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Showing papers on "Maxima and minima published in 1971"


Journal ArticleDOI
TL;DR: In this paper, a process with analytical criteria is described which sometimes finds smaller local minima in an algorithmic manner, under the assumption that a local minimum is known, and the process to be described sometimes finds the smaller local minimizers in an analytical manner.
Abstract: When a local minimum of a function of several variables has been found by use of an algorithm for finding such minima numerically, one often runs the same algorithm many times with different starting values in the hopes of finding a lower minimum. Here, under the assumption that a local minimum is known, a process with analytical criteria is described which sometimes finds smaller local minima in an algorithmic manner. Methods of descent are useful for minimizing functions of several variables. Generally, one can always obtain points (if such exist) for which the gradient vanishes, and moreover, points which are local minima. At saddle points. one can continue descent with second derivative information. A point which is a local minimum for a function may or may not be a global minimum. At this juncture one resorts to search techniques to attempt to further decrease the function. The process to be described sometimes finds smaller local minima in an algorithmic manner with analytical criteria. One has no general test, of course, for a global minimum. Consider first the problem of finding the global minimum for a 2nth degree polynomial P1(x) in one variable. The coefficient of x24 will be positive. Let xl be a local minimizer of P1. Then one may write

155 citations


Journal ArticleDOI
TL;DR: This article presents a method for solving linear programming problems in which elements of the tableau are stochastic, and concludes that the largest obstacle to examining related problems is in the limitations imposed by lack of data.
Abstract: This article presents a method for solving linear programming problems in which elements of the tableau are stochastic. Using least-cost poultry rations as an example, the authors demonstrate the procedure developed. Sufficiently accurate results are obtained with less time and complexity than required by alternative methods. The authors conclude that the largest obstacle to examining related problems is in the limitations imposed by lack of data, in this case with regard to biological minima.

45 citations


01 Dec 1971
TL;DR: In this article, the authors examined several global solutions for not necessarily convex programming problems with emphasis on the associated pitfalls, including penalty function methods, Lagrangian methods, grid methods, heuristic methods, random methods, and a branch and bound technique for separable programming problems.
Abstract: : Proposals for obtaining global solutions to not necessarily convex programming problems are examined with emphasis on the associated pitfalls. Included are penalty function methods, Lagrangian methods, grid methods, heuristic methods, random methods, and a branch and bound technique for separable programming problems.

27 citations


Journal ArticleDOI
TL;DR: Algebraic expressions for the generalized oscillator strengths of hydrogen-like atoms for the transitions involving s-, p-, and d-states with the principal quantum number n of 2, 3, and 4 were obtained in this article.
Abstract: Algebraic expressions are obtained for the generalized oscillator strengths of hydrogen-like atoms for the transitions involving s-, p-, and d-states with the principal quantum number n of 2, 3, and 4 Zero minima are found for all the transitions involving s-states, and their positions are shown to be proportional to the effective charge, the coefficients being tabulated The Transitions from the 1s-state sometimes show zero minima if different effective charges are used for the initial and the final states Even nodeless wave functions can produce zeros owing to the nodes of the spherical Bessel function Some excitation processes of Ne, Na, Mg, and Ar caused by high-energy charged particles are investigated in the first Born hydrogen-like approximation, and the possibility of finding zero minima in the differential cross sections is discussed; the 3s→4p transitions of Na and Mg have the most conspicuous zeros of all that are studied

20 citations


Book
01 Jan 1971
TL;DR: A survey of related characterizations of constrained extrema can be found in this paper, where the authors discuss the problems of Bolza with equality constraints and inequalities as added constraints.
Abstract: Preface.- I. Introductory Survey.- 1: A Survey of Derivative Characterization of Constrained Extrema.- II. Finite Dimensional Problems.- 2: Equality Constraints.- 3: Inequalities As Added Constraints.- 4: Extensions and Applications.- III. Variational Problems.- 5: The Problem of Bolza with Equality Constraints.- 6: The Problem of Bolza with Equality-Inequality Constraints.- 7: Extensions and Applications.

15 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that the kinetic theory of nonspherical molecules is unable to account for sharp dips in the composition dependence of the thermal diffusion factor and thermal conductivity of He-H2 mixtures; at most only shallow minima can be accommodated.
Abstract: The kinetic theory of nonspherical molecules is shown to be unable to account for sharp dips in the composition dependence of the thermal diffusion factor and thermal conductivity of He–H2 mixtures; at most only shallow minima can be accommodated. No maxima or minima in the thermal diffusion factor of Xe–CO2 can be accounted for. In view of these results and the recent careful measurements on He–H2 by Taylor and Weissman, it is concluded that the anomalies reported previously are probably spurious.

12 citations


Patent
22 Apr 1971
TL;DR: In this article, the authors proposed a method and a device for measuring minima or maxima of the thickness of a dielectric layer on electric conductor, where a measuring head scans the surface of the layer and minima and maxima are detected, stored and indicated.
Abstract: This invention relates to a method and a device for measuring minima or maxima of the thickness of a dielectric layer on electric conductor means having prominent portions, wherein a measuring head scans the surface of the layer and minima or maxima of the thickness of the dielectric layer are detected, stored and indicated. Several minima or maxima may simultaneously be indicated and compared with each other whereby it is possible to determine the general position of the conductor means in a dielectric and absolute extremum values of the thickness.

6 citations


Book ChapterDOI
01 Jan 1971
TL;DR: In this article, the authors analyze the isoperimetric constraints in the calculus of variations and present a simplified proof of the multiplier rule for fixed end point for the Lagrange problem.
Abstract: This chapter focuses on constraints in classical theory. In the theory of ordinary maxima and minima, the most interesting and realistic and also the most difficult problems are those in which constraints are present. Constraints also appear in many applications of the calculus of variations. Constraints add to the complexity both of the mathematical theory and of the associated computational methods. This chapter analyzes the isoperimetric constraints. The name isoperimetric arose in the historical development of the calculus of variations as a result of interest in the problem of finding the closed curve of given length (constant perimeter), which encloses the greatest area. The answer is a circle, and it is of some interest that this result seems to have been known since earliest times, long before the development of the calculus of variations. The chapter explores the problem of Lagrange in detail and presents a simplified proof of the multiplier rule for fixed end point.

3 citations