Abstract: An efficient method is developed to evaluate the function w ( z )= e - z 2 (1+(2 i /√ π )∫ z 0 e t 2 dt ) for the complex argument z = x + iy . The real part of w(z) is the Voigt function describing spectral line profiles; the imaginary part can be used to compute derivatives of the spectral line shapes, which are useful, e.g. in least-squares fitting procedures. As an example of the method a simple and fast FORTRAN subroutine is listed in the Appendix from which w(z) in the entire y ⩾ 0 half-plane can be calculated, the maximum relative error being less than 2 × 10 -6 and 5 × 10 -6 for the real and imaginary parts, respectively.

Abstract: Levels of Well-Being are social preferences, or weights that members of society associate with time-specific states of function. A Weighted Life Expectancy, which can be used to measure program outputs, is created by summing the levels across diverse cases and multiplying them by probable transitions (prognoses) among the states and levels. This operation requires however, that the Levels of Well-Being be measured on underlying metric scale. The present analysis compares preference measurements from a simple category rating procedure with those obtained using the more complex and difficult magnitude estimation method which has been claimed to yield ratio level measures. In a randomly counterbalanced design, 65 college students rated 30 case descriptions representing the range of the Well-Being continuum. The results exhibit the classical logarithmic relation observed for a prothetic continua. When transformed to a meaningful 0-1 unit scale, however, the magnitude responses are compressed at the lower end of the scale near death. Such results are inconsistent not only with category rating, but also with intuitive notions of the relative importance of the function states, with the results of rating procedures that simulate social choice, and with evidence that confirms the interval properties of the category ratings themselves. Furthermore, the ease of administration of category rating means that multiple attributes of cases can be considered jointly, avoiding the need to aggregate scale values for different attributes by arbitrary rules. In sum, magnitude estimation is inappropriate as a measurement method for a Health Status Index and is probably also inappropriate for other measures of utility and social choice.

Abstract: Inspired by the linear filter method introduced by D. P. Ghosh in 1970 we have developed a general theory for numerical evaluation of integrals of the Hankel type: Replacing the usual sine interpolating function by sinsh (x) =a· sin (ρx)/sinh (aρx), where the smoothness parameter a is chosen to be “small”, we obtain explicit series expansions for the sinsh-response or filter function H*. If the input function f(λ exp (iω)) is known to be analytic in the region o < λ < ∞, |ω|≤ω0 of the complex plane, we can show that the absolute error on the output function is less than (K(ω0)/r) · exp (−ρω0/Δ), Δ being the logarthmic sampling distance. Due to the explicit expansions of H* the tails of the infinite summation ((m−n)Δ) can be handled analytically. Since the only restriction on the order is ν > − 1, the Fourier transform is a special case of the theory, ν=± 1/2 giving the sine- and cosine transform, respectively. In theoretical model calculations the present method is considerably more efficient than the Fast Fourier Transform (FFT).

Abstract: A hardware logic simulation machine comprised of an array of specially designed parallel processors, with there being no theoretical limit to the number of processors which may be assembled into the array. Each processor executes a logic simulation function wherein the logic subnetwork simulated by each processor is implicitly described by a program loaded into each processor instruction memory. Logic values simulated by one processor are communicated to other processors by a switching mechanism controlled by a controller. If the array consists of i processor addresses, the switch is a full i-by-i way switch. Each processor is operated in parallel, and the major component of each processor is a first set of two memory banks for storing the simulated logic values associated with the output of each logic block. A second set of two memory banks are included in each processor for storing logic simulations from other processors to be combined with the logic simulation stored in the first set of memory banks.

Abstract: We show how interval analysis can be used to compute the minimum value of a twice continuously differentiable function of one variable over a closed interval. When both the first and second derivatives of the function have a finite number of isolated zeros, our method never fails to find the global minimum.

Abstract: This paper studies belief functions, set functions which are normalized and monotone of order 8. The concepts of continuity and condensability are defined for belief functions, and it is shown how to extend continuous or condensable belief functions from an algebra of subsets to the corresponding power set. The main tool used in this extension is the theorem that every belief function can be represented by an allocation of probability i.e., by a n-homomorphism into a positive and completely additive probability algebra. This representation can be deduced either from an integral representation due to Choquet or from more elementary work by Revuz and Honeycutt.

Abstract: Publisher Summary This chapter discusses small-angle scattering experiments with particles in solution—i.e., the particles are nonoriented. A large number of particles contribute to the scattering and the resulting spatial average leads to a loss in information. The information contained in the three-dimensional electron density distribution is thereby reduced to the one-dimensional distance distribution function. This function is proportional to the number of lines with length, which connect any volume element i with any volume element k of the same particle. The spatial orientation of these connection lines is of no account to the function. The connection lines are weighted by the product of the number of electrons situated in the volume elements i and k , respectively. The correlation between the function and the structure of the particle is also discussed in the chapter. The connection between the distance distribution function and the measured experimental scattering curve is also shown. It is observed that the each distance between two electrons of the particle, which is part of the function, leads to an angular-dependent scattering intensity. This physical process of scattering can be mathematically expressed by a Fourier transformation, which defines the way in which the information in “real space” (distance distribution function) is transformed into “reciprocal space” (scattering function). The chapter also discusses monochromatization and the camera type developed in Graz.

Abstract: Cluster statistics in two- and three-dimensional site percolation problems are derived here by Monte Carlo methods. The average number n, of percolation clusters with s occupied sites each is calculated by up to 19 runs on a 4000 X 4000 triangular lattice near pc. Our data support the two-exponent scaling assumption n, as-'f(z'), where z'= ( p/pc - 1)s". At the percolation threshold p = pc we find for s up to lo6 a rough agreement with the expected power law n, as-' over 12 decades in n, ; we can approximate the leading correction term near 5-10' by n,cXs-'(l-1.2 s-~'~). If the ratio U, = n,(p)/n,(p,) is plotted against z', then all data follow the same curve U, = f(z') for different p. This scaling function f(z') has a finite slope at z' = 0, has a maximum f(zk, = -0.8) = 5 for p below pc, and decays rapidly for z'+*m. For 5-+m at fixed p this rapid decay corresponds to In n, Cc -s"' above pc and In n, a --s below pc. Apart from finite-size corrections we find the second moment x =Zs2n, diverges as 1 p -pC(-', with y = 2.4, on both sides of the phase transition; the amplitude ratio x(p p,) is about 200. The fraction of occupied sites belonging to the infinite cluster vanishes as (p -p,)'. with p -0.13. In three dimen- sions using system sizes up to 400 x 400 x 400 the two-exponent scaling function is also supported, with the same universal function f(z') valid for both the simple cubic and BCC lattices. f(z') has a maximum f(z;, = -0.8) = 1.6. The amplitude ratio is approximately 11. Our conclusions are in general consistent with but more complete than other recent Monte Carlo work by Stoll and Domb, Leath and Reich, and Nakanishi and Stanley.

Abstract: The objective in nonparametric regression is to infer a function $m(x)$ on the basis of a finite collection of noisy pairs $\{(X_i, m(X_i) + N_i)\}^n_{i=1}$, where the noise components $N_i$ satisfy certain lenient assumptions and the domain points $X_i$ are selected at random. It is known a priori only that $m$ is a member of a nonparametric class of functions (that is, a class of functions like $C\lbrack 0, 1\rbrack$ which, under customary topologies, does not admit a homeomorphic indexing by a subset of a Euclidean space). The main theoretical contribution of this study is to derive uniform convergence bounds and uniform consistency on bounded intervals for the Nadaraya-Watson kernel estimator and its derivatives. Also, we obtain the corresponding convergence results for the Priestly-Chao estimator in the case that the domain points are nonrandom. With these developments we are able to apply nonparametric regression methodology to the problem of identifying noisy time-varying linear systems.

Abstract: In the context of deterministic multicriteria maximization, a Pareto optimal, nondominated, or efficient solution is a feasible solution for which an increase in value of any one criterion can only be achieved at the expense of a decrease in value of at least one other criterion. Without restrictions of convexity or continuity, it is shown that a solution is efficient if and only if it solves an optimization problem that bounds the various criteria values from below and maximizes a strictly increasing function of these several criteria values. Also included are discussions of previous work concerned with generating or characterizing the set of efficient solutions, and of the interactive approach for resolving multicriteria optimization problems. The final section argues that the paper's main result should not actually be used to generate the set of efficient solutions, relates this result to Simon's theory of satisficing, and then indicates why and how it can be used as the basis for interactive procedures with desirable characteristics.

Abstract: We show how interval analysis can be used to compute the minimum value of a twice continuously differentiable function of one variable over a closed interval. When both the first and second deriva- tives of the function have a finite number of isolated zeros, our method never fails to find the global minimum. Consider a function f(x) in C 2. We shall describe a method for computing the minimum value of f(x) on a closed interval (a, b). We shall see that, if f'(x) and f"(x) have only a finite number of isolated zeros, our method always converges. In a subsequent paper, we shall show how the method can be extended to the case in which x is a vector of more than one variable. Moreover, it will be extended to the constrained case, and a modified method will remove the differentiability condition. The present paper serves to introduce the necessary ideas. In practice, we can only compute minima in a bounded interval. Hence, it is no (additional) restriction to confine our attention to a closed interval. The term global minimum used herein refers to the fact that we find the smallest value of f(x) throughout (a, b). We shall not mistake a local minimum for the global one. Indeed, our method will usually not find local minima, unless forced to do so. Its efficiency would then be degraded if it did. In our method, we iteratively delete subintervals of (a, b) until the remaining set is sufficiently small. These subintervals consist of points at which either f(x) is proved to exceed the minimum in value or else the derivative is proved to be nonzero.

Abstract: This paper provides a complete set of local duality results for a utility maximizing consumer (or single output cost minimizing firm). Given a continuous local expenditure function defined on a compact, convex set of positive prices we establish the existence of continuous local direct, indirect utility and distance functions. This procedure avoids troublesome continuity problems at the boundary of IR N. In addition it is shown that if two utility functions are second order approximations at some point, then their respective expenditure, distance, and indirect utility functions are also second-order approximations to each other at some point. This latter result provides additional impetus for using duality theory and substantial justification for the use of "flexible" functional forms which can provide second-order differential approximations to any twice continuously differentiable function at a point.

Abstract: T he strain-energy density function surface for the rubber tested by L. R.G. T reloak (1944a) is determined from bis stress-strain data. The data were given for the three pure homogeneous strain paths of simple elongation, pure shear, and equi-biaxial extension of a thin sheet. The surface is formed by plotting calculated points of the strain-energy function above a plane having the first and second strain invariants as rectangular cartesian coordinates. The strain-energy function is expressed as a double power series in the invariants expanded about the zero energy state which is the origin of coordinates. An analysis of this experimentally derived surface provides the information required for the rational selection of terms and the determination of the coefficients in the series expansion, thus defining a function within the Rivlin-type formulation. The function so determined is tested by employing it in the appropriate constitutive formulae to compute stresses for comparison with experimental values. Another test is made by utilizing the function to predict shapes of an inflated membrane for comparison with experimentally observed shapes. Excellent agreement between prediction and experiment is found. A second demonstration is given for another rubber tested by D.F. J ones and L.R.G. T reloar (1975). Again, excellent results are obtained.

Abstract: The adaptive line enhancer (ALE) is an adaptive digital filter designed to suppress uncorrelated components of its input, while passing any narrow-band components with little attenuation. The purpose of this paper is to analyze the second-order output statistics of the ALE in steady-state operation, for input samples consisting of weak narrow-band signals in white Gaussian noise. The ALE output is shown to be the sum of two uncorrelated components, one arising from optimum finite-lag Wiener filtering of the narrow-band input components, and the other arising from the misadjustment error associated with the adaptation process. General expressions are given for the output auto-correlation function and power spectrum with arbitrary narrow-band input signals, and the case of a single sinusoid in white noise is worked out as an example. Finally, the significance of these results to practical applications of the ALE is mentioned.

Abstract: This paper uses linear programming to compute an optimal policy for a stopping problem whose utility function is exponential. This is done by transforming the problem into an equivalent one having additive utility and nonnegative (not necessarily substochastic) transition matrices.

Abstract: Given an i.i.d. sequence X 1,X 2, … with common distribution function (d.f.) F, the usual non-parametric estimator of F is the e.d.f. Fn ; where Uo is the d.f. of the unit mass at zero. An admissible perturbation of the e.d.f., say , is obtained if Uo is replaced by a d.f. , where is a sequence of d.f.'s converging weakly to Uo. Such perturbed e.d.f.′s arise quite naturally as integrals of non-parametric density estimators, e.g. as . It is shown that if F satisfies some smoothness conditions and the perturbation is not too drastic then ‘has the Chung–Smirnov property'; i.e., with probability one, 1. But if the perturbation is too vigorous then this property is lost: e.g., if F is the uniform distribution and Hn is the d.f. of the unit mass at n–α then the above lim sup is ≦ 1 or = ∞, depending on whether or

Abstract: Fix a family C of continuous distributions on the line. Sufficient and (different) necessary conditions on C are given in order that the sample distribution function be an optimal estimator in the asymptotic minimax sense. The abstract results are illustrated by a variety of concrete families C that have arisen in the literature; some of these illustrations settle known, but previously unsolved, problems. Methods involve systematic consideration of statistical experiments whose parameter lies in a Hilbert space, and the theory of abstract Wiener spaces.

Abstract: W E shall motivate our investigation of exact index numbers by presenting an example of application of index number theory. We consider a profit-maximizing, price-taking firm endowed with a continuously differentiable non-decreasing and concave production function F(q). Suppose we have observations on the normalized prices p (nominal prices divided by the price of output) and the quantities of the inputs q at two distinct points in time, but not the outputs. An index number formula for the unobserved output is a function of the normalized prices and the quantities of the inputs at the two points, say I(P1,P2,q1,q2), which gives a measure of the ratio of the outputs at the two points, that is,

Abstract: A functional Fourier series is developed with emphasis on applications to the nonlinear systems analysis. In analogy to Fourier coefficients, Fourier kernels are introduced and can be determined through a cross correlation between the output and the orthogonal basis function of the stochastic input. This applies for the class of strict-sense stationary white inputs, except for a singularity problem incurred with inputs distributed at quantized levels. The input may be correlated if it is zero-mean Gaussian. The Wiener expansion is treated as an example corresponding to the white Gaussian input and this modifies the Lee-Schetzen algorithm for Wiener kernel estimation conceptually and computationally. The Poisson-distributed white input is dealt with as another example. Possible links between the Fourier and Volterra series expansions are investigated. A mutual relationship between the Wiener and Volterra kernels is presented for a subclass of analytic nonlinear systems. Connections to the Cameron-Martin expansion are examined as well The analysis suggests precautions in the interpretation of Wiener kernel data from white-noise identification experiments.

Abstract: SUMMARY Two graphical procedures for analysing distributions of survival time are compared. One works with the survivor function, or with the order statistics, and the other is based on estimates of log hazard and is designed to give points with independent errors of constant known variance. The distribution of survival time, T, may be described equivalently by a probability density function f(t), by a survivor function F(t) = pr (T> t), and by a hazard function p(t) = f(t)/l(t). One advantage of p(.) is that often, although not always, it varies slowly over all or most of its range. Constant p(.) corresponds to an exponential distribution and, even when it is not intended to base the analysis on an assumption of exponential form, that distribution often gives a natural base against which to judge distributional shape. There are a lot of ways of comparing data graphically with the exponential distribution; for a cryptic list of ten such, see Cox (1978, Table 2). Most are transformations of one another so that choice is partly a matter of taste. The present note contrasts two simple graphical techniques.

Abstract: We consider the marginal function of a vertically perturbed nonlinear mathematical program with equality and inequality constraints. Conditions are given to have this function locally Lipschitz and to obtain estimates for its generalized gradient.

Abstract: Lower bounds are derived on the number of comparisons to solve several well-known selection problems Among the problems are finding the t largest elements of a given set m order (Wt), finding the s smallest and t largest elements in order (We.t), and finding the tth largest element (Vt) The results follow from bounds for more general selection problems, where an arbitrary partml order is given The bounds for Wt and Vt generahze to the case where comparisons between hnear functions of the input are allowed The approach is to show that a comparison tree for a selection problem contains a number of trees for smaller problems, thus estabhshmg a lower bound on the number of leaves An equivalent approach uses an adversary, based on a numerical "chaos" function that measures the number of unknown relations

Abstract: 1. Introduction and summary. This paper offers yet another example of what probability theory can do for analysis. Using a Feynman-Kac formula derived in the theory of random evolutions (5), we find an expression (1) for the spectral radius r(A) of a finite square non-negative matrix A. This expression makes it very easy to study how r(A) behaves as a function of the diagonal elements of A.