{"title":"Bounds for the number of generators of a finite group","authors":"M. Newman","doi":"10.6028/JRES.071B.034","DOIUrl":"https://doi.org/10.6028/JRES.071B.034","url":null,"abstract":"","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"51 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1967-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124939132","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 matrix A is said to be symmetri zable by V when V is positive definite and AV is hermitian. Several le mmas regard ing symmetrizability are given. For three classes of generalized inverses it is s hown that if A is s mmetrizable by V the re exists a generali zed inverse in each class which is sy mmetrizable by V. The Moore·Penrose inverse (or pseudo-inverse) of a matrix symmetrizable by V is also symmetrizable by V if and only if the matrix and the pseudo-inverse com mute.
{"title":"Symmetrizable Generalized Inverses of Symmetrizable Matrices","authors":"J. Hearon","doi":"10.6028/JRES.071B.031","DOIUrl":"https://doi.org/10.6028/JRES.071B.031","url":null,"abstract":"The matrix A is said to be symmetri zable by V when V is positive definite and AV is hermitian. Several le mmas regard ing symmetrizability are given. For three classes of generalized inverses it is s hown that if A is s mmetrizable by V the re exists a generali zed inverse in each class which is sy mmetrizable by V. The Moore·Penrose inverse (or pseudo-inverse) of a matrix symmetrizable by V is also symmetrizable by V if and only if the matrix and the pseudo-inverse com mute.","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1967-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123273497","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}
L It is a result familiar in the theory of automorphic form s that an entire automorphic form of positive dimension on an H-group is identically zero (see sec. 2 for the definitions). This follows immediately , for exa mple, from the well-known exac t formula for the Fourier coefficie nts of automorphic forms of positive dimension ([1], p. 314).1 Another proof is by means of a formula for the numbe r of zeros minus the number of poles of an automorphic form in a fundamental domain. This formula (obtained by contour integration around the fundamental domain) shows that whe n the dime nsion of the form is positive, thi s difference is negative, and he nce such a form mu st have poles. In section 3 of thi s note we give what appears to be a new proof of this result by using the method Hecke e mployed to estimate the Fourier coeffi cients of cusp forms of negative dimension ([1] , p. 281). I This proof is simpler and more direc t than the proofs mentioned above. In sections 45 we give two variations of this method. The me thod of section 5 is applicable to a larger class of groups than the H-groups , and in particular applies to compact groups and groups conjugate to H-groups. 2. A group r of real linear fractional transformations acting on :J't', the upper half-plane 1m 7 > 0, is an H-group provided (i) r is discontinuous on :J't', but is not di scontinuous at any point of the real line, (ii) r is finitely generated, and (ii i) r contains translations. With each transformation v~r we associate a real
{"title":"Notes on automorphic functions: an entire automorphic form of positive dimension is zero","authors":"M. Knopp","doi":"10.6028/JRES.071B.022","DOIUrl":"https://doi.org/10.6028/JRES.071B.022","url":null,"abstract":"L It is a result familiar in the theory of automorphic form s that an entire automorphic form of positive dimension on an H-group is identically zero (see sec. 2 for the definitions). This follows immediately , for exa mple, from the well-known exac t formula for the Fourier coefficie nts of automorphic forms of positive dimension ([1], p. 314).1 Another proof is by means of a formula for the numbe r of zeros minus the number of poles of an automorphic form in a fundamental domain. This formula (obtained by contour integration around the fundamental domain) shows that whe n the dime nsion of the form is positive, thi s difference is negative, and he nce such a form mu st have poles. In section 3 of thi s note we give what appears to be a new proof of this result by using the method Hecke e mployed to estimate the Fourier coeffi cients of cusp forms of negative dimension ([1] , p. 281). I This proof is simpler and more direc t than the proofs mentioned above. In sections 45 we give two variations of this method. The me thod of section 5 is applicable to a larger class of groups than the H-groups , and in particular applies to compact groups and groups conjugate to H-groups. 2. A group r of real linear fractional transformations acting on :J't', the upper half-plane 1m 7 > 0, is an H-group provided (i) r is discontinuous on :J't', but is not di scontinuous at any point of the real line, (ii) r is finitely generated, and (ii i) r contains translations. With each transformation v~r we associate a real","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"152 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1967-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129420128","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}
L The object of this note is to prove the THEOREM: The system of equations af + a~ + . + a~_1 = br + b~ + . . . + b~_l' r=2,3, .. . , n ; (1) has no nontrivial solutions in positive integers. In what follows, we write Ar for a~+ a;+ "'" 1; Br for bi+ b;+ + b~_ I' r"'" 1; and all small letters denote integers"'" 0 unless stated otherwise. 2. PROOF OF THE THEOREM: Let at, a2, ... , a,,_1 From (4) we have AI 1 0 0 0 A2 AI 2 0 0 A3 A2 AI 3 0 r!'r =
{"title":"A system of equations having no nontrivial solutions","authors":"H. Gupta","doi":"10.6028/JRES.071B.024","DOIUrl":"https://doi.org/10.6028/JRES.071B.024","url":null,"abstract":"L The object of this note is to prove the THEOREM: The system of equations af + a~ + . + a~_1 = br + b~ + . . . + b~_l' r=2,3, .. . , n ; (1) has no nontrivial solutions in positive integers. In what follows, we write Ar for a~+ a;+ \"'\" 1; Br for bi+ b;+ + b~_ I' r\"'\" 1; and all small letters denote integers\"'\" 0 unless stated otherwise. 2. PROOF OF THE THEOREM: Let at, a2, ... , a,,_1 From (4) we have AI 1 0 0 0 A2 AI 2 0 0 A3 A2 AI 3 0 r!'r =","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1967-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129039241","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}
Th e complex, not necessaril y square matrix A is called a partial isometry if the vectors x and A x have the same Euclidean norm whenever x is in the orthogonal complement of the null space of A. The main result s of the paper give necessary and sufficient conditions for a matrix to be a partial isometry, for a partial isometry to be normal and for the product of two partial isometries to be a partial isometry. A factorization for an arbitrary matrix involving partial isometries is given. The concept of a ge neralized inverse is used in establishing the primary results .
{"title":"Partially isometric matrices","authors":"J. Hearon","doi":"10.6028/JRES.071B.030","DOIUrl":"https://doi.org/10.6028/JRES.071B.030","url":null,"abstract":"Th e complex, not necessaril y square matrix A is called a partial isometry if the vectors x and A x have the same Euclidean norm whenever x is in the orthogonal complement of the null space of A. The main result s of the paper give necessary and sufficient conditions for a matrix to be a partial isometry, for a partial isometry to be normal and for the product of two partial isometries to be a partial isometry. A factorization for an arbitrary matrix involving partial isometries is given. The concept of a ge neralized inverse is used in establishing the primary results .","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1967-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121515684","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}
It is s hown in [1] I that e very graphic matroid is regular ([1], 5.63) and even ([IJ, 9.23). Moreover a regular matroid c an be charac terized as a binary one which has no minor of either of the types called BI and BII. ([11. 7.51). In the prese nt paper we es tablish a converse theore m: any ever: matroid whic h has no minor of Type BI mu st be graphi c. 1. Let Y be an atom of a binary matroid M, and suppose it to have the following properties. (i) Y is brid~e-separable (ii) If 8 is any bridge of Y in M , then M X (B U Y) is graphic. Then M is g raphic.
在[1]1中表明，非常图形化的矩阵是正则的([1]，5.63)和偶的([IJ, 9.23)。此外，正则矩阵c可以被表征为一个二元矩阵，它没有BI和BII这两种类型的子矩阵。([11。7.51)。本文建立了一个逆定理:任何不含BI型的任意矩阵都是图1。设Y是二元矩阵M的一个原子，并假定它具有下列性质。(i) Y是桥~e可分的(ii)如果8是M中Y的任意桥，则M X (B U Y)是图形的。那么M是图形的。
{"title":"On even matroids","authors":"W. T. Tutte","doi":"10.6028/JRES.071B.028","DOIUrl":"https://doi.org/10.6028/JRES.071B.028","url":null,"abstract":"It is s hown in [1] I that e very graphic matroid is regular ([1], 5.63) and even ([IJ, 9.23). Moreover a regular matroid c an be charac terized as a binary one which has no minor of either of the types called BI and BII. ([11. 7.51). In the prese nt paper we es tablish a converse theore m: any ever: matroid whic h has no minor of Type BI mu st be graphi c. 1. Let Y be an atom of a binary matroid M, and suppose it to have the following properties. (i) Y is brid~e-separable (ii) If 8 is any bridge of Y in M , then M X (B U Y) is graphic. Then M is g raphic.","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1967-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127609024","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}
So me purposes of thi s paper are: (1) To take se riously the term , " term rank. " (2) To ma ke an issue of not " rea rra nging rows a nd colu mns" by not "a rranging" the m in the firs t place. (3) To promote the nu merica l use of Cra mer 's rul e. (4) To ill us tra te that the re levance of " numbe r of s teps" to "a mount of wo rk" depends on the amount of work in a step. (5) To ca ll a tt ention to the com puta tional as pec t of SDR's, an aspect where the subjec t di ffe rs fro m bein g an instance of fa milia r li near algebra. (6) To describe a n SDR in s ta nce of a theory on extre mal co mbi nato rics tha t uses linea r algebra in ve ry dif· fe rent ways than does to tall y unimodular theory. (The preceding paper, Optimum Branc hings, de· sc ribes another instanc e of tha t theory.)
{"title":"Systems of distinct representatives and linear algebra","authors":"J. Edmonds","doi":"10.6028/JRES.071B.033","DOIUrl":"https://doi.org/10.6028/JRES.071B.033","url":null,"abstract":"So me purposes of thi s paper are: (1) To take se riously the term , \" term rank. \" (2) To ma ke an issue of not \" rea rra nging rows a nd colu mns\" by not \"a rranging\" the m in the firs t place. (3) To promote the nu merica l use of Cra mer 's rul e. (4) To ill us tra te that the re levance of \" numbe r of s teps\" to \"a mount of wo rk\" depends on the amount of work in a step. (5) To ca ll a tt ention to the com puta tional as pec t of SDR's, an aspect where the subjec t di ffe rs fro m bein g an instance of fa milia r li near algebra. (6) To describe a n SDR in s ta nce of a theory on extre mal co mbi nato rics tha t uses linea r algebra in ve ry dif· fe rent ways than does to tall y unimodular theory. (The preceding paper, Optimum Branc hings, de· sc ribes another instanc e of tha t theory.)","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1967-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116881340","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}
Simple bounds are established for the solutions of second-order homogeneous linear difference equations in ranges in which the solutions are exponential in character. The results are applied to a recent algorithm for the computation of subdominant solutions of second-order linear difference equations, homogeneous or otherwise. Strict and extremely realistic bounds are obtained for the truncation error associated with the algorithm in a number of examples, including Anger·Weber functions, Struve functions , and the solution of a differential equation in Chebyshev series.
{"title":"Bounds for the solutions of second-order linear difference equations","authors":"F. Olver","doi":"10.6028/JRES.071B.021","DOIUrl":"https://doi.org/10.6028/JRES.071B.021","url":null,"abstract":"Simple bounds are established for the solutions of second-order homogeneous linear difference equations in ranges in which the solutions are exponential in character. The results are applied to a recent algorithm for the computation of subdominant solutions of second-order linear difference equations, homogeneous or otherwise. Strict and extremely realistic bounds are obtained for the truncation error associated with the algorithm in a number of examples, including Anger·Weber functions, Struve functions , and the solution of a differential equation in Chebyshev series.","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1967-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132178778","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}
In high precision ca librations one measures diffe re nces be tween nominally equal objects or group of objects and establi shes a value for the individuals with refere nce to one or more standards. The so lutions to the class ica l tournament problem, which ca ll s fo r arranging v individual s into tea ms of I) players so that a player is teamed the same number of times with each of the other players and also that ea ch player is pitted equally often aga inst each of the other players, provide balanced designs for scheduling the meas ure ments. These designs are useful in weighing and ot he r meas ure me nt s wh en the objects to be measured ca n be combin ed into groups without loss of prec ision or acc uracy in the co mparisons.
{"title":"Calibration designs based on solutions to the tournament problem","authors":"R. C. Bose, J. M. Cameron","doi":"10.6028/JRES.071B.020","DOIUrl":"https://doi.org/10.6028/JRES.071B.020","url":null,"abstract":"In high precision ca librations one measures diffe re nces be tween nominally equal objects or group of objects and establi shes a value for the individuals with refere nce to one or more standards. The so lutions to the class ica l tournament problem, which ca ll s fo r arranging v individual s into tea ms of I) players so that a player is teamed the same number of times with each of the other players and also that ea ch player is pitted equally often aga inst each of the other players, provide balanced designs for scheduling the meas ure ments. These designs are useful in weighing and ot he r meas ure me nt s wh en the objects to be measured ca n be combin ed into groups without loss of prec ision or acc uracy in the co mparisons.","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"51 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1967-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127013915","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}
Abstract : The Pseudo Primal-Dual Algorithm solves the pure integer programming problem in two stages, systemmatically violating and restoring dual feasibility while maintaining an all-integer matrix. The algorithm is related to the Gomory All-Integer Algorithm and the Young Primal Integer Programming Algorithm, differing from the former in the dual feasible stage by the choice of cuts and pivot variable, and from the latter in the dual infeasible stage by the use of a more rigid (and faster) rule for restoring dual feasibility. The net advance in the objective function value produced by the algorithm between two consecutive stages of dual infeasibility is shown to be at least as great as that produced by pivoting with the dual simplex method. Example problems are given that illustrate basic features and variations of the method. (Author)
{"title":"A PSEUDO PRIMAL-DUAL INTEGER PROGRAMMING ALGORITHM.","authors":"F. Glover","doi":"10.6028/JRES.071B.026","DOIUrl":"https://doi.org/10.6028/JRES.071B.026","url":null,"abstract":"Abstract : The Pseudo Primal-Dual Algorithm solves the pure integer programming problem in two stages, systemmatically violating and restoring dual feasibility while maintaining an all-integer matrix. The algorithm is related to the Gomory All-Integer Algorithm and the Young Primal Integer Programming Algorithm, differing from the former in the dual feasible stage by the choice of cuts and pivot variable, and from the latter in the dual infeasible stage by the use of a more rigid (and faster) rule for restoring dual feasibility. The net advance in the objective function value produced by the algorithm between two consecutive stages of dual infeasibility is shown to be at least as great as that produced by pivoting with the dual simplex method. Example problems are given that illustrate basic features and variations of the method. (Author)","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":"1967-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115773218","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}
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