Showing papers on "Maxima and minima published in 1976"
TL;DR: In this article, the potential energy surfaces of clusters of up to thirteen atoms interacting through the Lennard-Jones and Morse potentials were investigated and a survey of the distinct minima available to systems of this size for the potentials used was provided.
Abstract: Mechanical and thermodynamic systems of a very few atoms (N≤ 100) interacting through central-forces are considered in the light of their role in an ideal model for “precipitation”. We discuss the stability of such systems at low energies and present detailed numerical investigations of the potential-energy surfaces of “clusters” of up to thirteen atoms interacting through Lennard-Jones and Morse potentials. Our main results comprise what we believe to be an almost exhaustive survey of the distinct minima available to systems of this size for the potentials used. Each such stable structure obtained is vibrationally and rotationally analysed and its symmetry is examined.The most striking feature of these results is the extreme sensitivity of the number of possible stable configurations to the range and softness of the pair potential. Thus, of no fewer than 988 minima for 13 Lennard-Jones atoms, only some 36 are supported by the (α= 3) Morse potential. The minima available are also classified geometrically and it is shown that non-crystallographic configurations predominate in structures of both greatest and least binding energy.A preliminary account of the statistical mechanics of cluster systems based on the rigid-rotor/harmonic oscillator approximation is given.
108 citations
TL;DR: In this paper, the first-order Taylor series expansions of the dynamic response functions with respect to the design variables were used for the unconstrained minimization of the exterior penalty function.
Abstract: Dynamic response quantities, treated parametrically in time, are included in the problem for the first time. The method is made efficient by the use of the following approximation concepts: (1)Design variable linking; (2)time parametric constraint deletion; and (3)use of the first-order Taylor series expansions of the dynamic response functions with respect to the design variables. It is found that the feasible design space in an optimum structural design problem in the dynamic response regime is usually disjoint. The Davidon-Fletcher-Powell algorithm is used for the unconstrained minimizations. To successfully implement the exterior penalty function with approximate constraints, dummy boundaries and a new concept of move limits are introduced. It is found that the sinusoidal contributions in the relation between design variables and dynamic response functions can cause infeasible minima of the exterior penalty function.
55 citations
TL;DR: In this paper, a simple proof of a theorem concerning functions whose local minima are global is presented and some closedness properties of this class of functions are discussed; see Section 2.
Abstract: In this note, a simple proof of a theorem concerning functions whose local minima are global is presented and some closedness properties of this class of functions are discussed.
24 citations
TL;DR: In this paper, the problem of minimum compliance of solid plates is formulated in statical terms, and necessary conditions for optimality that distinguish local minima from local maxima are derived from the second variation on the compliance.
17 citations
TL;DR: Conditions are given which are sufficient to prove the existence of a unique unconstrained minimizer in a convex compact set.
Abstract: Conditions are given which are sufficient to prove the existence of a unique unconstrained minimizer in a convex compact set. An exact formula is given for the amount by which a value of the function exceeds its global minimum.
14 citations
TL;DR: In this article, the angular location of Mie scatterers within the m range 1.050 −1.333 was calculated for the simple Rayleigh-Gans-Debye case of (m−1) → 0.
Abstract: Comprehensive data were collected on the angular location of the various types of maxima and minima of Mie scatterers within the m range 1.050–1.333. Detailed comparison with the extrema calculated for the simple Rayleigh–Gans–Debye case of (m−1) →0 led to a relatively simple correction equation which allows one to calculate with fairly good accuracy the angular location of Mie extrema quickly and easily and within a wide range of α and m values the limits of which are defined. Application of the results is discussed for fast particle size determinations in monodisperse systems of isotropic homogeneous spheres.
7 citations
TL;DR: It is shown that under a general formulation no favorable mathematical properties can be deduced beyond the “center of gravity”, and thus that computational algorithms are seemingly the best resort for resolving the problem.
7 citations
TL;DR: A linear approximation to the second-derivative matrix in terms of first derivatives of residuals is found to be very accurate in the neighborhood of minima; it provides a way of introducing second-Derivative information that is significantly superior to the use of variable-metric algorithms.
6 citations
01 Dec 1976
TL;DR: In this article, a generalized leaset-squares method for post-flight test analysis of small submerged craft is presented. But the method is not suitable for the analysis of large vessels.
Abstract: This paper discuses the application of a generalized leaset-squares method for post-flighttest analysis of small submerged craft. Specifically, the purpose is to identify the transfer functions characterizing the longitudinal and lateral dynamics from exprimental data pertaining to appropriate maneuvers. The method utilizes a set of signals, called measurement signals, generated by processing the input-output variables by a cascade of know digital filters. An enginvector of the Gram matrix of the measurement signales yields, after a linear transformation, the estimated coefficients of the desired transfer function. The method offers the advantages of global minima and robustness of estimates relative to data noise.
5 citations
4 citations