The algorithm is most closely related to three existing procedures: the simplex method of G. B. Dantzig for linear programming problems, the Gumory all-integer integer programming algurithm, and the direct algorithm for integer prugramming uf Ben-Israel and Charnes. The algorithm is similar to the Gomory all·integer algorithm in these respects: (i) it is an all-integcr algorithm; (ii) it uses t he same c ut generation proccdure; (iii) it uses the cut row as the pivot row; and (iv) the pivot coefficient always has unit value. While the dual method provides the vehicle for moving from tableau to tableau in the Gomory all-integer algorithm. the simplex method has the analag()us role in the primal algorithm. Thus in a general sense this algorithm is a primal analog t() the (dual) Comory all-integer algo rithm. The direct algorithm of Be n-Israel and Charnes also has the above similarities to the Gomory all-integer al~orithm, but has one significant difference: an iteration or cycle of the direct algorithm must frequently include the solution of an "auxiliary problem" (which is itself an int eger prugramming problem) or a determination that no solution to the "auxiliary problem" exists. In contrasl. the cycles of the primal algorithm include only the adjoining of a ComUl'y cut and the executiun of the change of basis procedure of the simplex method. The procedure uf the algorithm and the proof of finit eness are founded on a classification of cycles of the algorithm and on two theore ms. Two types of prorf'dural restrictions are imposed as a basis for proving fin iteness: (a) selection of the incoming variable is subjected to regulation (beyond that required by t he simplex met hod). and t he rules applied are a function uf the type of cycle being executed; (b) selection of the row Llsed as the source of the data for the
该算法与现有的三个程序最密切相关:G. B. Dantzig的线性规划问题的单纯形法、Gumory全整数规划算法和Ben-Israel和Charnes的整数规划直接算法。该算法在以下几个方面与Gomory全整数算法相似:(1)它是一个全整数算法;(ii)使用相同的c - t生成程序;(三)以切割行为枢轴行;(iv)支点系数总是有单位值。而对偶方法在Gomory全整数算法中提供了从一个表移动到另一个表的载体。单纯形法在原始算法中具有analag()的作用。因此，在一般意义上，该算法是原始模拟t()(对偶)Comory全整数算法。Be n-Israel和Charnes的直接算法也与Gomory的全整数算法有上述相似之处，但有一个显著的区别:直接算法的迭代或循环必须经常包含“辅助问题”的解(它本身就是一个整数编程问题)，或者确定“辅助问题”不存在解。在contrasl。原始算法的循环只包括相邻的ComUl切割和单纯形法的基变换过程的执行。算法的程序和有限性证明是建立在算法的循环分类和两个理论基础上的。作为证明有限性的基础，施加了两种类型的证明限制:(a)输入变量的选择受到调节(超出了单纯形方法所要求的)。所应用的规则是所执行循环类型的函数;(b)选择第ll行作为数据的来源
{"title":"A primal (all-integer) integer programming algorithm","authors":"R. D. Young","doi":"10.6028/JRES.069B.025","DOIUrl":"https://doi.org/10.6028/JRES.069B.025","url":null,"abstract":"The algorithm is most closely related to three existing procedures: the simplex method of G. B. Dantzig for linear programming problems, the Gumory all-integer integer programming algurithm, and the direct algorithm for integer prugramming uf Ben-Israel and Charnes. The algorithm is similar to the Gomory all·integer algorithm in these respects: (i) it is an all-integcr algorithm; (ii) it uses t he same c ut generation proccdure; (iii) it uses the cut row as the pivot row; and (iv) the pivot coefficient always has unit value. While the dual method provides the vehicle for moving from tableau to tableau in the Gomory all-integer algorithm. the simplex method has the analag()us role in the primal algorithm. Thus in a general sense this algorithm is a primal analog t() the (dual) Comory all-integer algo rithm. The direct algorithm of Be n-Israel and Charnes also has the above similarities to the Gomory all-integer al~orithm, but has one significant difference: an iteration or cycle of the direct algorithm must frequently include the solution of an \"auxiliary problem\" (which is itself an int eger prugramming problem) or a determination that no solution to the \"auxiliary problem\" exists. In contrasl. the cycles of the primal algorithm include only the adjoining of a ComUl'y cut and the executiun of the change of basis procedure of the simplex method. The procedure uf the algorithm and the proof of finit eness are founded on a classification of cycles of the algorithm and on two theore ms. Two types of prorf'dural restrictions are imposed as a basis for proving fin iteness: (a) selection of the incoming variable is subjected to regulation (beyond that required by t he simplex met hod). and t he rules applied are a function uf the type of cycle being executed; (b) selection of the row Llsed as the source of the data for the","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"7 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":"127990860","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":"The use of finite polynomial rings in the factorization of the general polynomial","authors":"D. Lloyd","doi":"10.6028/JRES.069B.024","DOIUrl":"https://doi.org/10.6028/JRES.069B.024","url":null,"abstract":"","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"9 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":"128449767","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}
Given a finit e number of points PI, ., PI." in th e plane, consider the proble m of findin g a point x that minimizes the sum of Euclidean distances 'Ld(p; , x) . More general versions of thi s proble m arise in s patial economics, concerning opti mal locations for a central office , plant, or ware ho use (compare [3]). Mos t of these will be based on me trics d more general than th e Euclidean metric. Among them, th e class of metri cs that are convex fun ctions in each variable co mmand partic ular interest: in thi s case, local minima are automati cally global minima, facilitating minimization decisively. We s hall show in thi s paper that convex metrics are invariant under translation, and therefore arise from a norm. For th e concepts of topologies, metrics , and norms in linear spaces see, for instance, [1)2 and [2]. 1. Metrics and norms. Le t L be a linear s pace over the fi eld R of real numbers. A fun ction f: L -7 R is convex if
给定平面上有限个点(PI， .， PI.)，考虑求一个点x使欧几里德距离之和最小的问题m (p;， x)。这个问题的更一般的版本出现在局部经济学中，涉及到中心办公室、工厂或仓库的最佳位置(比较b[3])。其中大部分将基于比欧几里得度规更通用的度量。其中，在每个变量中都是凸函数的度量类引起了特别的兴趣:在这种情况下，局部最小值自动成为全局最小值，从而促进了决定性的最小化。我们将在本文中证明凸度量在平移下是不变的，因此是由范数产生的。关于线性空间中的拓扑、度量和范数的概念，例如参见[1)2和[2]。1. 度量标准和规范。设L是实数域R上的线性s步。一个有趣的函数f: L - 7r是凸函数
{"title":"On convex metrics","authors":"C. Witzgall","doi":"10.6028/JRES.069B.021","DOIUrl":"https://doi.org/10.6028/JRES.069B.021","url":null,"abstract":"Given a finit e number of points PI, ., PI.\" in th e plane, consider the proble m of findin g a point x that minimizes the sum of Euclidean distances 'Ld(p; , x) . More general versions of thi s proble m arise in s patial economics, concerning opti mal locations for a central office , plant, or ware ho use (compare [3]). Mos t of these will be based on me trics d more general than th e Euclidean metric. Among them, th e class of metri cs that are convex fun ctions in each variable co mmand partic ular interest: in thi s case, local minima are automati cally global minima, facilitating minimization decisively. We s hall show in thi s paper that convex metrics are invariant under translation, and therefore arise from a norm. For th e concepts of topologies, metrics , and norms in linear spaces see, for instance, [1)2 and [2]. 1. Metrics and norms. Le t L be a linear s pace over the fi eld R of real numbers. A fun ction f: L -7 R is convex if","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"12 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":"114721518","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}
For a va riant of the two-d imensional Ri e mann int egra l suggested by S. Marcus , it is shown that the only integrab le fun c tions which are continuous (o r merely continuou s se parately in one of th e variables) are the cons ta nt fun ctions . The int egrab le di scontinuou s functions a re proven to be cons ta nt except poss ib ly on a se t which is "s ma ll" in a sense made precise in the paper.
{"title":"A variant of the two-dimensional Riemann integral","authors":"A. J. Goldman","doi":"10.6028/JRES.069B.023","DOIUrl":"https://doi.org/10.6028/JRES.069B.023","url":null,"abstract":"For a va riant of the two-d imensional Ri e mann int egra l suggested by S. Marcus , it is shown that the only integrab le fun c tions which are continuous (o r merely continuou s se parately in one of th e variables) are the cons ta nt fun ctions . The int egrab le di scontinuou s functions a re proven to be cons ta nt except poss ib ly on a se t which is \"s ma ll\" in a sense made precise in the paper.","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"114 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":"116040186","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}
This paper presents a unified account of the theory of least squares and its adaptations to statis· tic al models more complicated than the classical one. Firs t comes a developme nt of the properties of weak general ized matrix inverses, a useful variant of the more familiar pseudo·inverse. These properties are e mployed in a proof of the usual Gauss theorem, and in analyzin g the case in which known linear res traints are obeyed by the para me ters. Anothe r s itu ation treated is that of a s ingular variance-co variance matrix for the observations . Applications include the case of equi-correlated variables (i ncluding es timation despite ignorance of the corre lation), linear " res tra ints" subject to random error, and step wise linear es timation.
{"title":"Weak generalized inverses and minimum variance linear unbiased estimation","authors":"A. J. Goldman, M. Zelen","doi":"10.6028/JRES.068B.021","DOIUrl":"https://doi.org/10.6028/JRES.068B.021","url":null,"abstract":"This paper presents a unified account of the theory of least squares and its adaptations to statis· tic al models more complicated than the classical one. Firs t comes a developme nt of the properties of weak general ized matrix inverses, a useful variant of the more familiar pseudo·inverse. These properties are e mployed in a proof of the usual Gauss theorem, and in analyzin g the case in which known linear res traints are obeyed by the para me ters. Anothe r s itu ation treated is that of a s ingular variance-co variance matrix for the observations . Applications include the case of equi-correlated variables (i ncluding es timation despite ignorance of the corre lation), linear \" res tra ints\" subject to random error, and step wise linear es timation.","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"59 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1964-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115041056","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":"Equivalence of Certain Inequalities Complementing Those of Cauchy-Schwarz and Holder","authors":"J. B. Díaz, A. J. Goldman, F. Metcalf","doi":"10.6028/JRES.068B.020","DOIUrl":"https://doi.org/10.6028/JRES.068B.020","url":null,"abstract":"","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1964-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122915501","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 fri ctional forc e on a Brownian motion particle can be expressed by means of the time· correlation of the flu ctuating force on the particle . We s how that this method, applied to a spherical particle in a vi scous in compressibl e fluid , leads to Stokes ' Law. The c al culation is based on the theory of hydrodynamic flu ctuations due to Landau and Lifs hitz, and on a hydrodynamic theorem due to Faxe n. The subj ect of thi s article is th e connecti on be tween two diffe rent me th ods for calculating th e fri cti onal force on a Brownia n moti on particle. One method is based on th e s tati s ti cal mech ani cal th eory of irre ve r sible processes, and involv es evalua ti on of a certain time-correlati on formula. The other me thod is based on macroscopi c hydrodynami cs, and requires solution of th e N avi er- S tokes equ a ti on. F or simplicity we r estri c t the di scussion to the s pecial case of a spheri cal p article of radiu s a moving slowly through a vi scous in compressible fluid. The vi scosity coeffi cient is T/ . In the hydrodynamic theor y, th e fri c ti onal force F on a sphere movin g with constant velocity v is giv e n by Stokes' law,
作用在布朗运动粒子上的摩擦力e可以用作用在粒子上的作用力的时间相关性来表示。我们知道这种方法如何应用于可压缩流体中的球形粒子，从而得出斯托克斯定律。计算是基于朗道和利夫斯希兹的流体力学波动理论，以及法克斯的一个流体力学定理。本文的主题是计算布朗尼亚运动粒子上的摩擦力的两种不同方法之间的联系。一种方法是基于非可逆过程的力学理论，对某一时间相关公式进行计算。另一种方法基于宏观流体力学，需要求解N - N - S - S方程。为简单起见，我们把讨论转到半径为1的球形物体在可压缩流体中缓慢通过一个孔道的特殊情况。学校系数是T/。在流体力学理论中，作用于匀速运动的球体上的摩擦力F由斯托克斯定律给出。
{"title":"Hydrodynamic Fluctuations and Stokes' Law Friction","authors":"R. Zwanzig","doi":"10.6028/JRES.068B.019","DOIUrl":"https://doi.org/10.6028/JRES.068B.019","url":null,"abstract":"The fri ctional forc e on a Brownian motion particle can be expressed by means of the time· correlation of the flu ctuating force on the particle . We s how that this method, applied to a spherical particle in a vi scous in compressibl e fluid , leads to Stokes ' Law. The c al culation is based on the theory of hydrodynamic flu ctuations due to Landau and Lifs hitz, and on a hydrodynamic theorem due to Faxe n. The subj ect of thi s article is th e connecti on be tween two diffe rent me th ods for calculating th e fri cti onal force on a Brownia n moti on particle. One method is based on th e s tati s ti cal mech ani cal th eory of irre ve r sible processes, and involv es evalua ti on of a certain time-correlati on formula. The other me thod is based on macroscopi c hydrodynami cs, and requires solution of th e N avi er- S tokes equ a ti on. F or simplicity we r estri c t the di scussion to the s pecial case of a spheri cal p article of radiu s a moving slowly through a vi scous in compressible fluid. The vi scosity coeffi cient is T/ . In the hydrodynamic theor y, th e fri c ti onal force F on a sphere movin g with constant velocity v is giv e n by Stokes' law,","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1964-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124683069","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 simple non.equilibrium thermodynamics is developed and a particular example is studied. The theory is formulated to describe a viscoelastic fluid, capable of finite deformation, which need not be locally in or near a state of thermodynamic equilibrium. This fluid may support shear stresses only when away from local thermodynamic equilibrium _ A notion of time-temperature superposition is contained in the formulation of the constitutive equations. Conservation of energy is obeyed and the second law of thermudynamics is satisfied as a consequence of simple requirements on the constitutive relations. In an adiabatic isochoric motion the temperature increases when work is done on the material and decreases when the material does work. For given volume and temperature, e ntropy decrea ses whe n the mate rial is deformed from equ il ibrium . It is s hown in what general way viscos it y depends upon te mperature . For infinites imal s train, the special form of the stress-s train relations are derived in order to de termine how t.e mpe rature and t ime-t e mperature supe rpos ition e nte r in thi s case.
一个简单的否定。发展了平衡热力学，并研究了一个具体的例子。该理论是用来描述一种粘弹性流体，具有有限变形能力，不需要局部处于或接近热力学平衡状态。这种流体只有在远离局部热力学平衡时才能承受剪切应力。本构方程的公式中包含了时间-温度叠加的概念。由于对本构关系有简单的要求，就满足了能量守恒和热力学第二定律。在绝热等时运动中，当对物质做功时温度升高，当物质做功时温度降低。对于给定的体积和温度，当材料从等平衡状态变形时，其熵值减小。它显示了粘度一般是如何取决于温度的。为无限imal年代火车,stress-s训练关系的特殊形式派生为了de termine t.e mpe涂改和t ime-t e mperature挂表rpo过渡e《r在thi s。
{"title":"Thermodynamics of perfect elastic fluids","authors":"B. Bernstein, E. Kearsley, L. J. Zapas","doi":"10.6028/JRES.068B.016","DOIUrl":"https://doi.org/10.6028/JRES.068B.016","url":null,"abstract":"A simple non.equilibrium thermodynamics is developed and a particular example is studied. The theory is formulated to describe a viscoelastic fluid, capable of finite deformation, which need not be locally in or near a state of thermodynamic equilibrium. This fluid may support shear stresses only when away from local thermodynamic equilibrium _ A notion of time-temperature superposition is contained in the formulation of the constitutive equations. Conservation of energy is obeyed and the second law of thermudynamics is satisfied as a consequence of simple requirements on the constitutive relations. In an adiabatic isochoric motion the temperature increases when work is done on the material and decreases when the material does work. For given volume and temperature, e ntropy decrea ses whe n the mate rial is deformed from equ il ibrium . It is s hown in what general way viscos it y depends upon te mperature . For infinites imal s train, the special form of the stress-s train relations are derived in order to de termine how t.e mpe rature and t ime-t e mperature supe rpos ition e nte r in thi s case.","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1964-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117213247","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":"Generation and composition of functions","authors":"A. J. Goldman","doi":"10.6028/JRES.068B.015","DOIUrl":"https://doi.org/10.6028/JRES.068B.015","url":null,"abstract":"","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":"1964-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116300890","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 electromagnetic fields produced by an electric dipole immersed in an aniso tropic medium are considered. Various approaches to the problem are outlined with special reference to a cold plasma. An attempt is made to show the close relationship between previously published work on this subject. It is shown that information on the radiation field in an anisotrop ic media may be obtained directly from the shape of the refractive index surface.
{"title":"Theory of radiation from sources immersed in anisotropic media","authors":"James R. Wait","doi":"10.6028/JRES.068B.018","DOIUrl":"https://doi.org/10.6028/JRES.068B.018","url":null,"abstract":"The electromagnetic fields produced by an electric dipole immersed in an aniso tropic medium are considered. Various approaches to the problem are outlined with special reference to a cold plasma. An attempt is made to show the close relationship between previously published work on this subject. It is shown that information on the radiation field in an anisotrop ic media may be obtained directly from the shape of the refractive index surface.","PeriodicalId":408709,"journal":{"name":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","volume":"129 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1964-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132024646","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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