An explicit description is giv e n for th e uniqu e gra ph with as few arcs (eac h bearin g a positive length) as pos s ibl e, whi c h has a presc rib ed mat rix of s hortest-p ath di stan ces be twee n pa irs of distinct vertices. The sam e is d one in th e case wh e n the ith diago na l matrix e ntr y, in s te ad o f be ing zero , represents th e. le ngth of a s hort est c losed path co ntainin g th e ith vertex. Ke y Word s: Graph, di s ta nce ma trix , s hortes t path. Le t G be a finite oriented graph with verti ces {Vi}~', wh e re n > 2. To avoid unn ecessary co mpli cation s, we res tric t attention to connected graph s, i. e., if i r!= j then G co ntain s a directed path from Vi to Vj. As add iti onal s tru cture, we assume associated to G a positive-valu ed fun cti on lc ass ignin g lengths lc(i, j) to the arcs (Vi, Vj) of G. The distance matrix Dc of G has e ntri es dc;(i , i) = ° on th e main diago nal; a typi c al off-diago nal e ntry dc(i, J) re pers e nts the le ngth of a s hortes t directed path in G from Vi to Vj. An arc of G is called redundant if its deletion leaves Dc un changed. Th e graph G will be called irreducible if it co ntain s no redundant arcs. A real square matrix D with e ntri es d(i , j) is called realizable if there is a grap h G s uc h that D = Dr;. Hakimi and Yau t showed that necessary and s uffi cie nt conditions for th e realiza bility of Dare The necessity of the se conditions should be clear. To prove sufficiency one need only take the arcs of G to be all (Vi. Vj) with i r!= j , and define le by le/i, J) = d(i , j) ; it follows readily from (1) to (3) that Dc= D. If matrix D is realizable, it clearly has a realization by …
{"title":"Realizing the distance matrix of a graph","authors":"A. J. Goldman","doi":"10.6028/JRES.070B.013","DOIUrl":"https://doi.org/10.6028/JRES.070B.013","url":null,"abstract":"An explicit description is giv e n for th e uniqu e gra ph with as few arcs (eac h bearin g a positive length) as pos s ibl e, whi c h has a presc rib ed mat rix of s hortest-p ath di stan ces be twee n pa irs of distinct vertices. The sam e is d one in th e case wh e n the ith diago na l matrix e ntr y, in s te ad o f be ing zero , represents th e. le ngth of a s hort est c losed path co ntainin g th e ith vertex. Ke y Word s: Graph, di s ta nce ma trix , s hortes t path. Le t G be a finite oriented graph with verti ces {Vi}~', wh e re n > 2. To avoid unn ecessary co mpli cation s, we res tric t attention to connected graph s, i. e., if i r!= j then G co ntain s a directed path from Vi to Vj. As add iti onal s tru cture, we assume associated to G a positive-valu ed fun cti on lc ass ignin g lengths lc(i, j) to the arcs (Vi, Vj) of G. The distance matrix Dc of G has e ntri es dc;(i , i) = ° on th e main diago nal; a typi c al off-diago nal e ntry dc(i, J) re pers e nts the le ngth of a s hortes t directed path in G from Vi to Vj. An arc of G is called redundant if its deletion leaves Dc un changed. Th e graph G will be called irreducible if it co ntain s no redundant arcs. A real square matrix D with e ntri es d(i , j) is called realizable if there is a grap h G s uc h that D = Dr;. Hakimi and Yau t showed that necessary and s uffi cie nt conditions for th e realiza bility of Dare The necessity of the se conditions should be clear. To prove sufficiency one need only take the arcs of G to be all (Vi. Vj) with i r!= j , and define le by le/i, J) = d(i , j) ; it follows readily from (1) to (3) that Dc= D. If matrix D is realizable, it clearly has a realization by …","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"380 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":"121917717","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 previous pape r gave the ge ne ra l theory and phys ica l princ iples involved in so modifying the author's s phe roidal potential for an ob late plane t as to pe rmit exact inc lus io n of the e ffects of the third zonal harmonic of the pl ane t's gravit ati ona l fi e ld. The present pape r ca rri es out the computationa l details necessary to de rive the res ulting orbit , whi c h now corres ponds t.o a pote nti a l fitte d exactl y through the third zona l ha rmonic and to a bout two·third s of th e fourth . The accuracy of the orbit it self, as a so lution fo r the give n po te nt.i a l, depends o n the acc uracy of so lutio n of a ce rtain c ubi c e qu a tion. T he pa pe r wo rks out thi s so lution t hrough te rm s of the thi rd orde r in .f" the coe ffi c ient of the seco nd zo na l ha rmo nic, but it s acc urac y, a nd thus tha t. of t.h e sec ula r te rm s, may be inc reased a t. will. Pe ri odi c te rms a re ca rried t.hrough the seco nd o rde r, but the ir ac· c uracy may a lso be in c reased. An o bvious ad va nt age of accounti ng for I . in thi s way is the absence of s ma ll deno min a tors in e o r sin J tha t occur in a pe rtu r ba tio n theory. Anuthe r is the res ult.in g inc rease in acc uracy, through te rms in i i, of t. he long·pe ri od ic third ha rmo nic te rm s.
《ge ne ra previous pape r给我理论和phys ica princ iples风险在如此modifying《作家s phe roidal潜在的美国t ob晚的飞机来说pe rmit准确inc . io家伙n e ffects之第三区域调和定律》的pl ane t的gravit ona l菲e肝ld。现在pape r ca rri冰房《res computationa l细节必要去de成河ulting轨道,whi现在用吨计算吨轨道的准确之处在于它是自我的T他爸pe r wo rks出去thi s所以lution T te hrough罗thi rd r。f的秩序》《o和c的蛋蛋coe ient佐na l哈rmo尼克,但它的acc的urac y、a,因此tha T。s T h e证交会乌拉te r罗威尔,may be公司reased a T。但我们的交流和交流是如此的真实。不要为我计算年龄。在某种程度上，这是我的疏忽。下图是秘诀。他长得像第三个国王。
{"title":"Inclusion of the third zonal harmonic in an accurate reference orbit of an artificial satellite.","authors":"J. Vinti","doi":"10.6028/JRES.070B.003","DOIUrl":"https://doi.org/10.6028/JRES.070B.003","url":null,"abstract":"The previous pape r gave the ge ne ra l theory and phys ica l princ iples involved in so modifying the author's s phe roidal potential for an ob late plane t as to pe rmit exact inc lus io n of the e ffects of the third zonal harmonic of the pl ane t's gravit ati ona l fi e ld. The present pape r ca rri es out the computationa l details necessary to de rive the res ulting orbit , whi c h now corres ponds t.o a pote nti a l fitte d exactl y through the third zona l ha rmonic and to a bout two·third s of th e fourth . The accuracy of the orbit it self, as a so lution fo r the give n po te nt.i a l, depends o n the acc uracy of so lutio n of a ce rtain c ubi c e qu a tion. T he pa pe r wo rks out thi s so lution t hrough te rm s of the thi rd orde r in .f\" the coe ffi c ient of the seco nd zo na l ha rmo nic, but it s acc urac y, a nd thus tha t. of t.h e sec ula r te rm s, may be inc reased a t. will. Pe ri odi c te rms a re ca rried t.hrough the seco nd o rde r, but the ir ac· c uracy may a lso be in c reased. An o bvious ad va nt age of accounti ng for I . in thi s way is the absence of s ma ll deno min a tors in e o r sin J tha t occur in a pe rtu r ba tio n theory. Anuthe r is the res ult.in g inc rease in acc uracy, through te rms in i i, of t. he long·pe ri od ic third ha rmo nic te rm s.","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1966-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133297716","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":"Tables for the evaluation of the Faxen approximation to the solution of the Lamm equation.","authors":"M. Dishon, G. Weiss","doi":"10.6028/JRES.070B.008","DOIUrl":"https://doi.org/10.6028/JRES.070B.008","url":null,"abstract":"","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":"1966-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126040450","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}
Gravitational potential of oblate planet, expressed in terms of oblate spheroidal coordinates, generalized by means of transformation of associated Cartesian system
用椭球坐标系表示的扁行星引力势，用相关笛卡尔坐标系的变换进行了推广
{"title":"Invariant properties of the spheroidal potential of an oblate planet.","authors":"J. Vinti","doi":"10.6028/JRES.070B.002","DOIUrl":"https://doi.org/10.6028/JRES.070B.002","url":null,"abstract":"Gravitational potential of oblate planet, expressed in terms of oblate spheroidal coordinates, generalized by means of transformation of associated Cartesian system","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1966-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131883947","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 random walk lattice model of adsorption of an iso la ted polymer c hain at a so lu t ion surface is inves ti gated . The model is a modifi cation of a s imple c ubic lattice in which the re is a co rre la tion between success ive s te ps. The direc tion of each s tep is at right angles to the direc tion of the preceding step (a ll bo nd angles are 90°). O ne-dime nsional c harac te ri s ti cs of the monomer unit di s tribution a re de termined analyti call y in the limit of long polyme r cha ins neglec tin g the self-excluded vo lume. T he mean numbe r of monomer units adsorbed in the surface laye r V(() , N) is de termined assuming th a t one end of the po.lymer chain li es in the s urface layer, where N is the mean nu mber of monome r unit s in the c ha in and () is the adsorption ene rgy of each monomer unit in the surface layer measured in units of kT . In addition, the mean di s tan ce of the free e nd of the cha in from the s urface laye r z(() , N) is dete rmined. Th e properties of thi s corre lated step model a re qualita ti ve ly s imila r to the pro perti es whi ch ha ve been fo und in unco rre la ted s te p models. In pa rti c ula r, there is a c riti ca l va lue of the adsorpt ion ene rgy ()c such that for () > ()c, vii!, N) is proportiona l to N. Num erical va lu es of N'v(() , N) and z( () , N) are p resented for () > ()c = In (V51).
{"title":"A random walk model of chain polymer adsorption at a surface. II. Effect of correlation between neighboring steps","authors":"R. Rubin","doi":"10.6028/JRES.069B.030","DOIUrl":"https://doi.org/10.6028/JRES.069B.030","url":null,"abstract":"A random walk lattice model of adsorption of an iso la ted polymer c hain at a so lu t ion surface is inves ti gated . The model is a modifi cation of a s imple c ubic lattice in which the re is a co rre la tion between success ive s te ps. The direc tion of each s tep is at right angles to the direc tion of the preceding step (a ll bo nd angles are 90°). O ne-dime nsional c harac te ri s ti cs of the monomer unit di s tribution a re de termined analyti call y in the limit of long polyme r cha ins neglec tin g the self-excluded vo lume. T he mean numbe r of monomer units adsorbed in the surface laye r V(() , N) is de termined assuming th a t one end of the po.lymer chain li es in the s urface layer, where N is the mean nu mber of monome r unit s in the c ha in and () is the adsorption ene rgy of each monomer unit in the surface layer measured in units of kT . In addition, the mean di s tan ce of the free e nd of the cha in from the s urface laye r z(() , N) is dete rmined. Th e properties of thi s corre lated step model a re qualita ti ve ly s imila r to the pro perti es whi ch ha ve been fo und in unco rre la ted s te p models. In pa rti c ula r, there is a c riti ca l va lue of the adsorpt ion ene rgy ()c such that for () > ()c, vii!, N) is proportiona l to N. Num erical va lu es of N'v(() , N) and z( () , N) are p resented for () > ()c = In (V51).","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"66 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1965-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130494122","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 me thod to determine a basis of the group of rational integral sy mmetri c posi ti ve definite uni· modular IlXII c irculants for any II. is presented . Th is method uses tllP co rrespondence between unin.vdular eirculants and units of the algebraic number field R(D, where ~ is a primitive nth root of unity. Know n resu lt s are used to obta in gene rato rs of ce rtain ape riodic subgroups of the abe li an, finitely generated gro up of un it s in RU;l. The co rres ponde nce, then , ~) ruduces the desired bas is eJements. The number of bas is elements for each II is proved to be r ~J + 1o-,,(n), where o-o(n) is the numbe)" uf posi ti ve diviso rs of n . In addition, an upper bound for t~1e number of congruence c lasses of these circulant s is obta ined , where co ngrue nce is re lative to rationa l symmetri c unimodular Ilxn circulant s.
{"title":"Groups of unimodular circulants","authors":"R. Austing","doi":"10.6028/JRES.069B.031","DOIUrl":"https://doi.org/10.6028/JRES.069B.031","url":null,"abstract":"A me thod to determine a basis of the group of rational integral sy mmetri c posi ti ve definite uni· modular IlXII c irculants for any II. is presented . Th is method uses tllP co rrespondence between unin.vdular eirculants and units of the algebraic number field R(D, where ~ is a primitive nth root of unity. Know n resu lt s are used to obta in gene rato rs of ce rtain ape riodic subgroups of the abe li an, finitely generated gro up of un it s in RU;l. The co rres ponde nce, then , ~) ruduces the desired bas is eJements. The number of bas is elements for each II is proved to be r ~J + 1o-,,(n), where o-o(n) is the numbe)\" uf posi ti ve diviso rs of n . In addition, an upper bound for t~1e number of congruence c lasses of these circulant s is obta ined , where co ngrue nce is re lative to rationa l symmetri c unimodular Ilxn circulant s.","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":"1965-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130473075","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 analysis of phase-integral methods .II. Application to wave-penetration problems","authors":"F. Olver","doi":"10.6028/JRES.069B.029","DOIUrl":"https://doi.org/10.6028/JRES.069B.029","url":null,"abstract":"","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1965-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122890774","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":"Table of Dedekind sums","authors":"R. D. Shipp","doi":"10.6028/JRES.069B.026","DOIUrl":"https://doi.org/10.6028/JRES.069B.026","url":null,"abstract":"","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"5 8","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1965-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114023835","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}
By reve rsing the usual d irection of appl ication, a co mmon procedure for solv ing integr a l eq ua tion s numeri call y is used to obtain th e asymptotic P-condition numbe rs of two weU-known tes t matrices. Todd [1]1 has recently sugges ted th e matrix AI7 collection of test matrices [2]. The P-co ndition numbers of the matrices are used as a measure of th eir difficulty for num e rical purposes. Where th e co ndi-tion numbers are not explicitly known, the asym ptotic behavior in 11 is give n. Lehmer's matrix A7 I is exceptional in that the correct order in /J is not known_ A si mple id ea will allow us to obtain the asymptotic cond ition number of A 17 and the correc t order for A 7-Hilbert 's first method for integral equation s [3] approximates the e ige nvalues of the kern el K(x, y), o ,s; x, y ,s; 1, by t hose of t he matrix If K is bounded and Riemann integrable, then the eigen-values of th e matrices tend to those of the integral equation as n tends to 00. We reverse this procedure. We wish to es timate the behavior of the eigenvalues of a set of matri ces as n tends to 00. If we can regard them as arising from the application of Hilbert's first method to a fixed ke rnel, then we may hope for an asymptotic res ult. To estimate the largest eigenvalue of A 17 , let us form *This work was SUI)por'!cd by the United States Atomic Energy Commission. Rcp rodu c-I lion in whole or in part is pe rmitt ed for any purpose of the U.S. Government. ] Fi gures in brac ke ts ind ica te the literature references at the end of this paper. We regard them as arising from the approximation of the kernel K(x , y)= Ix-yl. A simple co mputation gives the largest eigenvalue of K as A =! Z-2 I 0 where Zo is the unique real root of coth z = z. This equation has bee n studied [4] and Zo is approximately Zo = 1.9967864. We conclude that the largest eigen-value of A 17 is asymptotically It is easy to es timate the accuracy of the approximation using the bounds of [5] but we s …
通过改变通常的应用方向，利用求解整数方程的一般方法，得到两个已知矩阵的渐近p条件数。Todd[1]1最近提出了测试矩阵的矩阵AI7集合[2]。矩阵的P-co条件数被用作衡量其难度的数字目的。th e公司ndi-tion数字并不明确知道,这个asym内窥镜行为11给n。黄土的矩阵A7我异常的正确顺序/ J不是known_ si mple id ea将使我们得到渐近条件数过渡17和correc t一个7-Hilbert的第一积分方程方法年代[3]接近e的ige nvalues kern el K (x, y), o,年代;X, y,s;如果K是有界且黎曼可积的，那么当n趋于00时，e矩阵的特征值趋向于积分方程的特征值。我们把这个过程反过来。我们希望估计当n趋于00时一组矩阵的特征值的行为。如果我们认为它们是由希尔伯特第一种方法应用于一个固定的通道而产生的，那么我们可能希望得到一个渐近的结果。为了估计a17的最大特征值，让我们形成*This work was SUI) poor '!由美国原子能委员会批准。美国政府不得将全部或部分Rcp产品用于任何目的。括号内的数字是本文最后的文献参考文献。我们认为它们是由核函数K(x, y)= Ix-yl的近似引起的。一个简单的co计算给出了A =!时K的最大特征值z - 2i0，其中Zo是coth z = z的唯一实根，这个方程已经被研究过[4]，Zo近似于Zo = 1.9967864。我们得出结论，a17的最大特征值是渐近的。使用[5]的边界很容易估计近似的精度，但我们…
{"title":"The Condition of Certain Matrices","authors":"L. Shampine","doi":"10.6028/JRES.069B.034","DOIUrl":"https://doi.org/10.6028/JRES.069B.034","url":null,"abstract":"By reve rsing the usual d irection of appl ication, a co mmon procedure for solv ing integr a l eq ua tion s numeri call y is used to obtain th e asymptotic P-condition numbe rs of two weU-known tes t matrices. Todd [1]1 has recently sugges ted th e matrix AI7 collection of test matrices [2]. The P-co ndition numbers of the matrices are used as a measure of th eir difficulty for num e rical purposes. Where th e co ndi-tion numbers are not explicitly known, the asym ptotic behavior in 11 is give n. Lehmer's matrix A7 I is exceptional in that the correct order in /J is not known_ A si mple id ea will allow us to obtain the asymptotic cond ition number of A 17 and the correc t order for A 7-Hilbert 's first method for integral equation s [3] approximates the e ige nvalues of the kern el K(x, y), o ,s; x, y ,s; 1, by t hose of t he matrix If K is bounded and Riemann integrable, then the eigen-values of th e matrices tend to those of the integral equation as n tends to 00. We reverse this procedure. We wish to es timate the behavior of the eigenvalues of a set of matri ces as n tends to 00. If we can regard them as arising from the application of Hilbert's first method to a fixed ke rnel, then we may hope for an asymptotic res ult. To estimate the largest eigenvalue of A 17 , let us form *This work was SUI)por'!cd by the United States Atomic Energy Commission. Rcp rodu c-I lion in whole or in part is pe rmitt ed for any purpose of the U.S. Government. ] Fi gures in brac ke ts ind ica te the literature references at the end of this paper. We regard them as arising from the approximation of the kernel K(x , y)= Ix-yl. A simple co mputation gives the largest eigenvalue of K as A =! Z-2 I 0 where Zo is the unique real root of coth z = z. This equation has bee n studied [4] and Zo is approximately Zo = 1.9967864. We conclude that the largest eigen-value of A 17 is asymptotically It is easy to es timate the accuracy of the approximation using the bounds of [5] but we s …","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1965-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123923239","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":"Some number-theoretic calculations","authors":"K. Kloss","doi":"10.6028/JRES.069B.035","DOIUrl":"https://doi.org/10.6028/JRES.069B.035","url":null,"abstract":"","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"62 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1965-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121833607","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：