166823,"journal":{"name":"Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1973-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132599563","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
Performance testing of a FORTRAN library of mathematical function routines - a case study in the application of testing techniques 数学函数例程的FORTRAN库的性能测试-测试技术应用的一个案例研究
Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences Pub Date : 1973-07-01 DOI: 10.6028/JRES.077B.011
D. Lozier, L. Maximon, W. Sadowski
{"title":"Performance testing of a FORTRAN library of mathematical function routines - a case study in the application of testing techniques","authors":"D. Lozier, L. Maximon, W. Sadowski","doi":"10.6028/JRES.077B.011","DOIUrl":"https://doi.org/10.6028/JRES.077B.011","url":null,"abstract":"Thi s pape r describes a n app lication of tes ting methodology a nd te c hniques de ve loped by th e a uthors to a id in impro vi ng the qu a lity of mathem ati ca l soft wa re. These techni q ues diffe r rad ica ll y in se vera l as pects from techni q ues pre vious ly used . the most import a nt d iffere nce be ing th at th e testin g is not based e xc lus ivel y Oil random a rgume nt s. Ins tead , throughou t the range of eac h fun c tio n use is ma de of s pec ia l a rgum e nt s that a re des ig ned to d etec t progra mming e rrors and to tes t the pe rfnrm a nce of an algorithm in diffe re nt regions. T he fu nction values a re tes ted aga ins t refe re nce valu es whic h are s tored o n re fe re nce tapes ge ne ra ted b y a hi ghl y au the nti ca ted s yste m of s ubro utin es. S ince the eff ec tive ness of s uch a tes tin g sys te m in di scove ring e rrors and pe rform a nce limit ati ons can be fu ll y ascert a ined o nl y th rough ac tu a l use , we report the res ult s of e m ploying our s ys tem to tes t an exi stin g FORTRAN library of ma the matica l fun c tion routines. S pec if, c a s pects of the num e ri ca l accurac y of the libJ·a ry used in thi s te st case are di sc ussed in orde r to ill :~5tra t e the e ffec ti ve ness of a we ll -d es igned tes ting s yste m as a n ana lyti c tool for the e valuation of mathe matica l so ft wa re. S ince docum e nt ati on prov ides information necessary to perform testing and cont ain s s pecifications th at re flect the result s of testin g, our stud y inc ludes com me nts on th e doc ume nt at ion. No information on timing or storage re quire me nts is present ed in thi s case s tud y.","PeriodicalId":166823,"journal":{"name":"Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1973-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125152984","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: 6
Tables and graphs of the stable probability density functions 稳定概率密度函数的表和图
Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences Pub Date : 1973-07-01 DOI: 10.6028/JRES.077B.017
D. Holt, E. Crow
{"title":"Tables and graphs of the stable probability density functions","authors":"D. Holt, E. Crow","doi":"10.6028/JRES.077B.017","DOIUrl":"https://doi.org/10.6028/JRES.077B.017","url":null,"abstract":"","PeriodicalId":166823,"journal":{"name":"Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences","volume":"66 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1973-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114872480","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: 92
Bessel functions of real argument and integer order 实参数和整数阶贝塞尔函数
Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences Pub Date : 1973-07-01 DOI: 10.6028/JRES.077B.014
David J. Sookne
{"title":"Bessel functions of real argument and integer order","authors":"David J. Sookne","doi":"10.6028/JRES.077B.014","DOIUrl":"https://doi.org/10.6028/JRES.077B.014","url":null,"abstract":"","PeriodicalId":166823,"journal":{"name":"Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences","volume":"70 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1973-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126555623","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: 22
Triangles generated by powers of triplets on unit circle 由单位圆上三元组的幂生成的三角形
Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences Pub Date : 1973-07-01 DOI: 10.6028/jres.077b.016
Charles R. Johnson, M. Newman
{"title":"Triangles generated by powers of triplets on unit circle","authors":"Charles R. Johnson, M. Newman","doi":"10.6028/jres.077b.016","DOIUrl":"https://doi.org/10.6028/jres.077b.016","url":null,"abstract":"Le t a, 13, y be th ree d is tin c t complex numbers of mod ulus 1. It is shown that th ere is essent ially one e xception to the followin g s tat e ment: For some posi t ive int eger Ill , 0 is in th e c losed convex hull of a\"', 13\"' , y\"'. The exception occurs for the normalized tripl e 1, e'l 1T i/7, e2krri/7, whe re k = 3 or 5. Thi s qu es tion was motivated by the proble'm of determining when a pos itive intege r III and a nonze ro n X l vee'tor X e xist such tha i X * A IIIX = 0, whe re A is a given matrix of M,,(C). Key word s: Convex hull; lInit ci rcle; V ey l's Theorem.","PeriodicalId":166823,"journal":{"name":"Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences","volume":"86 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1973-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120835556","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
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