Showing papers on "Maxima and minima published in 1970"

A method of unconstrained global optimization

Abstract: The method that is defined in the following finds the maximum or minimum of a real-valued function of many variables even if the function has local maxima or minima. The methods is iterative and guaranteed to converge for polynomials in several variables up to fourth degree. It can also be used successfully for other types of functions. The method approximates a function automatically if it is not a polynomial of degree four or less. It is shown that a global optimization method for fourth degree polynomials can solve systems of polynomial equations in many variables of any degree. The speed of convergence is analysed theoretically and empirically. The method has been applied by the author and collaborators to the solution of systems of nonlinear equations in many variables (up to 100 variables), determination of rate constants in nonlinear differential equations (systems identification), chemical equilibrium equations, curve fitting of sums of exponentials, pattern recognition, and analysis of spectra with nonlinear superposition (Bremermann and Lam [11]). The applications will be reported elsewhere.

The application of the magnetophonon effect to a study of hot electron phenomena in insb

Abstract: The magnetoresistance of three samples of high purity n-type insb(nd-na varying from 5*1013 to 5*1014 cm-3) is studied at lattice temperatures of 11 and 20 k with electric field strengths sufficient (>100 mv cm-1) to heat the electron distribution appreciably above the temperature of the lattice and at magnetic field strengths such that the effect of landau quantization is apparent. Two distinct types of magnetophonon extrema are observed, each having a different dependence on electric field and temperature. The first consists of a series of minima which are only observed when the electron temperature is in excess of about 20 k and which arises from the emission of long wavelength longitudinal optical (lo) phonons.

Visual Masking by Low-Frequency Sinusoidally Modulated Light

Abstract: Threshold energy of foveal, 2-msec white test flashes was plotted as a function of their timing with respect to sinusoidal fluctuations of nearly achromatic background light. With frequencies of 3.1–10 Hz, modulation near 100%, and peak retinal illuminance of 1280 td, the resulting masking curves are non-sinusoidal. Threshold maxima precede background maxima, whereas threshold minima coincide with background minima. The phase lead of the threshold maxima increases with frequency, but tends to stabilize at about 90° above 5 Hz. Maximal and minimal threshold energies also increase with frequency, but their ratio is nearly constant between 3.1 and 8 Hz. Steady light added to the 1280-td fluctuation elevates threshold minima, but often lowers maxima, and introduces secondary maxima that accompany decrements of background luminance. Generally, thresholds are higher with fluctuating backgrounds than with peak-equivalent steady backgrounds. The data for 1280-td and 128-td fluctuations confirm that masking is a nonlinear function of modulated background luminance, and suggest the presence of unidirectional rate sensitivity. Of several possible physiological correlates discussed, ganglion-cell discharge patterns are most nearly compatible with the masking curves.

Joint optimization of automatic equalization and carrier acquisition for digital communication

Abstract: In this paper, we analyze single-sideband amplitude-modulation digital communication systems to develop a method for jointly and optimally setting the carrier phase and the automatic transversal equalizer of such systems. Mean-square equalization error is used as the performance criterion. We develop a simple receiver structure and study the convergence of the method. Exact locations of the stationary points in the parameter space are determined and the classifications of the stationary points are obtained. We show that the mean-square equalization error has only global minima and saddlepoints, but not local minima and maxima. Thus, the mean-square equalization error will converge to the absolute minimum by the proposed method, regardless of the initial settings of the parameters. A simple condition on the step sizes of the adjustments is also obtained which ensures the convergence of the process. Explicit formulas of the joint optimum parameter settings and of the corresponding minimum mean-square error are obtained. For illustration purposes, a single-sideband digital communication system using a five- or nine-tap transversal equalizer is simulated on a computer. Both theory and simulation show that the equalization error depends critically on the carrier phase when the number of equalizer taps is not large, and that the minimum equalization error can be obtained by using the proposed method.

Definition And Dynamic Portrait Of Large SpaceStructures For Control System Synthesis

Abstract: A new definition of large space structures (LSS) is given, yielding a mathematical model of minimal order for three-axis attitude control system synthesis. Then, the dynamic portrait is introduced, allowing the structure to be designed with minimal excitation of certain vibration modes by the control variables. The theory is developed by considering space structures having a branched configuration near the centre of which are located attitude sensors and actuators collocated with an orthogonal control axis set to be orientated. It is well known that the complete set of space structures comprises two subsets, one in which rigid body dynamics may be assumed and the other, referred to as the Large Space Structures (LSS), for which one or more flexure modes, typically with very low natural frequencies, must be taken into account. This paper provides a much needed quantitative boundary between the two subsets, given by the definition that a structure is a LSS if the inequalities, k > 2.47 and co < 2k , are satisfied for any i, where co and k are, respectively, the natural frequency and excitability coefficient of the i* flexure mode. The approach is based on a comparison of the flexure mode motion with the rigid-body mode motion in the phase double-plane (i* modal phase-plane superposed on the rigid-body phase-plane) of a structural model in the modal state representation to which is applied a step control variable. Hence, the model derived is suitable for designing control systems employing discontinuous on-off thrusters as well as continuous actuators. The Lagrangian and Modal dynamic models of the structure are then used together to derive the dynamic portrait as a set of Transactions on the Built Environment vol 19, © 1996 WIT Press, www.witpress.com, ISSN 1743-3509 256 Structures in Space graphs, [w ](%,) and [k.](X), where A, is a selected physical parameter of the spacecraft. The structure may be designed, where practicable, to correspond with minima in these graphs to simplify the control problem. The new method is demonstrated by examples.

Two Centers of Libration

Abstract: The problem where the Hamiltonian of a one-dimensional system has two maxima and two minima, in the angular variable, is briefly discussed when certain conditions are assumed. A tentative of generalization of Garfinkel’s formalism is presented. Analogy to the 1/1 resonance case of the restricted problem is discussed.

Use of Directional Derivative in Locating Extrema.

Abstract: IT IS possible to use the directional derivative effectively in locating relative extrema of a function of two variables Why this method has been virtually over looked is a mystery The criteria given in most texts, if given at all, for testing whether a critical point is a maximum or a minimum are usually developed from a Taylor's series for functions of two vari ables It has been my experience that most first-year calculus students get lost some where in this development The method I am going to propose employs methods used in functions of a single variable and hence is more meaningful, I've found, with such students There are two approaches one can take The first is to develop an expression for the second directional derivative and use this expression in each individual problem The second is to use this expression for the second directional derivative to develop the previously mentioned criteria We shall look at both approaches in the order stated The directional derivative for a func tion of two variables, w = f(x, y), at the point (a, b) in a direction with the posi tive x-axis, is* defined as

Optimale wahl der abszissenwerte für die lineare regression bei gegebener kostenfunktion

Abstract: The choice of the abscissa-values in linear regression is considered as a decision problem and it is treated in the present paper for the case that this choice is connected with given costs and that the risk is calculable by means of a given quadratic lossfunction. Three optimisation problems are formulated: i) minimal costs with bounded risk - ii) minima! risk with bounded costs - iii) minimal s u n of costs and risk. For these problems the existence of solutions is proved under assumptions concerning the cost function, assumptions being always satisfied in practice. For the first two problems alternative theorems are stated facilitating their treatment essentially.

On optimal stochastic orbital transfer strategy

Abstract: The optimal stochastic orbit transfer strategy is defined as the sequence of guidance corrections which will minimize a statistical measure of final error, subject to the constraint that the total correction capability expended be less than a specified number. The dynamic programming algorithm is employed to solve this problem. It is assumed that the state of the system at any time can be described by the correction capability remaining and the maximum likelihood estimates of the orbit parameters. The numerical difficulty of storing the many values of the optimized performance index corresponding to every discrete value of the state variables is overcome by representing the performance surface only in the neighborhood of local minima and by introducing the effect of the correction capability state variable constraint in a simplified way.

“Weather-type” Frequencies and Temperature Fluctuation: a Reply

Abstract: LAWRENCE'S information on possible factors affecting the temperature record at Eskdalemuir Observatory, which we had unfortunately overlooked, is appreciated. It is, however, difficult to find suitable neighbouring stations, with homogeneous data series in a conveniently available form. With respect to the possible shelterbelt effects, the tendency would be towards higher maxima and lower minima in 1957–67 than in 1925–35. In fact, for anticyclonic type, which should best reflect this influence, the maxima and minima increase or decrease together in all four months investigated.