The classical tournament problem caiJs for arranging v individuals into teams of /) players so that a player is teamed the same number of times with each of the other players and also that each player is pitted equally often against each of the other players. The play of the tuurnament results in the determination of difference in performance of the various pairings of the groups. In the special case when p = 2 each team consists of two players and the desigris are called bridge tournament designs. In high precisiun calibration one can measure only the difference between two nominally equal groups S(l that if u objects are to be inte rcompared in groups of fJ objects, then the solutions tu the tour nament proble m provide schedules for the gl'lluping. These designs are useful in weighing a nd any othe r measul'emcnts where the objects tu be measured can be combined intu gl'llups without loss of precision or accuracy ill the comparisons. Tilis paper presents general methuds fill' constructing of bridge lournament designs, i.e., for Lht' case whcn p = 2, fur all /1 ~ 50. Key Wurds: Calibration, calibration designs. combinatorial analysis, difference sets, experiment designs', incomplete block designs, tourname nts, weighing designs.
{"title":"The bridge tournament problem and calibration designs for comparing pairs of objects","authors":"R. C. Bose, J. M. Cameron","doi":"10.6028/JRES.069B.033","DOIUrl":"https://doi.org/10.6028/JRES.069B.033","url":null,"abstract":"The classical tournament problem caiJs for arranging v individuals into teams of /) players so that a player is teamed the same number of times with each of the other players and also that each player is pitted equally often against each of the other players. The play of the tuurnament results in the determination of difference in performance of the various pairings of the groups. In the special case when p = 2 each team consists of two players and the desigris are called bridge tournament designs. In high precisiun calibration one can measure only the difference between two nominally equal groups S(l that if u objects are to be inte rcompared in groups of fJ objects, then the solutions tu the tour nament proble m provide schedules for the gl'lluping. These designs are useful in weighing a nd any othe r measul'emcnts where the objects tu be measured can be combined intu gl'llups without loss of precision or accuracy ill the comparisons. Tilis paper presents general methuds fill' constructing of bridge lournament designs, i.e., for Lht' case whcn p = 2, fur all /1 ~ 50. Key Wurds: Calibration, calibration designs. combinatorial analysis, difference sets, experiment designs', incomplete block designs, tourname nts, weighing designs.","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":"129609365","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. I. General theory for simple turning points","authors":"F. Olver","doi":"10.6028/JRES.069B.028","DOIUrl":"https://doi.org/10.6028/JRES.069B.028","url":null,"abstract":"","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"222 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":"122361142","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 theore m of M. Newman states that if v, k, A, are the parameters for a diffe rence se l D, and k-A = P or 2p (p a prime) then p is a multiplier of D. This theorem is genera li ze d 10 th e case of an abelian diffe rence set and several consequences are noted.
纽曼的一个定理m指出，如果v, k, A是差分集D的参数，并且k-A = P或2p (P A ')，则P是D的乘子。这个定理在阿贝尔差分集的情况下得到了推广，并指出了几个结果。
{"title":"A generalization of a result of Newman on multipliers of difference sets","authors":"R. McFarland","doi":"10.6028/JRES.069B.032","DOIUrl":"https://doi.org/10.6028/JRES.069B.032","url":null,"abstract":"A theore m of M. Newman states that if v, k, A, are the parameters for a diffe rence se l D, and k-A = P or 2p (p a prime) then p is a multiplier of D. This theorem is genera li ze d 10 th e case of an abelian diffe rence set and several consequences are noted.","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"95 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":"115252932","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":"On Kirchhoff's law and its generalized application to absorption and emission by cavities","authors":"F. Kelly","doi":"10.6028/JRES.069B.019","DOIUrl":"https://doi.org/10.6028/JRES.069B.019","url":null,"abstract":"","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"7 3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1965-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124751804","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 the theore ti cal solution rece ntly obtained for stationary spati al-cohere nce functions over radiating apertures, [I]' the evaluation of the two-dime nsional Fourie r integral of the far-fi eld inte nsity di s tribution is required. Since the appearance of s uc h integrals is also quite common in other areas of mathe matical physics, it would be useful to render their evaluation amenable to the application of ex tensively tabulated result s available in the literature. (For examples of suc h sources see [2] , [4] , [5], and [6].) For fun ctions of one variable, comprehe nsive tables of Fourier tran sforms exis t [2], and although it is possible to reduce the k dimensional Fourier tran sform of radial functions 2 [3] to Hankel transforms [3 , p. 69] for which extensive tables [5] are available , there are no tables giving the Fourier transform for k > 1 of arbitrary functions (i. e., nonradi al) even in the case of k = 2. In this paper, the two-dimensional Fourier tran sform is reduced to a form which facilitates its evaluation by th e use of existing tables [4 , 5] and also yields a result which is an extension of that given in Bochner and Chandrasekharan [3], for the case k=2, to functions which are not necessarily radial. It will be shown that if g(a, (3 ) is the two-dimensional Fourier transform of fix, y), i.e.,
{"title":"On a relation between two-dimensional Fourier integrals and series of Hankel transforms","authors":"J. Cornacchio, R. P. Soni","doi":"10.6028/JRES.069B.020","DOIUrl":"https://doi.org/10.6028/JRES.069B.020","url":null,"abstract":"In the theore ti cal solution rece ntly obtained for stationary spati al-cohere nce functions over radiating apertures, [I]' the evaluation of the two-dime nsional Fourie r integral of the far-fi eld inte nsity di s tribution is required. Since the appearance of s uc h integrals is also quite common in other areas of mathe matical physics, it would be useful to render their evaluation amenable to the application of ex tensively tabulated result s available in the literature. (For examples of suc h sources see [2] , [4] , [5], and [6].) For fun ctions of one variable, comprehe nsive tables of Fourier tran sforms exis t [2], and although it is possible to reduce the k dimensional Fourier tran sform of radial functions 2 [3] to Hankel transforms [3 , p. 69] for which extensive tables [5] are available , there are no tables giving the Fourier transform for k > 1 of arbitrary functions (i. e., nonradi al) even in the case of k = 2. In this paper, the two-dimensional Fourier tran sform is reduced to a form which facilitates its evaluation by th e use of existing tables [4 , 5] and also yields a result which is an extension of that given in Bochner and Chandrasekharan [3], for the case k=2, to functions which are not necessarily radial. It will be shown that if g(a, (3 ) is the two-dimensional Fourier transform of fix, y), i.e.,","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"689 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1965-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126511160","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":"A development of theory of errors with reference to economy of time","authors":"M. D. Hersey","doi":"10.6028/JRES.069B.015","DOIUrl":"https://doi.org/10.6028/JRES.069B.015","url":null,"abstract":"","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1965-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123063196","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 : A matroid M = (E, F) is a finite set E of elements and a family F of subsets of E, called independent sets, such that (1) every subset of an independent set is independent, and (2) for every set A belonging to E, all maximal independent subsets of A have the same cardinality, called the rank r(A) of A. The concept of 'matroid' thus generalizes that of 'matrix' or, in particular, that of 'graph.' This paper treats a variety of partition problems involving independent sets of matroids.
{"title":"Transversals and Matroid Partition","authors":"J. Edmonds, D. R. Fulkerson","doi":"10.6028/JRES.069B.016","DOIUrl":"https://doi.org/10.6028/JRES.069B.016","url":null,"abstract":"Abstract : A matroid M = (E, F) is a finite set E of elements and a family F of subsets of E, called independent sets, such that (1) every subset of an independent set is independent, and (2) for every set A belonging to E, all maximal independent subsets of A have the same cardinality, called the rank r(A) of A. The concept of 'matroid' thus generalizes that of 'matrix' or, in particular, that of 'graph.' This paper treats a variety of partition problems involving independent sets of matroids.","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1965-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133522225","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 Theorems on the Permanent","authors":"R. Brualdi, M. Newman","doi":"10.6028/JRES.069B.018","DOIUrl":"https://doi.org/10.6028/JRES.069B.018","url":null,"abstract":"","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":"1965-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126734067","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 contraction theorem of Banac h remains the most fruitful means for proving and analyzing the convergence of iterative processes. For thi s reason, extension s of the theorem are of continuing interes t. The present paper describes '>ome extensions to a class of functions called local contrac tio ns . For comple te ness, we include he re a r esum e of the relevant defi nitions. A metric space (X, p) consists of a none mpty set X and a nonnegative-valued fun c tion p defined on X X X and satisfying
Banac h的收缩定理仍然是证明和分析迭代过程收敛性的最有成果的方法。因此，定理的推广具有连续的意义。本文描述了一类称为局部收缩的函数的一些推广。为了完整起见，我们包括了相关定义的概要。度量空间(X, p)由非空集X和定义在X X X上且满足的非负值函数p组成
{"title":"Some extensions of Banach's contraction theorem","authors":"P. R. Meyers","doi":"10.6028/JRES.069B.022","DOIUrl":"https://doi.org/10.6028/JRES.069B.022","url":null,"abstract":"The contraction theorem of Banac h remains the most fruitful means for proving and analyzing the convergence of iterative processes. For thi s reason, extension s of the theorem are of continuing interes t. The present paper describes '>ome extensions to a class of functions called local contrac tio ns . For comple te ness, we include he re a r esum e of the relevant defi nitions. A metric space (X, p) consists of a none mpty set X and a nonnegative-valued fun c tion p defined on X X X and satisfying","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1965-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122391597","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 L2 Markoff inequalities","authors":"L. Shampine","doi":"10.6028/JRES.069B.017","DOIUrl":"https://doi.org/10.6028/JRES.069B.017","url":null,"abstract":"","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"107 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1965-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132291516","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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