Showing papers on "Function (mathematics) published in 1972"

The analytic signal of two‐dimensional magnetic bodies with polygonal cross‐section: its properties and use for automated anomaly interpretation

Abstract: This paper presents a procedure to resoive magnetic anomalies due to two-dimensional structures. The method assumes that all causative bodies have uniform magnetization and a crosssection which can be represented by a polygon of either finite or infinite depth extent. The horizontal derivative of the field profile transforms the magnetization effect of these bodies of polygonal cross-section into the equivalent of thin magnetized sheets situated along the perimeter of the causative bodies A simple transformation in the frequency domain yields an analytic function whose real part is the horizontal derivative of the field profile and whose imaginary part is the vertical derivative of the field profile. The latter can also be recognized as the Hilbert transform of the former. The procedure yields a fast and accurate way of computing the vertical derivative from a given profile. For the case of a single sheet, the amplitude of the analytic function can be represented by a symmetrical function maximizing exactly over the top of the sheet. For the case of bodies with poiygonal cross-section, such symmetrical amplitude functions can be recognized over each corner of each polygon. Reduction to the pole, if desired, can be accomplished by a simple integration of the analytic function, without any cumbersome transformations. Narrow dikes and thin ilat sheets, of thickness less than depth, where the equivalent magnetic sheets are close together, are treated in the same fashion using the field intensity as input data, rather than the horizontal derivative. The method can be adapted straightforwardly for computer treatment. It is also shown that the analytic signal can be interpreted to represent a complex “field intensity,” derivable by differentiation from a complex “potential.” This function has simple poles at each polygon corner. Finally, the Fourier spectrum due to finite or infinite thin sheets and steps is given in the Appendix.

Electron Correlations at Metallic Densities. V

Abstract: In this paper we present a modification of an earlier theory of Singwi et al of electron correlations at metallic densities The modification consists in allowing for the change of the pair correlation function in an external weak field via the density derivative of the equilibrium pair correlation function This results in a new expression for the local-field correction The present theory has the merit of satisfying almost exactly the compressibility sum rule and of giving a satisfactory pair correlation function Results of self-consistent numerical calculations for the static pair correlation function, correlation energy, compressibility, and plasmon dispersion relation for the electron liquid in the metallic-density range are presented For those interested in the application of the results of the present paper, numerical values of the local-field correction as a function of wave number have been tabulated in the density range ${r}_{s}=1\ensuremath{-}6$

A Sequential Method Seeking the Global Maximum of a Function

Abstract: In this paper a sequential search method for finding the global maximum of an objective function is proposed. The method is applicable to an objective function of a single variable defined on a closed interval and such that some bound on its rate of change is available. The method is shown to be minimax. Computational aspects of the method are also discussed.

Non-Parametric Function Fitting

Abstract: SUMMARY In this note we consider the problem of fitting a general functional relationship between two variables. We require only that the function to be fitted is, in some sense, "smooth", and do not assume that it has a known mathematical form involving only a finite number of unknown parameters.

Widely convergent method for finding multiple solutions of simultaneous nonlinear equations

Abstract: A new method has been developed for solving a system of nonlinear equations g(x) = 0. This method is based on solving the related system of differential equations dg/dt±g(x)= 0 where in the sign is changed whenever the corresponding trajectory x(t) encounters a change in sign of the Jacobian determinant or arrives ata solution point of g(x)= 0. This procedure endows the method with much wider region of convergence than other methods (occasionally, even global convergence) and enableist to find multiple solutions of g(x)= 0 one after the other. The principal limitations of the method relate to the extraneouss ingularities of the differential equation. The role of these singularities is illustrated by several examples. In addition, the extension of the method to the problem of finding multiple extrema of a function of N variables is explained and some examples are given.

An algorithm for finding the absolute extremum of a function

Abstract: A GENERAL algorithm for finding the absolute minimum of a function to a given accuracy is described and special aspects of its application are illustrated by examples involving functions of one or more variables, satisfying a Lipschitz condition.

Multilinear Extensions of Games

Abstract: The multilinear extension of an n-person game v is a function defined on the n-cube IN which is linear in each variable and which coincides with v at the conrners of the cube, satisfying fx = v{i ∣

The maximally achievable accuracy of linear optimal regulators and linear optimal filters

Abstract: A linear system with a quadratic cost function, which is a weighted sum of the integral square regulation error and the integral square input, is considered. What happens to the integral square regulation error as the relative weight of the integral square input reduces to zero is investigated. In other words, what is the maximum accuracy one can achieve when there are no limitations on the input? It turns out that the necessary and sufficient condition for reducing the regulation error to zero is that 1) the number of inputs be at least as large as the number of controlled variables, and 2) the system possess no right-half plane zeros. These results are also "dualized" to the optimal filtering problem.

Easily Testable Realizations ror Logic Functions

Abstract: Desirable properties of "easily testable networks" are given. A realization for arbitrary logic function, using AND and EXCLUSIVE-OR gates, based on Reed-Muller canonic expansion is given that has many of these desirable properties. If only permanent stuck-at-0 (s-a-0) or stuck-at-1 (s-a-1) faults occur in a single AND gate or only a single EXCLUSIVE-OR gate is faulty, the following results are derived on fault detecting test sets for the proposed networks: 1) only (n/4) tests, independent of the function being realized, are required if the primary inputs are fault-free; 2) only 2n, additional inputs (which depend on the function realized) are required if the primary inputs can be faulty, where n, is the number of variables appearing in even number of product terms in the Reed-Muller canonical expansion of the function; and 3) the additional 2ne inputs are not required if the network is provided with an observable point at the output of an extra AND gate.

On a response characteristic of a mathematical neuron model.

Abstract: A mathematical neuron model in the form of a nonlinear difference equation is proposed and its response characteristic is investigated. If a sequence of pulses with a fixed frequency is applied to the neuron model as an input, and the amplitude of the input pulses is progressively decreased, the firing frequency of the neuron model, regarded as the output, also decreases. The relationship between them is quite complicated, but a mathematical investigation reveals that it takes the form of an extended Cantor's function. This result explains the “unusual and unsuspected” phenomenon which was found by L. D. Harmon in experimental studies with his transistor neuron models. Besides this, as an analogue of our mathematical neuron model, a very simple circuit composed of a delay line and a negative resistance element is presented and discussed.

Synthesis of recursive digital filters using the minimum p-error criterion

Abstract: The problem of designing a stable recursive digital filter to have an arbitrarily prescribed frequency response may be considered as an approximation problem. Using the minimum p - error criterion, a new problem of minimizing a function of n variables results, which is successfully solved using the Fletcher-Powell algorithm. An important theorem guaranteeing the existence of a stable optimum for a large class of synthesis problems is stated, and necessary modifications to the Fletcher-Powell algorithm to assure stability are considered. Finally a number of results of the application of this method are given.

Basic Concepts Underlying Singular Perturbation Techniques

Abstract: In many singular perturbation problems multiple scales are used. For instance, one may use both the coordinate x and the coordinate $x^ * = \varepsilon ^{ - 1} x$. In a secular-type problem x and $x^ * $ are used simultaneously. This paper discusses layer-type problems in which $x^ * $ is used in a thin layer and x outside this layer. Assume one seeks approximations to a function $f(x,\varepsilon )$, uniformly valid to some order in $\varepsilon$ for x in a closed interval D. In layer-type problems one uses (at least) two expansions (called inner and outer) neither of which is uniformly valid but whose domains of validity together cover the interval D. To define “domain of validity” one needs to consider intervals whose endpoints depend on $\varepsilon $. In the construction of the inner and outer expansions, constants and functions of e occur which are determined by comparison of the two expansions “matching.” The comparison is possible only in the domain of overlap of their regions of validity. Once overlap is established, matching is easily carried out. Heuristic ideas for determining domains of validity of approximations by a study of the corresponding equations are illustrated with the aid of model equations. It is shown that formally small terms in an equation may have large integrated effects. The study of this is of central importance for understanding layer-type problems. It is emphasized that considering the expansions as the result of applying limit processes can lead to serious errors and, in any case, hides the nature of the expansions.

Modelling Inter-city Road Distances by Mathematical Functions

Abstract: This study describes an analysis of seven distance functions with reference to their ability to estimate road distances between cities. The optimal parameters of each function are determined in relation to two samples of data, using two distinct goodness-of-fit criteria. A set of general statistical methods were chosen and applied to the comparison of the estimating power of the functions. Some properties of the lp distance function are also given with reference to its use in facilities location models.

Analytical and Numerical Studies of the Relative Convergence Phenomenon Arising in the Solution of an Integral Equation by the Moment Method

Abstract: The relative convergence phenomenon that occurs in the numerical solution of the integral equation for the iris discontinuity problem is studied both analytically and numerically. It is shown that the solution for the aperture field can be highly dependent upon the manner in which the kernel and the unknown function are approximated in the process of constructing a matrix equation by the moment method. An analytical explanation is provided for the above phenomenon and the theoretical predictions are verified numerically. Also incIuded is a suggested numerical algorithm for detecting and alleviating the relative convergence behavior for more general problems.

Application of the convolution equation to stream-aquifer relationships

Abstract: Flow and head variations in stationary linear stream-aquifer systems are obtained through application of the convolution equation. Four highly idealized cases involving finite and semi-infinite aquifers, with and without semipervious stream banks, are considered. Equations for the instantaneous unit impulse response function, the unit step response function, and the derivative of the unit step response function are given for each case. Head fluctuations in the aquifer due to an arbitrarily varying flood pulse are obtained for the cases involving a finite aquifer with and without a semipervious stream bank. Flow in and out of the aquifer at the stream bank is determined for the same cases and demonstrates the value of the convolution equation in evaluating the base flow. Head variations, and to a lesser extent flow variations, are apparently relatively insensitive to variations in aquifer diffusivity.

Predictive equations for self‐diffusion in liquids: A different approach

Abstract: Simple relationships have been obtained which can be used to predict self-diffusion coefficients of liquids with an average error of ±4%. In addition to the customary parameters, one of these equations contains the critical volume whereas the other uses Lennard-Jones potential parameters Both relations were derived by using a general relation between transport coefficients of pure fluids and a molecular-kinetic-model of liquids. No use has been made either of the Stokes-Einstein relation or the absolute rate theory. By applying the same relation between transport coefficients of pure fluids to gases, an equation has been obtained which can be used to calculate consistently molecular diameters in gases as a function of temperature, using Lennard-Jones potential parameters.

Theory of Surface Polarons

Abstract: We attempt to determine the binding energy and the wave function of the ground state of an electron that is attracted to the surface of an ionic crystal by its image potential and is repelled from the interior of the solid. For ionic crystals, such as LiF, the electrostatic theory is inadequate and the solid must be treated as a dynamical system. For shallow levels, the correction to the electrostatic approximation is small and behaves asymptotically as ${z}^{\ensuremath{-}3}$, where $z$ is the distance from the surface. The mass of a shallow electron is not enhanced. For deep levels the ground-state energy is calculated by a variational procedure in the limits of both weak and strong electron-phonon coupling. For the ground-state energy of an electron trapped on the LiF surface we find the value -0.29 eV. The mass of the deeply bound electron is enhanced.

Statistical Theory of Wave Propagation in a Random Medium and the Irradiance Distribution Function

Abstract: The complete statistical description of the waves in a random medium can be obtained from the characteristic functional or the cumulant functional of the wave function. The basic equations of these functionals are found first for the gaussian medium and then for the nongaussian medium. As an application of those equations, the approximate equations of the arbitrary-order coherence functions are derived, which are found to be equivalent to those recently obtained by several authors using different methods. The operational method is introduced to solve the equation of the νth-order moment of the irradiance, and an exact solution is obtained for the particular correlation function of the medium, assuming the gaussian form of the incident wave. The irradiance probability-density function is obtained without approximation by the use of the νth-order moment of irradiance for this particular medium, and is found to be exactly the Rice–Nakagami distribution with respect to the log irradiance. This distribution approaches the log-normal distribution in the outside domain of the wave beam, and is also checked in several points. Finally, the operator representation of the physical variables is introduced, and it is shown that the various equations, e.g., the wave equation and the energy-conservation equation, in the statistical system of the wave plus the random medium can be represented by the same equations as the corresponding equations in the deterministic medium. The discussion is also extended on the basis of DeWolf’s result for the irradiance-distribution function.

Unconstrained Inverse Electrocardiography: Epicardial Potentials

Abstract: The inverse problem in electrocardiography is attacked via the development of a model appropriate for the computation of epicardial potentials from a knowledge of heart and torso geometry as well as surface potentials. The model takes the form of an integral equation of the first kind in which the kernel is interpreted as a Green's function. A theoretical investigation of system independence in the presence of error is developed, and two techniques for the theoretical consideration of system independence are examined. Application of these two techniques to concentric spherical systems indicates that spheres with ratios of inner-to-outer radii less than 0.5 contain less than twenty independent parameters in the presence of realistic noise levels. The number of independent parameters deteriorates rapidly as this ratio falls below 0.5. These results suggest that it is not feasible to determine epicardial potentials from torso potentials by using unconstrained solutions.

Device for the control of shifting members of automatic change-speed transmissions

Abstract: A control system for regulating or controlling the actuating force of shifting elements which in automatically shifted change-speed transmissions act on friction elements that selectively brake, hold fast and release a structural element of such a change-speed transmission as a function of operating parameters; one of the operating parameters which is thereby used in the system of this invention is the predetermined change of the engine rotational speed as a function of time (dn/dt).

On hearing the shape of a drum: an extension to higher dimensions

Abstract: The inverse eigenvalue problem for vibrating membranes (4), may also be examined in three or more dimensions. Let us suppose that λn are the eigen values of the problemwhere Ω is a closed convex region or body in En and S is the bounding surface of Ω. The basic problem is to determine the precise shape of Ω on being given the spectrum of eigenvalues λn. In analogy with the membrane problem, it is clear that the trace function may be constructed in identical fashion; thuswhere G(r, r', t) is the Green's function of the diffusion equationand satisfies the Dirichiet condition G(r, r', t) = 0, r∈S, and the initial condition G(r, r', t) → δ(r–r') as t → 0.

Reexamination of Experimental Strength‐vs‐Grain‐Size Data for Ceramics

Abstract: The Orowan, Petch, and Knudsen equations were examined against 46 sets of strength-vs-grain-size data. For the 30 sets which are most discriminating, represented by 229 averaged-data points, the variances of the Orowan-Petch and Knudsen treatments are ∼4.5 and ∼17.1 kpsi2, respectively. Statistical considerations give preference to the Orowan-Petch treatment at a high confidence level in these 30 cases, showing that the Knudsen equation does not remove systematic variations of strength as a function of grain size. The Orowan-Petch treatment also appears to provide a sounder basis for extrapolation. The identification of the Orowan and Petch equations with postulated physical models permits interpretation of data in terms of probable underlying causes. Several examples are discussed and in most cases agree at least qualitatively with present understanding. A satisfactory physical model for Petch behavior in “nonyielding” ceramics is needed.

Stability Analysis of Power Systems

Abstract: A procedure for locating directly the lowest saddle point of the potential energy function associated with a power transmission network is presented. This procedure is used to compute a margin of stability that specifies the maximum asynchronous transient energy that can be retained by the system while in synchronism. The potential function is shown to be convex in the principal region of the angle space. A unique solution is shown to exist for the load flow problem if the algorithm of solution is initiated at the origin.

Lattice Dynamics of MgO, Al2MgO4 and Mg2SiO4 Spinel

Abstract: Summary The method of long waves is used to compute the volume dependence of the elastic properties of MgO, γ-Al2MgO4 (spinel), and γ-Mg2SiO4 (spinel structure) from a rigidion, central force model. A comparison of theoretical and experimental elastic constants for the first two compounds shows that the neglect of non-central forces and ionic polarization severely limits the application of the model for oxides and silicates. The Coulombic contributions to the elastic constants for both the 3–2 spinel (γ-Al2MgO4) and the 2–4 spinel (γ-Mg2SiO4) are expressed as dimensionless Madelung-like constants and are computed as a function of the oxygen parameter. The distortion of the two spinels from c.c.p. is shown to be primarily an electrostatic effect. The elastic properties of γ-Mg2SiO4 are predicted using the bond parameters for Mg-O found from data on periclase and for Si-O found from stishovite. Although the rigid-ion model is too restrictive to give geophysically useful predictions, it shows how lattice dynamics may be used as a framework through which laboratory data may be used to interpret seismic velocity and density profiles.

General transport theory of noise in pn junction-like devices—I. Three-dimensional Green's function formulation

Abstract: A general collective treatment of noise in three-dimensional junction devices of arbitrary geometry is presented, using Green's functions as in recent transport noise theories. The low-injection theory is extended to open-circuited devices. The density spectra are given in a form in which the volume part is linear in the Green's function and the covariance function, while the surface part is quadratic in the Green's function. The density covariance function for the short-circuited junction is Poissonian for low injection, except for a surface singularity. The noise input e.m.f and output current generator, as well as their cross correlation, are found directly for the hybrid transistor model and are expressed in the h ′ parameters, without the usual network transformation. The exact results indicate distributed effects; in particular, the current gain in the noise expressions (α noise ) is not equal to the small signal current gain α. The one-dimensional standard results are recovered in a lumped model approximation. For high injection, only the case of quasi band-band recombination (the Shockley-Read levels have equal capture probabilities for electrons and holes) is considered in this paper. The covariance function is then as for low injection but of half strength. The terminal noise depends, besides on the admittance or impedance and the current, on the emitter efficiency γ, the mobility ratio b , and the ratio of the junction admittance and the bulk admittance resulting from modulation effects. As a byproduct of this study, all pertinent network parameters are expressed in Green's functions.