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.

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 ${r}_{s}=1\ensuremath{-}6$

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.

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.

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 ∣

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.

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.

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.

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.

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.

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).

Abstract: The log-normal distribution function is a useful function that could be profitably introduced to the undergraduate and used more frequently in chemical research because of its relatively simple analytical form and the diverse physical mechanisms and models that can generate it.

Abstract: It is known that the nonlinear Volterra integral equation \[ \varphi (t)\pi ^{( - 1 / 2)} \,\int_0^t (t - s)^{{ - 1 / 2} } [ {f(s) - \varphi ^n (s)} ]ds,\quad t\geqq 0,\geqq n\geqq 1,\] has a continuous solution $\varphi (t) \geqq 0$ which is unique for each bounded and locally lntegrable function $f(t) \geqq 0$ Our prior investigation considered the asymptotic behavior, as $t \to \infty $, of $\varphi (t)$ when $f(t) \sim \gamma _0 t^{ - a_0 } + \cdots ,\gamma _0 > 0,a_0 \geqq 0$. The goal here is to complete this analysis so that the behavior of $\varphi (t)$, as $t \to \infty $, is provided for all $a_0 \geqq 0,n \geqq 1$. In achieving this, we must treat some cases with properties that are markedly different from those previously considered.

Abstract: By applying dynamic corrections a seismic trace recorded at a distance x from the energy source should be varied in such a way as to obtain a trace which would be recorded at zero-distance, i.e. at the source itself. Only such a zero-offset-trace contains the correct sequence of reflection coefficients (reflectivity function), whilst all other traces contain a distorted reflectivity function. In the simplest case, the reflectivity function is compressed over a shorter time whereas in more complicated cases a partial inversion of the reflectivity function results. This happens when some of the reflection hyperbolae intersect one another. The reconstruction of the true zero-offset reflectivity function by the application of dynamic corrections can only be an approximative process. In the first case mentioned we must expect a decrease in accuracy of the corrected trace in comparison with a zero-offset-trace. In the second case, where intersections of the hyperbolae occur, accurate reconstruction is clearly impossible. The problems are discussed with the help of theoretical and practical examples.

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.

Abstract: Let h be any rapidly increasing function recursive in the halting problem. One can find a double recursive program of size n for a zero—one valued function of finite support whose smallest primitive recursive program is larger than h ( n ). One can find a general recursive program of size n for a zero—one valued function of finite support such that any general recursive program of size at most h ( n ) for the function runs extremely slowly on all large arguments.

Abstract: : The purpose of the book is to introduce the basic techniques for locating extrema (minima or maxima) of a function of several variables. Such a need arises naturally in various design optimization and planning problems. A standard set of techniques for unconstrained function extremization are presented. Small-step and large-step gradient methods: methods involving second partial derivatives of the function, such as the Newton-Raphson method and the Davidon-Fletcher-Powell method; and several other direct search methods are discussed. There are also discussions on elementary aspects of function extremization subject to linear or nonlinear constraints-such as the concept of constraint qualification, Fritz-John and Kuhn-Tucker theorems, penalty function method, etc. assuming differentiability and convexity of objective functions and constraint equations. In addition to presenting various standard algorithms for function extremization, the book also contains some simplified accounts of optimization problems drawn from various branches of engineering and operations research. (Author)

Abstract: The generalized power production function includes as special cases the Cobb-Douglas, the Transcendental, and the Cobb-Douglas with variable returns to scale. It can be estimated by ordinary least squares without simultaneous equation bias under the behavioral assumption of maximization of expected profits.

Abstract: It is well known that iteration of any number-theoretic function f, which grows at least exponentially, produces a new function f′ such that f is elementary-recursive in f′ (in the Csillag-Kalmar sense), but not conversely (since f′ majorizes every function elementary-recursive in f). This device was first used by Grzegorczyk [3] in the construction of a properly expanding hierarchy {ℰn: n = 0, 1, 2, …} which provided a classification of the primitive recursive functions. More recently it was shown in [7] how, by iterating at successor stages and diagonalizing over fundamental sequences at limit stages, the Grzegorczyk hierarchy can be extended through Cantor's second number-class. A problem which immediately arises is that of classifying all recursive functions, and an answer to this problem is to be found in the general results of Feferman [1]. These results show that although hierarchies of various types (including the above extensions of Grzegorczyk's hierarchy) can be produced, which range over initial segments of the constructive ordinals and which do provide complete classifications of the recursive functions, these cannot be regarded as classifications “from below”, since the method of assigning fundamental sequences at limit stages must be highly noneffective. We therefore adopt the more modest aim here (as in [7], [12], [14]) of characterising certain well-known (effectively generated) subclasses of the recursive functions, by means of hierarchies generated in a natural manner, “from below”.

Abstract: An algorithm for minimization of functions of many variables, subject possibly to linear constraints on the variables, is described. In it a subproblem is solved in which a quadratic approximation is made to the object function and minimized over a region in which the approximation is valid. A strategy for deciding when this region should be expanded or contracted is given. The quadratic approximation involves estimating the hessian of the object function by a matrix which is updated at each iteration by a formula recently reported by Powell [6]. This formula enables convergence of the algorithm from any feasible point to be proved. Use of such an approximation, as against using exact second derivatives, also enables a reduction of about 60% to be made in the number of operations to solve the subproblem. Numerical evidence is reported showing that the algorithm is efficient in the number of function evaluations required to solve well known test problems.

Abstract: We consider the correlation of three atoms in liquid neon from the neutron diffraction measurement of the isothermal density derivative of the pair correlation function Several closure approximations for the triplet correlation function are discussed Three of the approximations are representations of the triplet correlation function as a functional of the pair correlation function, and the other is expressed as a simple function of the pair correlation function

Abstract: Fifty Ss compared the subjective magnitudes of adjacent, objectively equal intervals between numbers by the method of triads. The numbers investigated were the integers from 1 to 10. For every comparison, the interval between the larger integers was more frequently judged closer, which supported previous findings that, for numbers used in magnitude estimation, subjective number is a negatively accelerated function of objective number. The nature of the psychological and physical variables in a number psychophysical function was discussed.

Abstract: We show how the steady-state distribution and the mean squared error of a delta modulator with an ideal integrator can be computed exactly when the input signal to the modulator is a stationary Gaussian process with a rational power spectral density. Curves are presented for the mean squared error as a function of the quantizer step size and the sampling interval for several different input spectra. The mathematical development makes use of the Markov properties of the system and involves series ex-expansions in n-dimensional Hermite functions. The key integral equation is generalized to treat the case of a realizable filter in the feedback path, but an analytic method of solving this equation has not been found.