Showing papers on "Function (mathematics) published in 1977"

Generalized sampling expansion

Abstract: It is shown that a band-limited function f(t) is uniquely determined in terms of the samples g_k(nT) of the responses g_k(t) of m linear systems with input f(t) , sampled at 1/m the Nyquist rate. Various known extensions of the sampling theorem follow as special cases of the resulting generalized sampling expansion of f(t) .

Internal set theory: A new approach to nonstandard analysis

Abstract: 1. Internal set theory. We present here a new approach to Abraham Robinson's nonstandard analysis [10] with the aim of making these powerful methods readily available to the working mathematician. This approach to nonstandard analysis is based on a theory which we call internal set theory (1ST). We start with axiomatic set theory, say ZFC (Zermelo-Fraenkel set theory with the axiom of choice [1]). In addition to the usual undefined binary predicate E of set theory we adjoin a new undefined unary predicate standard. The axioms of 1ST are the usual axioms of ZFC plus three others, which we will state below. All theorems of conventional mathematics remain valid. No change in terminology is required. What is new in internal set theory is only an addition, not a change. We choose to call certain sets standard (and we recall that in ZFC every mathematical object-a real number, a function, etc.-is a set), but the theorems of conventional mathematics apply to all sets, nonstandard as well as standard. In writing formulas we use A for and, V for or, ~ for not, =* for implies, and for is equivalent to. We call a formula of 1ST internal in case it does not involve the new predicate "standard" (that is, in case it is a formula of ZFC); otherwise we call it external. Thus "x standard" is the simplest example of an external formula. To assert that x is a standard set has no meaning within conventional mathematics-it is a new undefined notion. The fact that we have adjoined "standard" as an undefined predicate (rather than defining it in terms of E as is the case with all of the predicates of conventional mathematics) requires a readjustment of an engrained habit. We are used to defining subsets by means of predicates. In fact, it follows from the axioms of ZFC that if A(z) is an internal formula then for all sets x there is a set y = {z E x: A(z)} such that for all sets z we have z&y n E N A n standard. We may not use external predicates to define subsets. We call the violation of this rule illegal set formation. We adopt the following abbreviations:

Remarks on blow-up and nonexistence theorems for nonlinear evolution equations

Abstract: OVER the last 20 years a large literature has developed concerning evolution equations which for certain initial data possess solutions that do not exist for all time. The bulk of this literature relates to problems arising from partial differential equations. To establish nonexistence it is customary to argue by contradiction. One supposes that for given UQ and t0 a solution u(t) with u(ro)= u0 exists for all times t^t0; typically u takes values in some Banach space X and we will assume that this is the case. A function p : X —» R is then constructed, and by use of differential inequalities it is shown that lim p(u(()) = ° for some tle(t0, °°). This usually

A Survey of Optimal Recovery

Abstract: The problem of optimal recovery is that of approximating as effectively as possible a given map of any function known to belong to a certain class from limited, and possibly error-contaminated, information about it. In this selective survey we describe some general results and give many examples of optimal recovery.

On the integration of the BBGKY equations for the development of strongly nonlinear clustering in an expanding universe.

Abstract: The evolution of density correlations in an expanding universe can be described by the BBGKY equations. This approach has been the subject of several previous studies, but always under the assumption of small-amplitude fluctuations, where the hierarchy of equations has a natural truncation. Reslts of these studies cannot be compared to the present universe because the galaxy two-point correlation function xi (r) is much greater than unity at r9 or approx. =1h/sup -1/ Mpc, and the three-point function zeta is on the order of xi (r)/sup 2/. In this strongly nonlinear situation the hierarchy is dominated by terms ignored in the linear analysis. Our method of truncating the hierarchy is based on the empirical result that zeta can be represented to good accuracy as a simple function of xi. We solve the equations via the velocity-moment method, and we truncate the resulting velocity-moment hierarchy for the two-point function by assuming that the distribution in the relative velocity of particle pairs has zero skewness about the mean. The second equation in this velocity-moment hierarchy is our main equation for xi. It involves the three-point spatial correlation function zeta, which we write as a function of xi following the empirical result. Themore » third equation involves the first velocity moment of the three-point position and velocity correlation function. We model this term in a way consistent with our model for zeta and with a constraint equation that expresses conservation of triplets.The equations admit a similarity transformation if (1) the effects of the discreteness of particles can be ignored, (2) the initial spectrum of density perturbations assumes a power law shape, and (3) the universe is described by an Einstein-de Sitter model (..cap omega..approx. =1). The numerical results presented here are based on this similarity solution.« less

Resonantly interacting solitary waves

Abstract: Resonant (phase-locked) interactions among three obliquely oriented solitary waves are studied. It is shown that such interactions are associated with the parametric end points of the singular regime for interactions between two solitary waves. The latter include regular reflexion at a rigid wall, which is impossible for ϕi < (3α)½ (ϕ = angle of incidence, α = amplitude/depth [Lt ] 1), and it is shown that the observed phenomenon of ‘Mach reflexion’ can be described as a resonant interaction in this regime. The run-up at the wall is calculated as a function of ϕi/(3α)½ and is found to have a maximum value of 4αd for ϕi = (3α)½. This same resonant interaction also describes diffraction of a solitary wave at a corner of internal angle π − ψi, −(3α)½, and suggests that a solitary wave cannot turn through an angle in excess of (3α)½ at a convex corner without separating or otherwise losing its identity.

Children's Judgments of Numerical Inequality.

Abstract: SEKULER, ROBERT, and MIERKIEWICZ, DIANE. Children's Judgments of Numerical Inequality. CHILD DEVELOPMENT, 1977, 48, 630-633. When adults judge which of 2 digits is numerically larger, their response times decrease linearly with the numerical difference. For children in fourth and seventh grades, the function relating judgment time to numerical difference has the same slope as that of adults. For children in kindergarten and first grade the function is considerably steeper. This may reflect a developmental change in the internal representation of the number series.

Global optimization using special ordered sets

Abstract: The task of finding global optima to general classes of nonconvex optimization problem is attracting increasing attention. McCormick points out that many such problems can conveniently be expressed in separable form, when they can be tackled by the special methods of Falk and Soland or Soland, or by Special Ordered Sets. Special Ordered Sets, introduced by Beale and Tomlin, have lived up to their early promise of being useful for a wide range of practical problems. Forrest, Hirst and Tomlin show how they have benefitted from the last few years, as a result of being incorporated in a general mathematical programming system. Nevertheless, Special Ordered Sets in their original form require that any continuous functions arising in the problem be approximated by piecewise linear functions at the start of the analysis. The motivation for the new work described in this paper is the relaxation of this requirement by allowing automatic interpolation of additional relevant points in the course of the analysis. This is similar to an interpolation scheme as used in separable programming, but its incorporation in a branch and bound method for global optimization is not entirely straightforward. Two bt-products of the work are of interest. One is an improved branching strategy for general special-ordered-set problems. The other is a method for finding a global minimum of a function of a scalar variable in a finite interval, assuming that one can calculate function values and first derivatives, and also bounds on the second derivatives within any subinterval. The paper describes these methods, their implementation in the UMPIRE system, and preliminary computational experience.

A theoretical investigation of the rectangular microstrip antenna element

Abstract: A theoretical treatment of the rectangular microstrip radiating element has been performed. The element has been modeled as a line resonator with radiation taking place at the open-circuited ends. This has been verified by using a liquid crystal visual detector. With the simplified model, the input impedance and the far fields have been calculated for different resonant modes. The interaction between the radiating ends will effect the input impedance, and this has been considered by defining a mutual conductance. Also, a mutual conductance between microstrip elements has been expressed in far-field quantities and plotted as a function of spacing along the E - and H - planes. The directivity of an isolated element has been calculated as the directivity of one radiating end times the contribution due to the array factor.

Exact Green's function method of solar force-free magnetic-field computations with constant alpha. I - Theory and basic test cases

Abstract: Exact closed-form solutions to the solar force-free magnetic-field boundary-value problem are obtained for constant alpha in Cartesian geometry by a Green's function approach. The uniqueness of the physical problem is discussed. Application of the exact results to practical solar magnetic-field calculations is free of series truncation errors and is at least as economical as the approximate methods currently in use. Results of some test cases are presented.

The identification of a particular nonlinear time series system

Abstract: A nonlinear time series system is considered. The system has the property that the output series corresponding to a given input series is the sum of a noise series and the result of applying in turn the operations of linear filtering, instantaneous functional composition and linear filtering to the input series. Given a stretch of Gaussian input series and corresponding output series, estimates are constructed of the transfer functions of the linear filters, up to constant multipliers. The investigation discloses that for such a system, the best linear predictor of the output given Gaussian input, has a broader interpretation than might be suspected. The result is derived from a simple expression for the covariance function of a normal variate with a function of a jointly normal variate.

An algorithm for reasoning about equality

Abstract: A simple technique for reasoning about equalities that is fast and complete for ground formulas with function symbols and equality is presented. A proof of correctness is given as well.

On Some Existence Theorems for Semi-Linear Elliptic Equations.

Abstract: : Boundary value problems for semilinear elliptic equations are considered. If the nonlinear terms do not grow too fast as a function of the gradient of the dependent variable and ordered upper and lower solutions are known, then maximal and minimal solutions can be obtained by an iteration procedure. Other results concerning the existence of additional solutions follow from topological principles.

Solving nonlinear resistive networks using piecewise-linear analysis and simplicial subdivision

Abstract: In recent years numerous results of piecewise-linear analysis of nonlinear resistive networks have been derived. The applicability of the method relies on the fact that every nonlinear device is modeled by a piecewise-linear continuous function. In order to extend the applicability of piecewise-linear analysis to treat more general nonlinear networks, three steps need to be carried out: i) the subdivision of the domain of the multi-dimensional nonlinear network function; ii) the interpolation of a piecewise-linear continuous function on the subdivided domain; and iii) the application of piecewise-linear analysis. It turns out that the above three steps can be accomplished effectively by using simplicial subdivision. In addition, the difficulties encountered in the conventional piecewise-linear analysis are simplified. The memory space needed for the analysis is also greatly reduced. The complete analysis has been implemented in a program on CDC 6400.

Some interpretations of the self-tuning controller

Abstract: The self-tuning controller is extended to include rational transfer function (as opposed to polynomial) terms in the associated cost function. Two interpretations of the self-tuning controller are examined in some detail: a model reference adaptive control, and a self-tuning least-squares predictor in conjunction with conventional compensation. The former version is shown to give not only prespecified set point response, but also a closed-loop disturbance with largely prespecified spectral density. The latter version is compared with the method of O.J.M. Smith, and is shown to be less sensitive to uncertainty in some system parameters. Examples are given which illustrate the continuous-time performance of these discrete-time control laws.

Time evolution, correlations, and linear response of non-Markov processes

Abstract: We investigate the time evolution of stochastic non-Markov processes as they occur in the coarse-grained description of open and closed systems. We show that semigroups of propagators exist for all multivariate probability distributions, the generators of which yield a set of time-convolutionless master equations. We discuss the calculation of averages and time-correlation functions. Further, linear response theory is developed for such a system. We find that the response function cannot be expressed as an ordinary time-correlation function. Some aspects of the theory are illustrated for the two-state process and the Gauss process.

The integrated squared error estimation of parameters

Abstract: SUMMARY The paper develops in a more general setting the methods used' by Paulson, Holcomb & Leitch (1975) to estimate the parameters of a stable law The statistic considered minimizes a distance function determined, by the empirical characteristic function Consistency is established under the condition of differentiability of the characteristio function and the existence of bounded second derivatives is required to obtain a central limit theorem for the estimators of one or more parameters Questions concerning efficiency and robustness are discussed

Pair Correlation Theories

Abstract: In the simplest possible description of an n-electron system, one one-electron function (spin-orbital) is associated with each electron and the n -electron wave function is a Slater determinant built up from these spin-orbitals. The one-to-one correspondence between electrons and spin-orbitals gives an acceptable first-order description only for closed-shell and certain open-shell states. A one-electron theory that is applicable in general to open-shell states as well is characterized by assigning sets of electrons to sets of degenerate spin-orbitals, where the number of electrons within one set can be equal to or smaller than the dimension of the irreducible representation spanned by the degenerate set of spin-orbitals. An example is the well-known characterization of an atomic state by its configuration,(1) e.g., for the carbon ground state 1s22s22p2, without specifying the ms and ml values. (For a general discussion of closed- and open-shell states in the framework of rigorous quantum mechanics, see Refs. 2 and 3.)

On the utility theoretic foundations of mean‐variance analysis

Abstract: THE CHOICE MODEL in which alternatives are ordered in terms of the mean and variance of their return has been utilized in a variety of fields with the best developed, both theoretically and empirically, being that of portfolio selection. Since the pioneering works of Markowitz [22] and Tobin [32], a series of results has been obtained regarding the utility theoretic foundations of this model. While these results are widely known, they continue to be the source of discussion.' For example, Markowitz demonstrated that if the ordering of alternatives is to satisfy the von Neumann-Morgenstern (NM) [35] axioms of rational behavior, only a quadratic (NM) utility function is consistent with an ordinal expected utility function that depends solely on the mean and variance of the return. Consequently, even if the return for each alternative has a normal distribution, the mean-variance framework cannot be used to rank alternatives consistently with the NM axioms unless a quadratic NM utility function is specified. The implications of this restriction are disturbing not only because of the undesirable properties of a quadratic utility function but also because, for example, the indifference curves in the mean-standard deviation plane are concentric circles with the center on the mean axis. Furthermore, a quadratic utility function leads to a rather disquieting result in portfolio theory, since it implies that in equilibrium each investor holds an equal percentage of every security (Mossin [23, p. 69]). Also, Ekern and Wilson [11, p. 179] have shown that in a mean-variance portfolio model all shareholders prefer that a firm characterized by stochastic constant returns operate so that its market value is zero. Another disquieting result is that of Borch [7] who demonstrates that if preferences satisfy a monotonicity condition, an indifference curve in the ( ,a)-plane consists of a single point. A number of authors, including Samuelson [27] [29] and Tsiang [33], have argued that mean-variance analysis may be viewed as an approximation to a more general choice model. The appropriateness of such an approximation has been considered by Borch [8], Bierwag [6], Levy [20], and Tsiang [34], for example, and will not be considered further here. The purpose of this paper is instead to consider the foundations of mean-variance analysis and the consistency of the ordering of alternatives according to means and variances with the NM axioms. More specifically, given alternatives a, and a2 and their corresponding random returns X, and X2 with distribution functions F,(x) and F2(x), respectively, preferences satisfying the NM axioms imply the existence of a measurable, continuous utility function

Differential Stability in Nonlinear Programming

Abstract: This paper consists of a study of stability and differential stability in nonconvex programming. For a program with equality and inequality constraints, upper and lower bounds are estimated for the potential directional derivatives of the perturbation function (or the extremal-value function). These results are obtained' with the help of a constraint qualification which is shown to be necessary and sufficient to have bounded multipliers. New results on the continuity of the perturbation function are also obtained.