Showing papers on "Function (mathematics) published in 1983"

Deterministic phase retrieval: a Green’s function solution

Abstract: Equations for the propagation of phase and irradiance are derived, and a Green’s function solution for the phase in terms of irradiance and perimeter phase values is given A measurement scheme is discussed, and the results of a numerical simulation are given Both circular and slit pupils are considered An appendix discusses the local validity of the parabolic-wave equation based on the factorized Helmholtz equation approach to the Rayleigh–Sommerfeld and Fresnel diffraction theories Expressions for the diffracted-wave field in the near-field region are given

A complex gradient operator and its application in adaptive array theory

Abstract: The problem of minimising a real scalar quantity (for example array output power, or mean square error) as a function of a complex vector (the set of weights) frequently arises in adaptive array theory. A complex gradient operator is defined in the paper for this purpose and its use justified. Three examples of its application to array theory problems are given.

Jointly Constrained Biconvex Programming

Abstract: This paper presents a branch-and-bound algorithm for minimizing the sum of a convex function in x, a convex function in y and a bilinear term in x and y over a closed set. Such an objective function is called biconvex with biconcave functions similarly defined. The feasible region of this model permits joint constraints in x and y to be expressed. The bilinear programming problem becomes a special case of the problem addressed in this paper. We prove that the minimum of a biconcave function over a nonempty compact set occurs at a boundary point of the set and not necessarily an extreme point. The algorithm is proven to converge to a global solution of the nonconvex program. We discuss extensions of the general model and computational experience in solving jointly constrained bilinear programs, for which the algorithm has been implemented.

State of the Art-Location on Networks: A Survey. Part I: The p-Center and p-Median Problems

Abstract: Network location problems occur when new facilities are to be located on a network. The network of interest may be a road network, an air transport network, a river network, or a network of shipping lanes. For a given network location problem, the new facilities are often idealized as points, and may be located anywhere on the network; constraints may be imposed upon the problem so that new facilities are not too far from existing facilities. Usually some objective function is to be minimized. For single objective function problems, typically the objective is to minimize either a sum of transport costs proportional to network travel distances between existing facilities and closest new facilities, or a maximum of "losses" proportional to such travel distances, or the total number of new facilities to be located. There is also a growing interest in multiobjective network location problems. Of the approximately 100 references we list, roughly 60 date from 1978 or later; we focus upon work which deals directly with the network of interest, and which exploits the network structure. The principal structure exploited to date is that of a tree, i.e., a connected network without cycles. Tree-like networks may be encountered when having cycles is very expensive, as with portions of interstate highway systems. Further, simple distribution systems with a single distributor at the "hub" can often be modeled as star-like trees. With trees, "reasonable" functions of distance are often convex, whereas for a cyclic network such functions of distance are usually nonconvex. Convexity explains, to some extent, the tractability of tree network location problems.

Coefficient bounds for the inverse of a function with derivative in . II

Abstract: Coefficient bounds for functions with a positive real part are used in a rather novel way to find sharp bounds for the first six coefficients of a function which is inverse to a regular normalized univalent function whose derivative has a positive real part in the unit disk.

The Application of Optimal Control Theory to the General Life History Problem

Abstract: The application of optimal control theory to life history evolution in species with discrete breeding seasons and overlapping generations is discussed. For each age class, the objective functional maximized consists of an integral (total reproduction for that age class) plus a final function (residual reproductive value). A simple example, for which monocarpy is the optimal strategy, is given. The present results complement previous studies (e.g., Schaffer 1979) of life history evolution as a problem in static optimization. The works of Leon (1976), who applied control theory to the case of species with continuous reproduction, and Mirmirani and Oster (1978), who did the same organisms with annual life histories, are thus extended.

A class of singular stochastic control problems

Abstract: We consider the problem of tracking a Brownian motion by a process of bounded variation, in such a way as to minimize total expected cost of both ‘action' and ‘deviation from a target state 0'. The former is proportional to the amount of control exerted to date, while the latter is being measured by a function which can be viewed, for simplicity, as quadratic. We discuss the discounted, stationary and finite-horizon variants of the problem. The answer to all three questions takes the form of exerting control in a singular manner, in order not to exit from a certain region. Explicit solutions are found for the first and second questions, while the third is reduced to an appropriate optimal stopping problem. This reduction yields properties, as well as global upper and lower bounds, for the associated moving boundary. The pertinent Abelian and ergodic relationships for the corresponding value functions are also derived.

A unified structural approach to linear carbon polytypes

Abstract: Compounds obtained by high-temperature treatment of graphite are thought to be linear carbon polytypes with sp-configuration and carbon–carbon chains, either conjugated triple bonded or cumulated double bonded, parallel to the c-axis. At least 10 crystallographically documented linear carbon forms (carbynes) have been reported. To relate the contradictory and confusing information found in the literature, a simple classification is suggested here based on a linear relationship between number of atoms in a chain (n) and the unit cell parameters a0 and c0 of all carbyne forms known. It is assumed that chains are kinked, and that the distribution of kinked spacings may be a function of the temperature of formation of the carbyne forms in question.

Tensor Methods for Nonlinear Equations.

Abstract: A new class of methods for solving systems of nonlinear equations, called tensor methods, is introduced. Tensor methods are general purpose methods intended especially for problems where the Jacobian matrix at the solution is singular or ill-conditioned. They base each iteration on a quadratic model of the nonlinear function, the standard linear model augmented by a simple second order term. The second order term is selected so that the model interpolates function values from several previous iterations, as well as the current function value and Jacobian. The tensor method requires no more function and derivative information per iteration and hardly more storage or arithmetic per iteration, than a standard method based on Newton’s method. In extensive computational tests, a tensor algorithm is significantly more efficient than a similar algorithm based on the standard linear model, both on standard nonsingular test problems and on problems where the Jacobian at the solution is singular.

On the calculation of time correlation functions in quantum systems: Path integral techniquesa)

Abstract: A Monte Carlo method for calculating quantum mechanical time correlation functions is presented. In this method the time correlation function is calculated at several values along the pure imaginary axis of the complex time plane such that 0

Canonical piecewise-linear analysis

Abstract: Any continuous resistive nonlinear circuit can be approximated to any desired accuracy by a global piecewise-linear equation in the canonical form a + B x + \sum_{i=1}^{p}c_{i} |\langle \alpha_{i}, x \rangle - \beta_{i}|= 0 . All conventional circuit analysis methods (nodal, mesh, cut set, loop, hybrid, modified nodal, tableau) are shown to always yield an equation of this form, provided the only nonlinear elements are two-terminal resistors and controlled sources, each modeled by a one-dimensional piecewise-linear function. The well-known Katzenelson algorithm when applied to this equation yields an efficient algorithm which requires only a minimal computer storage. In the important special case when the canonical equation has a lattice structure (which always occur in the hybrid analysis), the algorithm is further refined to achieve a dramatic reduction in computation time.

Combinational logic structure using pass transistors

Abstract: PASS transistors are used to reduce the layout complexity of logic circuits by using PASS transistors connected to pass a first and second input function to an output node in response to selected CONTROL signals, thereby to generate a selected output function on the output node. The PASS transistor comprises a transistor capable of passing an input function in response to a CONTROL signal applied to the transistor thereby to generate an output function related to the input function. In general, the input function comprises less than all of a set of input variables and the CONTROL function comprises one or more of the remainder of the set of input variables.

Computing Forward-Difference Intervals for Numerical Optimization

Abstract: When minimizing a smooth nonlinear function whose derivatives are not available, a popular approach is to use a gradient method with a finite-difference approximation substituted for the exact gradient In order for such a method to be effective, it must be possible to compute “good” derivative approximations without requiring a large number of function evaluations Certain “standard” choices for the finite-difference interval may lead to poor derivative approximations for badly scaled problems We present an algorithm for computing a set of intervals to be used in a forward-difference approximation of the gradient

The existence and calculation of traffic equilibria

Abstract: The paper considers the assignment problem when there are junction interactions. We give an objective function which measures the extent to which a traffic distribution departs from equilibrium, and an algorithm which (under certain conditions) calculates equilibria by steadily reducing the objective function to zero. It is shown that the algorithm certainly works if the network cost-flow function is monotone and continuously differentiable, and a boundary condition is satisfied.

Interval estimation: Effect of processing demands on prospective and retrospective reports

Abstract: Undergraduate students performed one of three levels of processing on each word (15, 30, or 45) presented during a 120-sec interval. Subjects were told in advance that they would be required to estimate the length of the presentation interval (prospective condition) or were presented with an unexpected estimation task (retrospective condition). In the prospective condition, interval estimates were an inverse function of list length when relatively deep levels of processing were required, but were an increasing function of list length when shallow processing was required. In the retrospective condition, estimates were an increasing function of list length and were unaffected by different levels of processing. The interval estimation model proposed by Hicks, Miller, and Kinsbourne (1976) provided a better account of the data than did the storage-size hypothesis of Ornstein (1969).

Synthesis of digital designs from recursion equations

Abstract: The discipline of applicative program design style is adapted to the design of synchronous systems. The result is a powerful and natural methodology for engineering correct hardware implementations. This dissertation makes a formal connection between functional recursion and component connectivity that is pleasantly direct, suggesting that applicative notation is the appropriate basis for digital design. Synthesis is a theory for constructing concretely descriptive realizations from abstractly descriptive specifications. Functional recursion equations serve here as specifications; the realization language uses signal equations to describe circuit connectivity. A semantics for realizations assigns an infinite value sequence to each signal. The register equation X = a ! t, where t is an applicative term, defines signal X to be the output a register initialized to a. Synchronous systems are characterized by iterative specifications over a single function symbol (simple while-loops). Moreover, the transcription from F(x(,1),..., x(,n)) p (--->) r, F(t(,1),..., t(,n)) to the canonical realization {X(,i) = a(,i) ! t(,i)} is immediate and correct. Realizations can be compiled from arbitrary iterative specifications by a familiar construction. In non-iterative cases synthesis of iterative form is the main tactic used here for deriving realizations. This subgoal formalizes the conventional technique of decomposing a circuit into an architecture and a finite state controller. This approach suggests yet another way to implement "function recursion" with "data recursion." An interpreter has been implemented for Daisy, a demand-driven list processing language. By representing signals as infinite lists, expressed realizations can be directly interpreted, resulting in a simulation of logical behavior. Thus, the engineering notation serves also as a vehicle for experimentation. A non-trivial exercise in language-driven design reveals that global design factorization techniques, such as hierarchical decomposition and data abstraction, are inherited from the functional description style. Next, a specialized transformation system is defined to address a typical local refinement problem: serialization of inputs.

Chapter 5 Model choice and specification analysis

Abstract: Publisher Summary This chapter discusses statistical theories of model selection and traditional problems. One important source of model-selection problems is the existence of a priori opinion that constraints are “likely.” Statistical testing is then designed either to determine if a set of constraints is “true” or to determine if a set of constraints is “approximately true.” The solution to these two problems that might be supposed to be essentially the same, in fact diverge in two important respects: (1) the first problem leads clearly to a significance level—that is a decreasing function of sample size, whereas (2) the second problem selects a relatively constant significance level. The problem has a set of alternative models—that is determined entirely from a priori knowledge, whereas the second problem can have a data-dependent set of hypotheses. Quadratic loss functions are discussed in the chapter both with and without fixed costs. Quadratic loss does not imply a model-selection problem. The chapter also discusses problems that are not well known.

The aliasing problem in discrete-time Wigner distributions

Abstract: There is no straightforward way to proceed from the continuous-time Wigner distribution to a discrete-time version of this time-frequency signal representation. A previously given definition of such a function turned out to yield a distribution that was periodic with period π instead of 2π and this caused aliasing to occur. Various alternative definitions are considered and compared with respect to aliasing and computational complexity. From this comparison it appears that no definition leads to a function that is optimum in all respects. This is illustrated by an example.

Measuring compositional change along gradients

Abstract: A new procedure for measuring compositional change along gradients is proposed. Given a matrix of species-by-samples and an initial ordering of samples on an axis, the ‘gradient rescaling’ method calculates 1) gradient length (beta diversity), 2) rates of species turnover as a function of position on the gradient, and 3) an ecologically meaningful spacing of samples along the gradient. A new unit of beta diversity, the gleason, is proposed. Gradient rescaling is evaluated with both simulated and field data and is shown to perform well under many ecological conditions. Applications to the study of succession, phenology, and niche relations are briefly discussed.

Run Probabilities in Sequences of Markov-Dependent Trials

Abstract: The probability of the occurrence of a run R is obtained as a function of the composition of R, the number n of trials, and the probabilities of the v ≥ 2 possible outcomes at each trial. The run R can consist of any specified sequence of outcomes, and the probability that one or more of a given collection of runs occurs is also evaluated. The probabilities of the v possible outcomes can vary arbitrarily from trial to trial, and can be L-order Markov dependent on the L preceding outcomes. The practical application of these results is discussed.

Mean-Variance Approximations to Expected Logarithmic Utility

Abstract: In this paper, we investigate how closely functions of means and variances can approximate Von Neumann-Morgenstern expected utility modeled as a logarithmic utility-of-wealth function. Using historical security return data, we computed portfolios maximizing expected logarithmic utility and compared them to those maximizing appropriate mean-variance formulations. In all cases the approximations were very good, and in many cases the optimal portfolios were virtually identical. We conclude that the mean-variance model can serve as a useful surrogate to at least one popular alternative investment strategy.

Method and apparatus for use in processing signals

Abstract: New dialogue is post-synchronised with guide track dialogue by using signal processing apparatus in which the analog guide track signal x, (t) undergoes speech parameter measurement processing in a processor 43 to provide a speech parameter measurement processing in a processor 43 to provide a speech parameter vector A (kT). The new dialogue signal x 2 (t') is processed to give waveform data which can be stored on disc 25 and a speech parameter vector B (jT) from a parameter extraction processor 42. The variables k and 1 are data frame numbers, and T is an analysis interval. Some parameters of the vector B are used in a process 48 to classify successive passages of the new dialogue signal into speech and silence, to produce classification data f (jT). The vectors A and B and the classification data are utilized in a time warp processor SBC2 to determine a timewarping function w (kT) giving the values of i in terms of the values of k associated with the corresponding speech features, and thereby, indicating the amount of expansion or compression of the waveform data of the new dialogue signal needed to align the time-dependent features of the new dialogue signal with the corresponding features of the guide track signal. Editing instructions are generated in signal editor computer SBCI from the w (kT) data, feature classification data, pitch data p (jT) and the data stream x 2 (nD) so that the editing of x 2 (nD) can be carried out by the computer SBCI in which periods of silence or speech are lengthened or shortened to give the desired alignment. The edited data x 2 (nD) is converted to analog by a converter unit 29. and low pass filtered to provide an audio output signal to be recorded as the synchronised new dialogue.

Majorations Explicites Pour le Nombre de Diviseurs de N

Abstract: Let It is proved that the function f reaches its maximum for n = 6 983 776 800, and that maxn≥2 f(n) = 1.5379. The proof deals with superior highly composite numbers introduced by Ramanujan.