Showing papers on "Maxima and minima published in 1982"

Three algorithms for interpreting models consisting of ordinary differential equations: Sensitivity coefficients, sensitivity functions, global optimization

Abstract: This paper reports the development and application of three powerful algorithms for the analysis and simulation of mathematical models consisting of ordinary differential equations. First, we describe an extended parameter sensitivity analysis: we measure the relative sensitivities of many dynamical behaviors of the model to perturbations of each parameter. We check sensitivities to parameter variation over both small and large ranges. These two extensions of a common technique have applications in parameter estimation and in experimental design. Second, we compute sensitivity functions, using an efficient algorithm requiring just one model simulation to obtain all sensitivities of state variables to all parameters as functions of time. We extend the analysis to a behavior which is not a state variable. Third, we present an unconstrained global optimization algorithm, and apply it in a novel way: we determine the input to the model, given an optimality criterion and typical outputs. The algorithm itself is an efficient one for high-order problems, and does not get stuck at local extrema. We apply the sensitivity analysis, sensitivity functions, and optimization algorithm to a sixth-order nonlinear ordinary differential equation model for human eye movements. This application shows that the algorithms are not only practicable for high-order models, but also useful as conceptual tools.

Enlarging the region of convergence of newton's method for constrained optimization

Abstract: In this paper, we consider Newton's method for solving the system of necessary optimality conditions of optimization problems with equality and inequality constraints. The principal drawbacks of the method are the need for a good starting point, the inability to distinguish between local maxima and local minima, and, when inequality constraints are present, the necessity to solve a quadratic programming problem at each iteration. We show that all these drawbacks can be overcome to a great extent without sacrificing the superlinear convergence rate by making use of exact differentiable penalty functions introduced by Di Pillo and Grippo (Ref. 1). We also show that there is a close relationship between the class of penalty functions of Di Pillo and Grippo and the class of Fletcher (Ref. 2), and that the region of convergence of a variation of Newton's method can be enlarged by making use of one of Fletcher's penalty functions.

On the latitude drift of sunspot groups and solar rotation

Abstract: The N-S drift of sunspot groups has been studied in a different way than previously, using positions of recurrent groups of the years 1874–1976. The existence of the meridional motions, the general shape of the drift curves, and the dissimilarity between these curves around sunspot maxima and minima, are all confirmed. In addition, also for the angular velocity of the Sun the same material gives differences around the times of sunspot maxima and minima.

An improved method for the evaluation of transition states

Abstract: A simple but efficient coupled iteration procedure is proposed for searching saddle points and extrema along the line of constrained minimum energy paths by (analytical) calculation of the derivatives, i.e., the reduced forces and reduced force constants. The advantage of the method is shown with analytical potentials as well as a calculation of HCHHNC rearrangement, as working examples.

Global minimum test problem construction

Abstract: A method is presented for the construction of test problems for which the global minimum point is known. Given a bounded convex polyhedron inR n , and a selected vertex, a concave quadratic function is constructed which attains its global minimum at the selected vertex. In general, this function will also have many other local minima.

Extreme value theory for continuous parameter stationary processes.

Abstract: In this paper the central distributional results of classical extreme value theory are obtained, under appropriate dependence restrictions, for maxima of continuous parameter stochastic processes. In particular we prove the basic result (here called Gnedenko's Theorem) concerning the existence of just three types of non-degenerate limiting distributions in such cases, and give necessary and sufficient conditions for each to apply. The development relies, in part, on the corresponding known theory for stationary sequences. The general theory given does not require finiteness of the number of upcrossings of any levelx. However when the number per unit time is a.s. finite and has a finite meanμ(x), it is found that the classical criteria for domains of attraction apply whenμ(x) is used in lieu of the tail of the marginal distribution function. The theory is specialized to this case and applied to give the general known results for stationary normal processes for whichμ(x) may or may not be finite). A general Poisson convergence theorem is given for high level upcrossings, together with its implications for the asymptotic distributions ofr th largest local maxima.

Ab initio study of rotational isomerism in vinylcyclopropane

Abstract: The geometry of vinylcyclopropane has been completely optimized at each critical point by analytic gradient (force) methods at the minimal STO-3G and the split-valence 3-21G basis set levels. The geometries obtained for the various critical points have been used to generate potential energy curves for vinyl group rotation within the rigid rotor approximation. Comparison of these curves clearly demonstrates the importance of complete geometry optimization. The potential energy curve for vinyl group rotation, generated with the s-trans STO-3G optimized geometry, predicts secondary gauche minima which are an artifact of the rigid rotor approximation. With complete geometry optimization along the curve, the STO- 3G basis set computations predict only s-trans and s-cis minima. In contrast, the complete optimizations with the 3-21G basis set, in agreement with experiment, predict a three-fold rotational contour with two equivalent gauche minima. These minima lie 6.86 kJ mol−1 above th e s-trans minimum. The computed barrier to rotation for the s-trans → gauche interconversion is 13.3 kJ mol−1. The electric dipole moment computed with the 3-21G basis for the s-trans 3-21G optimized geometry is 0.446 D or about 10% less than the experimental value.

Nonlinear integer programming for various forms of constraints

Abstract: A theoretical and computational investigation is made of the performance of a dynamic-programming-based algorithm for nonlinear integer problems with various types of constraints We include linear constraints, aggregated linear constraints, separable nonlinear constraints and constraints involving maxima and minima Separability of the objective function is assumed The new feature of the algorithm is that two types of fathoming or pruning are used to reduce the size of tables and number of computations: fathoming by bounds and fathoming by infeasibility

A limiting infisup theorem

Abstract: We show that duality gaps can be closed under broad hypotheses in minimax problems, provided certain changes are made in the maximum part which increase its value. The primary device is to add a linear perturbation to the saddle function, and send it to zero in the limit. Suprema replace maxima, and infima replace minima. In addition to the usual convexity-concavity type of assumptions on the saddle function and the sets, a form of semireflectivity is required for one of the two spaces of the saddle function. A sharpening of the results is possible when one of the spaces is finite-dimensional. A variant of the proof of the previous results leads to a generalization of a result of Sion, from which the theorem of Kneser and Fan follows.

Maximum Frequency Indication: Minima Counting Versus Zero-Crossing Counting

Abstract: Certain Doppler blood flow measurements are well done by extracting the greatest momentarily present frequency; an alternative method involving counting of minima is presented, along with a circuit type that directly produces a volume-indicating integral.

Normal modes of magnetic domain wall motion in a confined stripe domain lattice

Abstract: We report the observation of standing wave modes in an array of stripe domains confined by a pair of parallel cracks in a Gd, Ga:YIG film. These modes appear in the response spectrum of the confined lattice as shallow minima or maxima at frequencies lower than that of the usual domain wall resonance peak. A simple model, analogous to the forced response of a membrane clamped at the edges, fits the spatial patterns of wall motion observed at the frequencies of the maxima and minima. Experimental frequency‐wave vector values, interpreted with guidance from this analogy, provide the first experimental dispersion curve for a stripe domain lattice. We compare this result with recent theoretical calculations. The experimental value of the uniform mode frequency is 41.5 ± 0.2 MHz, with a long wavelength group velocity of 330±50 m/sec. A surprising conclusion from the observed extrema of the spatial patterns is that the damping of the waves is an order of magnitude less than expected from the damping of the unifo...

Dynamical stability and potential energy

Abstract: The relationship between stability and potential energy of stable mechanical systems is examined by means of specific diverse examples. In the case of statically stable systems the equilibrium configurations are always associated with potential minima. On the other hand, dynamic stability may occur at potential minima, at potential maxima, at fixed potentials other than extrema, with continuously varying potential, or even without any potential dependence.

Interval bounds for stationary values of functionals

Abstract: : A number of important problems in applied mathematics can be reduced to finding stationary values of functionals (maxima, minima, and critical values). For functionals defined in terms of integrals, the method of interval integration provides a way to obtain interval (two-sided) bounds for these stationary values. As a special case of this method, upper and lower bounds for eigenvalues of linear operators can be obtained. The inclusion of stationary values in intervals is based on the use of interval functions which include the function for which the functional is stationary, and its derivatives. A simple way to construct such interval functions is given, and examples are presented of a minimum and an eigenvalue problem. The improvement of initial results by iteration is indicated. (Author)

Application of an optimization technique to the inversion of an articulatory speech production model

Abstract: A procedure is proposed to identify parameters specifying the vocal-tract area function. Conventionnal optimization techniques fail to perform such a task because the articulatory-to-acoustic transformation contains severe non-linearities. So the problem is inevitably complicated by a set of local minima of the matching criterion. This paper describes a technique combining a well-adapted least-square algorithm and a table-lookup procedure which takes into account the non-linearities of the model function. Preliminary results on synthetic vowels are presented.

Analytic pole-zero modelling of speech spectra

Abstract: This paper describes an analytic technique, whereby the transfer function of a pole-zero model of the speech production process may be derived from the speech signal. The process involves fitting the model to the smoothed short-time amplitude spectrum which is derived by processing the speech signal through a bank of fourth-order, bandpass filters and cepstrally smoothing the output. The order of the pole-zero model is defined by twice the number of spectral maxima in the derived spectrum. The fitting criteria are specified on a perceptual basis rather than the usual least-squared error, the model being constrained to fit exactly the maxima and minima of the spectral curve.

The behaviour of extremals of a variational problem with phase constraints

Abstract: AN INTEGRAL functional with several local extrema is considered. The dependence of the extremals on the deformation of the domain of integration is studied.

Lorentz-symmetric solutions of nonlinear equations

Abstract: Asymptotic properties of a real scalar self-interacting classical field depending on one variablez = t2−x2 are studied. The fieldϕ(z) approaches a minimum of the potentialU(ϕ) for z → + ∞ and a maximum forz→−∞ ifU(ϕ(0)) is larger than two minima and smaller than two maxima ofU neighbouring toϕ(0).