Showing papers on "Function (mathematics) published in 1968"

Variance Algorithm for Minimization

Abstract: An algorithm is presented for minimizing real valued differentiable functions on an TV-dimensional manifold. In each iteration, the value of the function and its gradient are computed just once, and used to form new estimates for the location of the minimum and the variance matrix (i.e. the inverse of the matrix of second derivatives). A proof is given for convergence within iV-iterations to the exact minimum and variance matrix for quadratic functions. Whether or not the function is quadratic, each iteration begins at the point where the function has the least of all past computed values.

Triggered Correlation

Abstract: This paper shows to what extent an average-response computer can be utilized for computing a cross-correlation function. This type of computer needs synchronization pulses, and the simplest methods of computation are those in which these pulses are directly derived from one of the signals (triggered correlation). The first method is to generate a synchronization pulse whenever the signal crosses a pre-set threshold in any direction. In this case, the computer output function is shown to be proportional to the true correlation function, for Gaussian signals. In a second method, synchronization pulses are produced when the signal crosses the threshold in a specified (e.g., positive) direction. Then the computer output is found to be contaminated by a systematic error, which, in turn, depends on the derivative of the correlation function. These two methods are described in detail, both with respect to the results and to the accuracy obtainable. Several other, less important, methods are only briefly described.

Quantum Noise. XI. Multitime Correspondence between Quantum and Classical Stochastic Processes

Abstract: A correspondence ${a}_{j}\ensuremath{\leftrightarrow}{\ensuremath{\alpha}}_{j}$ between operators $\mathrm{a}=[{a}_{1}, {a}_{2}, \ensuremath{\cdots}{a}_{f}]$ and $c$ numbers $\ensuremath{\alpha}=[{\ensuremath{\alpha}}_{1}, {\ensuremath{\alpha}}_{2}, \ensuremath{\cdots}{\ensuremath{\alpha}}_{f}]$ together with an arbitrary ordering rule $\mathcal{C}$ (e.g., in sequence from 1 to $f$) permit an association $M(\mathrm{a})=\mathcal{C}{M}^{(c)}(\ensuremath{\alpha})$ between a general operator $M(\mathrm{a})$ and an associated $c$ number function ${M}^{(c)}(\ensuremath{\alpha})$. A quasiprobability $P(\ensuremath{\alpha}, t)$ is then defined so that a general ensemble average can be written as an ordinary integration: $〈M(\mathrm{a}(t))〉=\ensuremath{\int}{M}^{(c)}(\ensuremath{\alpha})d\ensuremath{\alpha}P(\ensuremath{\alpha}, t)$. The equation for $\frac{\ensuremath{\partial}P(\ensuremath{\alpha}, t)}{\ensuremath{\partial}t}$ suggests that the $\ensuremath{\alpha}$ obeys a classical Markoff process. If this classical Markoff process is taken literally, multitime classical averages can be computed. Do these correspond to appropriate quantum averages? For the case of field operators such that $[b, {b}^{\ifmmode\dagger\else\textdagger\fi{}}]=1$, important in discussing laser statistics, we show that with ${a}_{1}={b}^{\ifmmode\dagger\else\textdagger\fi{}}$ and ${a}_{f}=b$, the classical multitime average is equivalent to the average of the corresponding quantum operators written in time-ordered, normal-ordered sequence. For the atomic operators in a laser problem, we obtain the desired correspondence, but find that the more complicated commutation rules necessarily lead to derivative correction terms when multitime averages are taken. Our derivation of multitime averages is based on the quantum regression theorem. We show that this theorem is equivalent to assuming the quantum system to be Markoffian, by showing that it leads to an appropriate factorization of a multitime density matrix and to a Chapman-Kolmogoroff-like condition on the conditional density matrix.

Asymptotic Normality of Simple Linear Rank Statistics Under Alternatives II

Abstract: This is a straightforward continuation of Hajek (1968). We provide a further extension of the Chernoff-Savage (1958) limit theorem. The requirements concerning the scores-generating function are relaxed to a minimum: we assume that this function is a difference of two non-decreasing and square integrable functions. Thus, in contradistinction to Hajek (1968), we dropped the assumption of absolute continuity. The main results are accumulated in Section 2 without proofs. The proofs are given in Sections 4 through 7. Section 3 contains auxiliary results.

Lower bounds for approximation by nonlinear manifolds

Abstract: which is the precision to which F approximates K. An instance of classical interest is that in which L is C([O, 1]), K is AO-i.e. those functions bounded by one, and whose modulus of continuity is dominated by the modulus of continuity function w-and F is the family P,, of polynomials of degree n-1. For this case it is known that there are positive constants A and B, independent of w and n, such that

A novel approach to linear model simplification

Abstract: New techniques for the model simplification problem are presented. If the given system is expressed by a high-order transfer function, the technique is to expand the function into a continued fraction and then ignore some quotients. If the system is in the state equations form, the method is realized by partitioning the matrix and discarding some parts. The new approach not only offers a simple procedure and a good approximation but also gives a unified viewpoint of the general analysis of linear systems.

Simultaneous optimum detection and estimation of signals in noise

Abstract: The problem of simultaneous detection and estimation of signals in noise is formulated in the language of statistical decision theory. Optimum structures and corresponding general measures of system performance are derived under the Bayes criterion of minimum average risk for the detectors and estimators appropriate to this type of joint operation: It is shown that whereas the structures of the resulting optimum detectors require a class of modified likelihood ratios, the structures of the optimum estimators, which act on the data when there is uncertainty as to the presence of a signal [p(H_{1}) , have a common canonical form for a wide variety of operating strategies. This form is identical with that obtained for estimation alone [p(H_{1}) , even though there is generally mutual coupling between detector and extractor. A simple structure is obtained for the estimation of amplitude and waveform in the case of a quadratic cost function (least mean-square error), where it is found that the estimator which is optimum here [p(H_{1}) is the product of the corresponding Bayes estimator in the "classical" case [P(H1) = 1 ] and a simple algebraic function of the generalized likelihood ratio. In this case, one can also show that estimators that are unbiased in the classical sense remain unbiased. In parallel with the classical theory, a generalized version of unconditional maximum likelihood estimation is obtained for the "simple" cost function when p(H_{1} . It is found that estimators that are linear in the classical case [p(H_{1}) = 1 ] are nonlinear in the more general situation [p(H_{1}) , where increased structural complexity is always the rule for both detectors and estimators. A specific example involving the coherent estimation of signal amplitude illustrates the approach. Analogous extensions to prediction and filtering are formulated, making it evident that a broad area for further generalizations of classical Bayes detection and extraction theory is available for systematic investigation.

Magnetic correlations and neutron scattering

Abstract: An introduction is given to the methods and results of some recent researches into statistical thermodynamics bearing upon the correlation functions of magnetic moments in Heisenberg-coupled spin-only magnets, and their intimate connection with neutron-scattering theory and practice is brought out. The interrelationships between the correlation function, the relaxation function, the generalized susceptibility, the power spectrum of the fluctuations and the neutron scattering are explained, and it is shown that insights into any one of these aspects can serve to illuminate the others. Different forms of approximate theory are seen to be suitable in approaching the topic through these different avenues; the method of moments for analysing the power spectrum and the connection with Green function theory are described in particular. Some examples are calculated in molecular-field theory and with various other simple approximations; expressions for the frequency spectrum of the susceptibility are obtained in various temperature ranges. The extreme forms of the frequency spectrum in appropriate conditions as a set of delta functions, as a pseudo-Gaussian and as a pseudo-Lorentzian are derived, and the transitions between these cases are considered. The approximation of spin diffusion and its limitations are analysed; the problem of the dynamical slowing-down of magnetic fluctuations near the phase-transition point is examined and the current position in this enquiry is set out.

A test of a two-stage model of magnitude judgment

Abstract: It has been suggested that the power law J = an, describing the relationship between numerical magnitude judgments and physical magnitudes, confounds a sensory or input function with an output function flawing to do with O’s use of numbers. Judged magnitudes of differences between stimuli offer some opportunity for separating these functions. We obtained magnitude judgments of differences between paired weights, as well as magnitude judgments of the weights making up the pairs. From the former we calculated simultaneously an input exponent and an output exponent, working upon Attneave’s assumption that both transformations are describable as power functions. The inferred input and output functions, in combination, closely predict the judgments of individual weights by the same Os. Although pooled data (geometric means of judgments) conform fairly well to a linear output function, individual data do not; i.e., individual Os deviate quite significantly fromlinearity and from one another in their use of numbers. Individual values of the inferred sensory exponent, k, show significantly better uniformity over Os than do values of the phenotypica! magnitude exponent previously found to describe interval judgments of weight.

Use of Analyticity in the Calculation of Nonrelativistic Scattering Amplitudes

Abstract: A new method of calculating nonrelativistic scattering amplitudes is presented. The scattering amplitude is first calculated as a function of the complex energy below the scattering threshold, and the numerical results are then analytically continued to the physical region. The method is used to calculate two-body and two-channel scattering amplitudes. The numerical analytic continuation is accomplished by a rational-fraction representation similar to the Pad\'e method. Several techniques of numerical analytic continuation by rational fractions are described, and some examples are discussed.

Adjusting Poles and Zeros of Dielectric Dispersion to Fit Reststrahlen of Pr Cl 3 and La Cl 3

Abstract: The dielectric dispersion function of a medium can be defined by the locations of the poles and zeros of that function in the complex-frequency plane. In this form the dispersion function is not restricted by special characteristics of any specific physical model from which dispersion might be derived. The locations of these poles and zeros are subject to several more or less fundamental physical restrictions which are described in this paper. The connections between the locations of the poles and zeros, the frequencies and damping constants of certain optical modes, and the Lyddane-Sachs-Teller relation are discussed. Adjustments of locations of poles and zeros, consistent with the physical restrictions, were performed to obtain least-squares fits of reststrahlen data from Pr${\mathrm{Cl}}_{3}$ and La${\mathrm{Cl}}_{3}$ using a small number of poles and zeros. In this way, approximate dispersion functions for the two dielectric tensor components of each crystal were obtained at room temperature and at lower temperatures.

Calculations on the 2 S Ground State of the Lithium Atom Using Wave Functions of Hylleraas Type

Abstract: A wave function was obtained for the $^{2}S$ ground state of the lithium atom using 60 basis functions of the Hylleraas type, i.e., with interelectronic distance coordinates. The energy obtained was -7.478025 atomic units as compared with the value -7.478069 calculated from experiments. The wave function was used to calculate the Fermi contact term. It was found that this basis set gave the value 2.906, which is in agreement with experiments, when both doublet spin functions were used, but a value that was 4% greater when only one spin function was used. In the first case, 100, and in the latter, 60, linear parameters were varied. The interelectronic distance coordinates are expanded according to a formula by Sack. The final integrals are evaluated analytically, and the resulting formulas, along with a short discussion of their convergence properties, are given in an Appendix.

Anomalies of the Energy of Positive Ions Extracted from High‐Frequency Ion Sources. A Theoretical Study

Abstract: Starting from the hypothesis of the modulation of energy of ions extracted by the high‐frequency voltage one calculates the width and the form of the energy spectrum of these ions. The variation of the energy deviation of the ions in function of the extraction voltage is obtained by applying to the extraction system the double floating probe theory. These results are in agreement with the experimental results.

Optimized polynomial expansion for scattering amplitudes

Abstract: In devising the most efficient way to determine a scattering amplitude from experimental data, it is important to make full use of the analyticity properties of the amplitude. The amplitude $f(x)$ considered here is given by data on a connected part of the real $x$ axis, and $f(x)$ is assumed to be analytic in a simply connected part of the $x$ plane; there are branch cuts on part of the remainder of the real axis. For convenience in practical calculations it is simplest to expand in polynomials, but for greater flexibility one may consider polynomials of some function $z(x)$. The polynomial expansion will converge as rapidly as possible if $z(x)$ maps the domain of analyticity in the $x$ plane onto the interior of a certain ellipse in the $z$ plane. More precisely, the expansion will then have the greatest possible geometric rates of convergence, both to $f(x)$ in the physical region, and also at any arbitrary point away from the physical region to which one may wish to extrapolate. Formulas are given that enable the mapping from a cut plane to an ellipse to be calculated quickly and easily. Some properties of the transformation that are relevant to partial-wave analysis are examined in detail. A method is suggested whereby the requirements of unitarity may be explicitly incorporated.

Letter to the Editor—-A Closed Form Solution of Certain Programming Problems

Abstract: A formula is given for the coordinates of the point that maximizes a given function F(x1, …, xn) over the closure of a bounded domain S in n-dimensional Euclidean space. The principal assumption made in deriving the formula is that F attains a global maximum at exactly one point of S. In certain cases the formula may be used to discuss the maximization problem as a function of the parameters involved. Some simple examples are given.

Analytic Capacity and Rational Approximation

Abstract: Peak points.- Analytic capacity.- Some useful facts.- Estimates for integrals.- Melnikov's theorem.- Further results.- Applications.- The problem of rational approximation.- AC capacity.- A scheme for approximation.- Vitushkin's theorem.- Applications of Vitushkin's theorem.- Geometric conditions.- Function algebra methods.- Some open questions.

Differentiation Formulas for Analytic Functions

Abstract: In a previous paper (Lyness and Moler (11), several closely related formulas of use for obtaining a derivative of an analytic function numerically are derived. Each of these formulas consists of a convergent series, each term being a sum of function evaluations in the complex plane. In this paper we introduce a simple generalization of the previous methods; we investigate the "truncation error" associated with truncating the infinite series. Finally we recommend a particular differentiation rule, not given in the previous paper.

The functional equation ()+(⁻¹)=2()() for groups

Abstract: wheref is a complex-valued function on a group G, for all x, y in G. On the line this functional equation is obviously satisfied by the cosine function and may be called the cosine equation. Of course this equation has a meaning on any group. One obvious way to solve the functional equation (A) on any group is by means of a homomorphism of G, say g, into the multiplicative group of nonzero complex numbers, K. If g is such a homomorphism, then the function defined by

On better-quasi-ordering transfinite sequences

Abstract: Let Q be a quasi-ordered set, i.e. a set on which a reflexive and transitive relation ≤ is defined. If, for every finite sequence q1, q2, … of elements of Q, there exist i and j such that i < j and qi ≤ qj then we call Q well-quasi-ordered. For any ordinal number α the set of all ordinal numbers less than α is called an initial set. A function from an initial set into Q is called a transfinite sequence on Q. If ƒ: I1 → Q, g: I2 → Q are transfinite sequences on Q, the statement ƒ ≤ g means that there is a one-to-one order-preserving function o:I1 → I2 such that f(α) ≤ g(o(α)) for every α ∈ I1. Milner has conjectured in (3) that, if Q is well ordered, then any set of transfinite sequences on Q is well-quasi-ordered under the quasi-ordering just defined. In this paper, we define so-called ‘better-quasi-ordered sets’, which are well-quasi-ordered sets of a particularly ‘well-behaved’ kind, and we prove that any set of transfinite sequences on a better-quasi-ordered set is better-quasi-ordered. Milner's conjecture follows a fortiori, since every well ordered set is better-quasi-ordered and every better-quasi-ordered set is well-quasi-ordered.

Grid for Scoring Visual Fields: II. Perimeter

Abstract: A device is presented for scoring peripheral acuity Values are expressed in percent analagous to what the Snellen scale does for central acuity Like the recently published scale for the tangent-screen field, it is based on function The grid consists of 100 units whose unequal size and distribution reflect the unequal functional value of different parts of the field—in effect a weighted or relative-value scale Because each unit equals 1%, a simple count of units yields the functional score in percent The grid improves on the American Medical Association method of scoring, although it is based on the same AMA standard isopter for the normal (100%) peripheral field The device, tested on 1,000 fields by 20 experienced ophthalmologists, yielded a 959% correspondence between their estimates and the grid scores It is simple, quick, inexpensive, consistent, and can, after brief instruction, be delegated by the ophthalmologist to a nonprofessional aide

Coherent states in the theory of superfluidity.

Abstract: It is shown that the coherent-state representation of a many-boson wave function may be identified with the order-parameter function conventionally used to describe a superfluid. The statistical mechanics of the many-boson system is reformulated in terms of the coherent states, and a theory of the Ginzburg-Landau form is recovered in an obvious approximation. The formalism is particularly useful for describing metastable states of finite superflow and the fluctuations which may cause spontaneous decay of such states.

Quasi-separable utility functions†

Abstract: This paper is concerned with a method for the assessment of utility functions of multi-numeraire consequences. It is proven that given von Neumann and Morgenstern's axioms of “rational behavior” and two additional assumptions, the utility function for (x, y) consequences can be written as U(x, y) = Ux(x) + Uy(y) + KUx(x) Uy(y). K is a constant that must be evaluated empirically. This form shall be designated as a quasi-separable utility function. It is more general than the separable utility function and is shown to be nearly as easy to use. Implications and ramifications of such a utility function and its requisite assumptions are discussed. A technique for practical application of this work is presented.

A Procedure for Detecting Intersections of Three-Dimensional Objects

Abstract: As a step toward the solution of the placement problem in engineering design, a procedure has been developed for detecting intersections of convex regions in 3-space by means of a pseudocharacteristic function. The mathematical techniques underlying the procedure are discussed, and a system of programs embodying these techniques is described. As a special case a solution is given for the hidden-line problem in graphic display.

A Note on Houthakker's Aggregate Production Function in a Multifirm Industry

Abstract: ONE OF THE common problems facing an economist dealing with production functions is the problem of aggregation of factors. In a rather neglected paper, Houthakker advances an ingenious approach for explaining the possibility of finding a neoclassical production function for an industry even when production within each of the firms (or "cells")3 is done according to a fixed coefficients production function. These fixed proportions vary in a regular way from one cell to another so that the overall input-output relationship takes the form of a regular neoclassical production function. As Solow notices in a survey article on production functions4 this paper has been forgotten and not followed in any direction. In this note we try to reverse Houthakker's procedure and to show how each neoclassical production function implies some density function or distribution function over the cells. We here do it for CES production functions, but it will be obvious that the same method applies to any production function. Following Houthakker we normalize the cells so that each of them is capable of producing one unit of output. Each cell has a requirement, say t, of the variable factor and this requirement varies from one cell to another. If the wage rate in terms of output produced is p then all the cells with tp < 1 will produce a unit of output, all others will be idle. Assume that we are given a density function of the various cells by g(t). Output produced will then be Q = f Pg(t)dt and the input used A f f'IP tg(t)dt. By eliminating i/p one gets a relationship between Q and A. In this way Houthakker has shown that a Pareto distribution implies a CobbDouglas production function. Notice that the relationship between Q and A the cumulated product and factor used-is the familiar Lorenz curve. Assume that the overall relationship between output and the variable factor follows a CES production function with elasticity of substitution (a) smaller than 1;

Many-Body Approach to the Atomic Hyperfine Problem. I. Lithium-Atom Ground State

Abstract: The Brueckner-Goldstone many-body perturbation method, previously utilized for calculations of atomic correlation energies and polarizabilities, has been extended to the study of the hyperfine structure. The correlation energy as well as the hyperfine coupling constant of the lithium atom are calculated and compared with the results of some earlier methods. The present method makes use of Feynman-like diagrams which facilitate the evaluation of the importance of various physical effects. Analysis of the hyperfine diagrams shows that the difference between the experimental and the Hartree-Fock values is mainly accounted for by spin polarization, although correlation effects are by no means negligible. Our result of 2.887 a. u. agrees very well with the experimental value of 2.9096 a. u. The excellent result for the total energy of -7.478 a. u., comparing with the corresponding experimental value of -7.47807 a. u., shows that the wave function is good over-all, as well as in the region near the nucleus.