Showing papers on "Function (mathematics) published in 1973"
TL;DR: This paper uses a technique called recursive doubling in an algorithm for solving a large class of recurrence problems on parallel computers such as the Iliac IV.
Abstract: An mth-order recurrence problem is defined as the computation of the series x 1 , x 2 , ..., X N , where x i = f i (x i-1 , ..., x i-m ) for some function f i . This paper uses a technique called recursive doubling in an algorithm for solving a large class of recurrence problems on parallel computers such as the Iliac IV. Recursive doubling involves the splitting of the computation of a function into two equally complex subfunctions whose evaluation can be performed simultaneously in two separate processors. Successive splitting of each of these subfunctions spreads the computation over more processors. This algorithm can be applied to any recurrence equation of the form x i = f(b i , g(a i , x i-1 )) where f and g are functions that satisfy certain distributive and associative-like properties. Although this recurrence is first order, all linear mth-order recurrence equations can be cast into this form. Suitable applications include linear recurrence equations, polynomial evaluation, several nonlinear problems, the determination of the maximum or minimum of N numbers, and the solution of tridiagonal linear equations. The resulting algorithm computes the entire series x 1 , ..., x N in time proportional to [log 2 N] on a computer with N-fold parallelism. On a serial computer, computation time is proportional to N.
1,376 citations
TL;DR: A simple theorem on generating functions is proved which can be used to establish the asymptotic normality of an(k) as a function of k and local limit theorems are turned to in order to obtain asymPTotic formulas for an( k).
405 citations
TL;DR: In this article, the welfare function of a survey by the Consumer Union in The Netherlands is estimated based on a lognormal distribution of the individual welfare function, and the results of the survey are compared with the results reported in this journal in 1971.
203 citations
TL;DR: This paper considers the problem of merging sets formed from a total of n items in such a way that at any time, the name of a set containing a given item can be ascertained.
Abstract: This paper considers the problem of merging sets formed from a total of n items in such a way that at any time, the name of a set containing a given item can be ascertained. Two algorithms using different data structures are discussed. The execution times of both algorithms are bounded by a constant times $nG(n)$, where $G(n)$ is a function whose asymptotic growth rate is less than that of any finite number of logarithms of n.
194 citations
TL;DR: In this paper, the problem of estimating the structural parameters of linear stochastic differential equations with respect to continuous time models is addressed, where the function relating the parameters of the continuous time model to those of the corresponding discrete time model is not, in general, bijective.
162 citations
TL;DR: In this paper, the critical temperature of a superconducting transition is calculated for a rather general form of the electron-electron interaction, and it is shown that even if both the energy and momentum dependence of the interaction is included, the equation determining the temperature coincides formally with the corresponding equation of the BCS theory.
Abstract: A critical temperatureT
c of a superconducting transition is calculated for a rather general form of the electron-electron interaction. It is shown that even if both the energy and momentum dependence of the interaction is included, the equation determiningT
c coincides formally with the corresponding equation of the BCS theory. The kernel of this equation is a smooth real function of its variables; it is expressed through ρ(k, E), the spectral density of the inverse dielectric function of the system. The expression forT
c is written in terms of ρ(k, E); this enables us to analyze the dependence of the critical temperature on the properties of the metal in a normal state. Some simple models illustrating the results are considered, and a discussion of the limits onT
c is given.
155 citations
TL;DR: In this article, a method for measuring ultrasonic attenuation as a function of frequency by spectrum analysis of broadband echoes in a buffer/specimen system is presented, and a new technique for diffraction corrections in this configuration is also presented.
Abstract: A method is presented for measuring ultrasonic attenuation as a function of frequency by spectrum analysis of broad‐band echoes in a buffer/specimen system. Also presented is a new technique for diffraction corrections in this configuration. The buffer rod/spectrum analysis method with diffraction corrections is applied to three experiments to demonstrate its accuracy and versatility in measuring attenuation and defining transducer radiation/reception efficiency.
116 citations
TL;DR: The characterization of directional derivatives for three major types of extremal-value functions is reviewed and the characterization for the completely convex case is used to construct a robust and convergent feasible direction algorithm.
Abstract: Several techniques in mathematical programming involve the constrained optimization of an extremal-value function. Such functions are defined as the extremal value of a related parameterized optimization problem. This paper reviews and extends the characterization of directional derivatives for three major types of extremal-value functions. The characterization for the completely convex case is then used to construct a robust and convergent feasible direction algorithm. Such an algorithm has applications to the optimization of large-scale nonlinear decomposable systems.
103 citations
TL;DR: In this article, an approach for efficient mathematical modeling of discretely measured irregular surfaces is presented, which is applicable to modeling of arbitrary surfaces; it is shown to be especially well suited for modeling of fine-structure topographic surfaces.
Abstract: A novel approach is presented for efficient mathematical modeling of discretely measured irregular surfaces. The technique is applicable to modeling of arbitrary surfaces; it is shown to be especially well suited for modeling of fine-structure topographic surfaces. The macroscopic features of the method are as follows. (1) Given a set of discrete coordinate measurements, an average least squares mathematical model for the surface geometry is determined. (2) The model consists of an arbitrarily large family of locally valid surface functions that join smoothly; nth-order continuity is satisfied everywhere. (3) Each locally valid surface function can typically be reduced to a low-degree polynomial of two variables; thus an efficient and consistent mathematical model for local surface calculations is provided. (4) The method sequentially operates on a moderate to small subset of the measured data; it is therefore applicable to an arbitrarily large set of observed data. These features and associated computational devices are discussed in the light of numerical results obtained by using actual geodetic data sets. These results demonstrate that the method is a versatile, accurate, and efficient means for obtaining general-purpose mathematical models of irregular surfaces.
80 citations
TL;DR: It is proved that the set of minimal true vertices and maximal false vertices of the expanded truth table constitutes a test set to detect any number of stuck-at-faults in a network belonging to a class of restricted networks, called unate gate networks.
Abstract: The problem of designing fault detecting test sets from the functional description rather than the structural description of the networks realizing the logic function is studied. The concept of an expanded truth table for logic functions is introduced. It is proved that the set of minimal true vertices and maximal false vertices of the expanded truth table constitutes a test set to detect any number of stuck-at-faults in a network belonging to a class of restricted networks, called unate gate networks. It is further indicated that even in the presence of redundancies in the network, the test sets given remain valid.
TL;DR: In this article, exact expressions for the change in cost because of a change in control are presented, which are useful for obtaining the conditions of optimality, particularly sufficient conditions, and for obtaining optimization algorithms, including D.D.P. algorithms.
Abstract: Publisher Summary This chapter presents exact expressions for the change in cost because of a change in control and indicates the unifying role that these expressions could play. The expressions are useful for obtaining the conditions of optimality, particularly sufficient conditions, and for obtaining optimization algorithms, including the powerful differential dynamic programming (D.D.P.) algorithms. The expressions enable two arbitrary controls to be compared, thus permitting the consideration of strong variations in control. The fact that the two controls might be arbitrary should facilitate the construction of new algorithms. The global sufficiency theorems that are based on the existence of a function satisfying the Hamilton-Jacobi-Bellman partial differential equation are similar. The main use of the technique has been to prove sufficiency theorems.
TL;DR: In this paper, the concept of spaciousness as an alternative criterion for window design is introduced, and observations in scale models differing in spaciousness under an artificial sky have been carried out, from these, spaciousness for different sky luminances has been shown to be a function of interior illuminance, room size and window size.
Abstract: Since traditional criteria for window design, e.g. the daylight factor, have lost their force, interest in re-evaluating the function of windows has been growing rapidly. The concept of spaciousness as an alternative criterion is introduced, and observations in scale models differing in spaciousness under an artificial sky have been carried out. From these, spaciousness for different sky luminances has been shown to be a function of interior illuminance, room size and window size. Observations in full scale interiors were also made to confirm the validity of the results. From additional experiments, it has been found that a lower limit of just acceptable spaciousness does exist. This would be a useful criterion for use in design.
TL;DR: In this article, the total energy of the symmetrical, bent O3 molecule was studied as a function of internuclear angle and internuclear distance, and the method used is an ab initio molecular orbital calculation with a...
Abstract: The total energy of the symmetrical, bent O3 molecule is studied as a function of internuclear angle and internuclear distance. The method used is an ab initio molecular orbital calculation with a ...
TL;DR: In this paper, it was shown that the same asymptotic result can be achieved without the knowledge of the density of the errors of estimates of function values, without the need to know the density.
Abstract: Anbar (1971) and, independently, Abdelhamid (1971) have shown that if the density $g$ of the errors of estimates of function values is known, a transformation of observations leads to stochastic approximation methods which under mild conditions produce asymptotically efficient estimators (the first author considers the RM case, the second the RM and KW cases). This paper treats the RM case and shows that the same asymptotic result can be achieved without the knowledge of the density $g$.
TL;DR: The static properties of the dielectric response function for an electron liquid, obtained previously by one of the authors, is investigated in this paper, where the resulting pair correlation function has the correct long-range and short-range behaviors so that the correlation energy calculated therefrom reproduces the known exact values to the order of e2, when an expansion with respect to the plasma parameter E is carried out.
Abstract: The static properties of the dielectric response function for an electron liquid, obtained previously by one of the authors, is investigated. It is shown that the resulting pair correlation function has the correct long-range and short-range behaviors so that the correlation energy calculated therefrom reproduces the known exact values to the order of e2, when an expansion with respect to the plasma parameter E is carried out. The compressibility sum rule is satisfied to the order of e2 ln E. It is found that within the accuracy stated above the short-range behavior of the pair correlation function is not ,sensitive to the form of the ternary correlation function.
TL;DR: In this paper, a theory for fitting weighted complex data to a given function is given, illustrated with two examples from the field of dielectrics: one for real data and the other for complex data.
Abstract: A theory is given for fitting weighted complex data to a given function. The method is illustrated with two examples from the field of dielectrics: one for real data and the other for complex data. It is shown that the inclusion of weighting factors alters the parameter estimates as well as reducing the confidence intervals.
TL;DR: In this article, it was shown that the image of an incoherent luminous object can be reconstructed from measurements of the second order correlation function of the optical field far from the source.
Abstract: As is well known from the work of Michelson, van Cittert and Zernicke, the image of an incoherent luminous object can be reconstructed from measurements of the second order correlation function of the optical field far from the source. The correlation function is a complex function that contains information both about the amplitude and the phase of the electromagnetic field. However, in most practical situations the phase of the correlation function is difficult to measure, so that experiments are usually limited to a determination of the modulus alone. The lack of phase information has then to be overcome in other ways.
TL;DR: In this paper, a tight binding approach to the electronic structure of disordered systems is developed for a simple one-orbital model of a liquid metal, and an equation is derived for a one-electron continuum Green's function from which the electronic density of states can be obtained.
Abstract: A tight-binding approach to the electronic structure of disordered systems is developed for a simple one-orbital model of a liquid metal. An equation is derived for a one-electron continuum Green's function from which the electronic density of states can be obtained. Utilizing an analogy between this Green's function and the $T$ matrix of multiple-scattering theory, results are obtained corresponding to the quasicrystalline approximation (QCA) of Lax and the self-consistent approximation (SCA) of Schwartz and Ehrenreich. Moments of the spectral function are also analyzed. Calculations were made using random and hard-sphere pair distribution functions. The QCA in this model is quite inadequate, and the SCA, while a considerable improvement, proves to involve a questionable approximation to the three-body distribution function.
TL;DR: In this article, a graph theoretic approach for deriving various stationary principles for single-element type nonlinear networks is presented, where the concepts of total content, co-content, and hybrid content are generalized to that of total parametric content for resistive non-linear networks containing multivalued elements.
Abstract: This paper presents a novel graph theoretic approach for deriving various stationary principles for single-element type nonlinear networks. The concepts of total content, co-content and hybrid content are generalized to that of total parametric content for resistive nonlinear networks containing multivalued elements. The results by Brayton and Moser on potential functions for complete m-ports are generalized to non-complete m-ports in terms of three pseudopotential functions; namely, the pseudo-content, pseudo- co-content and the pseudo-hybrid content. The precise criterion for which each of these pseudo-potential functions reduces to a legitimate potential function is shown to be the unique solvability of an implicit set of equations given explicitly in terms of standard topological matrices. Simple circuit theoretic sufficient conditions are given which require merely the positive-definiteness of the incremental resistance matrix of an auxiliary complete current-controlled p-port and the incremental conductance matrix of an auxiliary complete voltage-controlled q-port. These conditions also guarantee that a non-complete dynamic nonlinear network can be represented by a system of normal form differential equations in terms of a single mixed potential function.
TL;DR: In this paper, a systematic study of the cross sections for $K$-shell ionization by 2.0-MeV-electron impact has been made as a function of atomic number.
Abstract: A systematic study of the cross sections for $K$-shell ionization by 2.0-MeV-electron impact has been made as a function of atomic number. The cross sections for 32 elements from V ($Z=23$) to Bi ($Z=83$) were first measured relative to each other, and then were normalized to 43 b at Sn ($Z=50$). Thus systematic errors usually associated with absolute measurements were minimized and did not obscure minor variations in the $Z$ dependence of the cross sections. The measured values drop from 353 b at V to 9.9 b at Bi. The general trend of the $Z$ dependence of the data is in agreement with theoretical predictions of Kolbenstvedt. However, variations in cross sections by as much as 30% from one element to the next are not accounted for by the theory.
TL;DR: In this paper, lower and upper linear bounds are obtained on the renewal function of an ordinary renewal process, with slope equal to the reciprocal of the mean of the time between renewals, when iterated in the renewal equation.
Abstract: Lower and upper linear bounds are obtained on the renewal function of an ordinary renewal process. We show that any linear function, with slope equal to the reciprocal of the mean of the time between renewals, when iterated in the renewal equation will converge on the renewal function. However, convergence may not be monotonic for all t. We find the “best” linear bounds, which we define to be the sharpest bounds which, when iterated, converge monotonically to $M( t )$ for all t. We also show that the failure rate function of an equilibrium distribution has less variation than the failure rate of the underlying distribution. This fact is used to show that the bounds in this paper are an improvement over previously published linear bounds.
TL;DR: In this article, it was shown that the set of all experimental functions is an orthocomplemented partially ordered set with respect to the natural order of real functions with the complementationf′=1−f.
Abstract: Letp(A,α,E) be the probability that a measurement of an observableA for the system in a stateα will lead to a value in a Borel setE. An experimental function is a function f from the set of all statesI into [0,1] for which there are an observableA and a Borel setE such thatf(α)=p(A, α, E) for allα ∈I. A sequencef
1,f
2,... of experimental functions is said to be orthogonal if there is an experimental functiong such thatg+f
1+f
2+...=1, and it is said to be pairwise orthogonal iff
i+f
j⩽ 1 fori≠j. It is shown that if we assume both notions to be equivalent then the setL of all experimental functions is an orthocomplemented partially ordered set with respect to the natural order of real functions with the complementationf′=1−f, each observableA can be identified with anL-valued measureμ
A, each stateα can be identified with a probability measurem
α onL and we havep(A,α,E)=m
α oμA(E). Thus we obtain the abstract setting of axiomatic quantum mechanics as a consequence of a single postulate.
1 Dec 1973
TL;DR: In this paper, a parallel variable metric algorithm was proposed to exploit parallel computing or vector streaming (pipeline) capabilities of computers, and the convergence of the iterates to the solution was proved for a quadratic functional on a real separable Hilbert space.
Abstract: An algorithm, designed to exploit the parallel computing or vector streaming (pipeline) capabilities of computers is presented. When p is the degree of parallelism, then one cycle of the parallel variable metric algorithm is defined as follows: first, the function and its gradient are computed in parallel at p different values of the independent variable; then the metric is modified by p rank-one corrections; and finally, a single univariant minimization is carried out in the Newton-like direction. Several properties of this algorithm are established. The convergence of the iterates to the solution is proved for a quadratic functional on a real separable Hilbert space. For a finite-dimensional space the convergence is in one cycle when p equals the dimension of the space. Results of numerical experiments indicate that the new algorithm will exploit parallel or pipeline computing capabilities to effect faster convergence than serial techniques.
TL;DR: In this article, it was shown that the disk-packing function M(a, b, c; t) is a strictly convex function of (a, c, c) and that M is a symmetric function of these three variables.
Abstract: This result lends considerable weight to the heuristic estimate S ~ 1.306951, obtained by Melzak [6]. The improvement is a result of some new inequalities involving the disk-packing function M(a, b, c; t). Our principal new result is that M is a strictly convex function of (a, b, c). This, combined with the fact that M is a symmetric function of these three variables and an auxiliary lemma, allows us to show that
TL;DR: In this paper, the Lagrange Multiplier Rule was used in function space to obtain necessary conditions for optimal control problems with general nor&car retarded systems, where the trajectories must satisfy function space terminal conditions such as ~(0) = {(S), 0 E PI - h, ti], 4 being some given function.
TL;DR: In this paper, the output process in a buffer-multiplexer data communication system with N identical input lines and one output line is considered, and the main result is that active periods on the output line follow the same distribution function as the busy period in a certain M/G/1 queue.