Showing papers on "Operator (computer programming) published in 1985"

Fading memory and the problem of approximating nonlinear operators with Volterra series

Abstract: Using the notion of fading memory we prove very strong versions of two folk theorems. The first is that any time-invariant (TI) continuous nonlinear operator can be approximated by a Volterra series operator, and the second is that the approximating operator can be realized as a finite-dimensional linear dynamical system with a nonlinear readout map. While previous approximation results are valid over finite time intervals and for signals in compact sets, the approximations presented here hold for all time and for signals in useful (noncompact) sets. The discretetime analog of the second theorem asserts that any TI operator with fading memory can be approximated (in our strong sense) by a nonlinear moving- average operator. Some further discussion of the notion of fading memory is given.

Transformation relations for the conventional Okq and normalised O'kq Stevens operator equivalents with k=1 to 6 and -k⩽q⩽k

Abstract: Extensions of the conventional Stevens operators Okq to negative values of q and their transformation properties are reviewed and reconsidered. Transformation matrices with k=1 to 6 and -k

Finite element approximation of regularized solutions of the inverse potential problem of electrocardiography and applications to experimental data

Abstract: This paper investigates the problem of estimating the electric potential distribution in proximity of the heart from potential data given on the body and is here reformulated as a control problem in terms of a «transfer» operator and stabilized by means of a suitable regularization operator. The numerical approximation by means of the finite element method of the regularized problem is investigated; convergence results and error estimates are established.

Nest Algebras and Similarity Transformations

Abstract: In recent years the theory of algebras of operators on Hilbert space has been stimulated by developments in the theory of quasitriangularity. Andersen [1] has shown that up to unitary equivalence there is only one "continuous" quasitrianguilar algebra. We use this to provide the following answer to a question posed by J. R. Ringrose approximately 20 years ago: Similar continuous nests on separable Hilbert space can fail to be unitarily equivalent (Theorem 2.2). A consequence is the existence of a nonhyperintransitive compact operator (Corollary 2.3), which answers a question of Kadison and Singer [12] and of Gohberg and Krein [11]. We extend our initial theorem to show that arbitrary continuous nests are similar (Theorem 2.10), and that every maximal nest is similar to a multiplicity-one nest (Theorem 2.11). A consequence is that every compact operator is similar to a hyperintransitive compact operator (Corollary 2.12). The similarity transformation can be induced by an arbitrarily small compact perturbation of a unitary operator. The methods of Section 2 apply only to the continuous parts of nests. For general results the atomic core part must be dealt with. In Section 3 different methods are developed for this purpose, again utilizing Andersen's results. These are used in Section 4 to prove that a complete nest X admits an Arveson factorization for every positive invertible operator if and only if X is countable as a family of subspaces (Theorem 4.7). A consequence is that if X is an uncountable nest with atomic core then some similarity transformation of X has a continuous part. This could not be deduced from Section 2. These methods also yield a weak factorization result which concludes the paper. It should be noted that a negative resolution to the Ringrose question was conjectured in recent years by several mathematicians, including W. Arveson and J. Ringrose. Also, many of the results presented in this paper were announced in an A.M.S. Bulletin article [16]. Finally, we wish to thank the referee for suggesting that the original manuscript could be condensed and improved.

Absorbing boundary conditions for acoustic media

Abstract: By decomposing the acoustic wave equation into incoming and outgoing components, an absorbing boundary condition can be derived to eliminate reflections from plane waves according to their direction of propagation. This boundary condition is characterized by a first‐order differential operator. The differential operator, or absorbing boundary operator, is the basic element from which more complicated boundary conditions can be constructed. The absorbing boundary operator can be designed to absorb perfectly plane waves traveling in any two directions. By combining two or more absorption operators, boundary conditions can be created which absorb plane waves propagating in any number of directions. Absorbing boundary operators simplify the task of designing boundary conditions to reduce the detrimental effects of outgoing waves in many wave propagation problems.

Comments on "Digital Step Edges from Zero Crossings of Second Directional Derivatives

Abstract: In a recent paper,1 Haralick published an edge detection scheme that was supported, in part, by an evaluation against the Prewitt and the Marr-Hildreth (?2G) operators. This evaluation led to the conclusion that Haralick's method performed the best, and the ?2G operator performed the worst. The implementation of the ?2G operator, on which this evaluation was based, differed significantly from that used by Marr and Hildreth. Evaluation of the performance of the Marr-Hildreth implementation of the ?2G operator on similar images shows that this edge detection method in fact performs comparably to the Prewitt and Haralick operators.

On the choice of expansion and weighting functions in the numerical solution of operator equations

Abstract: One of the objectives of this paper is to discuss the mathematical requirements that the expansion functions must satisfy in the method of moments (MM) solution of an operator equation. A simple differential equation is solved to demonstrate these requirements. The second objective is to study the numerical stability of point matching method, Galerkin's method, and the method of least squares. Pocklington's integral equation is considered and numerical results are presented to illustrate the effect of various choices of weighting functions on the rate of convergence. Finally, it is shown that certain choices of expansion and weighting functions yield numerically acceptable results even though they are not admissible from a strictly mathematical point of view. The reason for this paradox is outlined.

Computational Aspects of the Choice of Operator And Sampling Interval For Numerical Differentiation In Large-Scale Simulation of Wave Phenomena

Abstract: Conventional finite-difference operators for numerical differentiation become progressively inaccurate at higher frequencies and therefore require very fine computational grids. This problem is avoided when the derivatives are computed by multiplication in the Fourier domain. However, because matrix transpositions are involved, efficient application of this method is restricted to computational environments where the complete data volume required by each computational step can be kept in random access memory. To circumvent these problems a generalized numerical dispersion analysis for wave equation computations is developed. Operators for spatial differentiation can then be designed by minimizing the corresponding peak relative error in group velocity within a spatial frequency band. For specified levels of maximum relative error in group velocity ranging from 0.03% to 3%, differentiators have been designed that have the largest possible bandwidth for a given operator length. The relation between operator length and the required number of grid points per shortest wavelength, for a required accuracy, provides a useful starting point for the design of cost-effective numerical schemes. To illustrate this, different alternatives for numerical simulation of the time evolution of acoustic waves in three-dimensional inhomogeneous media are investigated. It is demonstrated that algorithms can be implemented that require fewer arithmetic and I/O operations by orders of magnitude compared to conventional second-order finite-difference schemes to yield results with a specified minimum accuracy.

On the Regularity of the Critical Point Infinity of Definitizable Operators

Abstract: In this note necessary and sufficient conditions for the regularity of the critical point infinity of a definitizable operator A are given. Using these criteria it is proved that the regularity of the critical point infinity is preserved under some additive perturbations as well as for some operators which are related to A. Applications to indefinite Sturm-Liouville problems are indicated.

A note on the choice weighting functions in the method of moments

Abstract: The objective is to show from a mathematical standpoint that there are certain rules that must be followed in the choice of weighting functions used in the method of moments (MM). It is shown that for a particular problem it is the operator that dictates the method (Galerkin's method or another method such as the method of least squares) to be applied, and it is not computational considerations only. For example, it is shown that in solving Hallen's and Pocklington's equation by the method of moments, it is unnatural to choose the weighting functions which are zero at the ends of the domain of the solution. The deficiency of certain weighting functions is presented based on mathematical reasoning, and a numerical example is given to illustrate the effect of the choice of the weighting functions on the rate of the convergence of the solution.

On finite element approximation in the L∞-norm of variational inequalities

Abstract: We are interested in the approximation in theL ?-norm of variational inequalities with non-linear operators and somewhat irregular obstacles We show that the order of convergence will be the same as that of the equation associated with the non-linear operator if the discrete maximum principle is verified

Quantum-mechanical sum rules and gauge invariance: A study of the HF molecule

Abstract: The perturbed Hamiltonian for magnetic dipole transitions is rewritten in terms of the torque operator, instead of the angular momentum operator, which, owing to its nondifferential form, permits tactical advantages in actual calculations of magnetic susceptibility. The translational gauge invariance of the magnetic properties is used to obtain a large series of sum rules involving linear and angular momenta and torque, force, and position operators. These are found to be very general quantum-mechanical relations, restating in a synthetic and unitary form the Thomas-Reiche-Kuhn sum rule, the basic operator commutation properties, the hypervirial theorem, and the conservation of the current-density vector, which are reduced to the same theoretical framework. Accurate calculations of the magnetic properties of the HF molecule, based on the equation-of-motion approach, reveal that the gauge-invariant sum rules can be used for rigorous tests of the quality of approximate molecular wave functions.

The linearly driven parametric oscillator: Application to collisional energy transfer

Abstract: The time‐evolution operator is explicitly constructed for a general linearly driven parametric quantum oscillator, equivalent to a harmonic oscillator driven by linear plus quadratic potentials. The method is based on an algebra of operators which are bilinear in the position and momentum operators, and form a closed set with respect to commutation. The obtained result requires only integrals over time and the solution of two coupled first order linear differential equations related to the classical equations of motion. The model is used to obtain vibration‐translation probabilities in a collinear collision of an atom with a diatomic molecule. Numerical calculations have been performed for systems with several mass combinations and potential parameters. Approximation methods are compared, and criteria are established to determine when it is necessary to go beyond the popular linearly driven harmonic oscillator.

The Magnetic Exchange Current and the Spin-Orbit Interaction

Abstract: It is shown that model independent local isoscalar and isovector exchange current operators associated with the spin-orbit component of the nucleon-nucleon interaction may be constructed from such nucleon-nucleon interactions that have been derived from relativistic models (e.g. boson exchange) for the scattering amplitude. The resulting magnetic moment operators contain terms that cannot be obtained by direct gauging of the conventional spin-orbit interaction. It is shown that the model independent isoscalar exchange current associated with the spin-orbit interaction causes a quenching of the order 10% of the effective isoscalar magnetic moment operator in heavy nuclei. The corresponding isovector exchange current operator causes a somewhat smaller enhancement of the effective isovector magnetic moment operator, which counteracts the much larger quenching effect caused by isobar-hole screening of the nucleons. The isoscalar exchange current operator leads to a small modification of the calculated magnetic formfactor of deuterium, which counteracts the contribution from typical model dependent exchange current contributions.

Essential dimension as a well-defined number of degrees of freedom of finite-convolution operators appearing in optics

Abstract: Inversion of a finite-convolution operator is known to be an ill-posed problem. However, although the complete solution cannot be recovered to within any specified accuracy, certain components of the solution can be accurately determined. We present an estimate of the number of such components, termed here the essential dimension of the finite-convolution operator, that is dependent on the noise levels in the data, the desired accuracy in the solution, and the singular values of the finite convolution. We then show that the required singular values may be easily and accurately approximated so that the essential dimension is easily estimated and indicate its superiority over previously proposed measures of ill conditioning for this problem.

Nonstationary iterative image restoration

Abstract: This paper introduces different types of nonstationary constrained iterative image restoration algorithms. The adaptivity of the algorithm is introduced by the constraint operator which incorporates properties of the response of the human visual system. The properties of the visual system are represented by noise masking and visibility functions. A new way of computing the masking function is also introduced. The proposed algorithms are general and can be used for any type of linear constraint and distortion operators. The algorithms can also be used to restore signals different than images, since the constraint operator can be interpreted as adapting to the local signal activity.

Some aspects of self-consistent propagator theories

Abstract: A general method for constructing bases for operator manifolds for any propagator, which satisfy ``vacuum annihilation conditions'' (VAC's) is developed. This approach is based on the observation that if the transformation of the unperturbed ground state to the correlated ground state is represented as a rotation in the Fock space, the corresponding rotation induced in the basis of the concerned operator space would generate a basis which satisfies VAC's on the correlated ground state. The associated requirements for the Hermiticity of superoperator Hamiltonian would also be met in this new basis. The proposed method is noniterative in that, once the form of the ground-state function is specified, the expansion of the operator manifold satisfying VAC's on the ground state does not require any iterative readjustment. The resultant propagators in this approach are fully linked. It is shown that this theory is equivalent to a propagator formalism in terms of hole- and particle-creation and/or annihilation operators with a modified effective Hamiltonian. The self-consistent electron and polarization propagators are considered as examples, and their underlying perturbative structures are analyzed. The role of density shift operators and higher-rank operators are discussed.

Method and apparatus for sharing operators among assistance systems

Abstract: An arrangement for serving an operator assistance request detected in a first operator assistance system from an operator position in a second operator assistance system. If a first operator assistance system is overloaded or has no attended operator positions, the two operator assistance systems cooperate to establish a connection between the customer requesting operator assistance and the operator position in the second system. The attending operator at that position keys information into the operator position for transmission to the first system. Responsive to this data, the first system enters data, such as billing data, into its records, and sets up connections, such as a connection for a person-to-person call, in its switching network. The arrangement can be used to transfer operator assistance requests at night, during overload, or to specialized operators not available at the first office. The arrangement can also be used on a permanent basis, to transfer all operator assistance requests, when a serving team of operators cannot be economically justified for the first office.

Fine-structure analyses with orthogonal operators

Abstract: The single-electron operator representing the spin-orbit interaction in the 3d shell is augmented by a collection of operators, each of which acts on two electrons at a time and possesses identical ranks k in the spin and orbital spaces with k=1 or 2. Orthogonality is guaranteed by labelling each operator by a distinct set of irreducible representations of various Lie groups. The operators are able to represent all terms deriving from the non-relativistic limit of the Breit interaction for 3d electrons as well as those terms deriving from the leading types of electrostatically correlated spin-orbit (EL-SO) effects. Fits to the experimental data of Cr IV 3d3 and Ni IV 3d7 yield mean errors of only 1.91 and 3.87 cm-1 respectively. A comparison of the values of fine-structure parameters with those deduced from Hartree-Fock calculations gives good agreement for Cr IV 3d3 apart from a single parameter that is particularly sensitive to EL-SO.

Towards a Semigroup Pricing Theory

Abstract: In an arbitrage-free economy, there will always exist a set of linear operators which map future contingent dividends of securities into their current prices. It happens that such operators will also form an "evolution semigroup" as a consequence of intertemporal analysis of the no-arbitrage restriction. This paper summarizes some of the major implications of the semigroup properties, but avoids almost all of the technical discussion which underlies them. Instead, several practical examples are presented. Some wellknown continuous-time results are replicated by this alternative method, and certain new developments are explored. IT IS WELL KNOWN that in an arbitrage-free economy, there exists a nonnegative linear operator mapping dividend streams into current prices.1 What is less well known is that these operators, as a consequence of intertemporal no-arbitrage, must additionally possess a semigroup property. This paper is an overview of work in progress which deals with the semigroup features, but which is contained in rather longer and more complicated papers.2 Here I shall attempt to make the main results (and prospective results in some cases) accessible to a broader financial economics readership. At first blush, the terminology itself is intimidating: we shall be dealing with the apparently difficult notions of (1) linear operators; (2) infinitesimal generators; and (3) evolution semigroups. These terms become less daunting when it is realized that almost everyone has encountered a simple example of each via the "discounting" of future payments. Consider a world of certainty where the discount function between dates t and T is given as tQT. That is, if the current time is t and a future payment is to be received at time T in the amount A,, then the present value of the future payment would be given as Vt = tQ,A,. We see that tQ, may be considered an operator, since it maps future payments into current values; it is linear, since two (contemporaneous) payments may be added together first and then discounted, or discounted individually and added, with the same result; and finally it is nonnegative, since no positive future payment can ever have a present value of less than zero. Thus, the (certainty) discount function is an instance of a nonnegative linear operator. The infinitesimal generator may be thought of as the time derivative of the operator, in our certainty case the rate of change of the discount function. In other words, we