Showing papers on "Operator (computer programming) published in 1976"

Theory of self‐consistent electron pairs. An iterative method for correlated many‐electron wavefunctions

Abstract: A rapidly converging iterative method is presented to solve the many‐electron Schrodinger equation within a Hilbert space confined to functions with at most two electrons outside an internal space defined by the orbitals of a reference function. The wavefunction is given in terms of external two‐electron clusters represented by coefficients and density matrices referring directly to the basis functions. All matrix elements are obtained from generalized Coulomb and exchange operators. Only one operator per correlated electron pair is required for each iteration cycle.

Fractional Derivatives and Special Functions

Abstract: The fractional derivative operator is an extension of the familiar derivative operator $D^n $ to arbitrary (integer, rational, irrational, or complex) values of n. The most important representation...

Approximate Fokker–Planck collision operator for transport theory applications

Abstract: An analytically tractable approximation is developed for the linearized Fokker–Planck collision operator describing a plasma nearly in thermal equilibrium. This approximate operator preserves the symmetry properties of the exact collision integral which imply the physical conservation laws, self‐adjointness, and the H theorem. A renormalization procedure is developed to accurately treat collisions between particles of arbitrary masses. For large or small mass ratios, the approximate operator reduces to the standard expansions of the exact operator. In the case of identical particle collisions, the present approximation provides a significant improvement over the ’’model operator’’ previously given in the literature, yet retains the simplicity of former operators necessary for analytic work. The recalculation of the classical transport coefficients with this operator reduces to the solution of a coupled set of algebraic equations and indicates its reliability for use in complex neoclassical transport situations. The neoclassical electrical conductivity calculation demonstrates the new physical features of the approximate operator.

Heuristic Methods for Telephone Operator Shift Scheduling: An Experimental Analysis

Abstract: Heuristic methods are presented for scheduling telephone traffic exchange operators to meet demand that varies over a 24-hour operating period. Two types of heuristics are described 1 for determining the work shift types to be considered in preparing an operator shift schedule and 2 for constructing an operator shift schedule from a given set of work shift types. These heuristics are evaluated both in terms of solution quality and computational efficiency, using actual operating data.

On the principal eigenvalue of second‐order elliptic differential operators

Abstract: If the operator L is selfadjoint with respect to some reference measure on X , then I ( p ) can be readily computed and formula (1.2) reduces to the classical variational formula for the principal eigenvalue. If L is not selfadjoint, then in general I ( p ) is difficult to evaluate explicitly. In this paper we investigate the situation when X is Rd (it can be replaced by any connected manifold without boundary) and L is an elliptic secondorder differential operator. I ( p ) can be defined as the exact analogue of (1.1) but we need some compactness for (1.2) and therefore we consider the operator L + V only on bounded regions with Dirichlet boundary conditions. After establishing the analogue of (1.2) in such cases, we look at the problem of when two operators L1 and L2 of the type we, are dealing with can have identical I-functions. Equivalently, we seek the relation between L1 and L1 if

Sojourn Times and Asymptotic Properties of the Scattering Matrix

Abstract: and their inverses exist and are, therefore, unitary operators from H to H intertwing U and ^LZ0. the scattering operator is S=W (W~)", and it intertwines ^L£0 with itself. Now in the cases of interest for scattering theory HQ has a uniform continuous spectrum. This means there exists a Hilbert space K and an isomorphism of Hilbert spaces p:H-+L(R,K) such that p^lX^t) p~ is the operator "multiplication by e" for (7EE.R. Since S commutes with ^o? pSp~* commutes with multiplication by e for all t, and must then necessarily be of the form "multiplication by *S(0")" where S(fi^):IC—>K is for each ff^R a unitary operator. The subject of this talk will be the asymptotic behavior of 5((T) for large values of (T. Our purpose will be to examine this asymptotic behavior in special cases and attempt to discern some general pattern.

Extensions of Rank Conditions for Controllability and Observability to Banach Spaces and Unbounded Operators

Abstract: Generalizations of the familiar rank conditions for controllability and observability of linear autonomous finite-dimensional systems to the general case when both the state space and the control space are infinite-dimensional Banach spaces and the operator A acting on the state is only assumed to generate a strongly continuous semigroup (group) are sought. It is shown that a suitable version of the rank condition, although generally only sufficient for approximate controllability (observability), is however “essentially” necessary and sufficient in two important cases: (i) when A generates an analytic semigroup, (ii) when A generates a group. Such generalization of the rank condition is then used to derive, in turn, easy-to-check tests for approximate controllability (observability) for the important class of normal operators with compact resolvent. In the case of finite number of scalar controls (observations), the tests are expressed by a sequence of rank conditions, using the complete set of eigenvect...

Perturbations and Approximations for Generalized Inverses and Linear Operator Equations

Abstract: Publisher Summary This chapter explains the problems of perturbations and approximations of generalized inverses of linear operators. The approximation theory of generalized inverses of linear operators has many subtle points involving several modes of convergence, analytic and computational tractability, and techniques that are not merely extensions of those used in the matrix case. Often the study of approximations for a given mathematical object leads to a deeper understanding of the properties of that object, as suggested by Bertrand Russell. Resolution of the difficulties arising in the approximations leads to sharper insight into the properties of the exact object. The approximation methods include projectional and iterative methods and collectively-compact operator approximations. The chapter discusses these methods.

Some unsolved problems of unknown depth about operators on Hilbert space

Abstract: The paper presents a list of unsolved problems about operators on Hilbert space, accompanied by just enough definitions and general discussion to set the problems in a reasonable context. The subjects are: quasitriangular matrices, the resemblances between normal and Toeplitz operators, dilation theory, the algebra of shifts, some special invariant subspaces, the category (in the sense of Baire) of the set of non-cyclicoperators, non-commutative(i.e. operator) approximation theory, infinitary operators, and the possibility of attacking invariance problems by compactness or convexity arguments.

The Infinite-Dimensional Riccati Equation for Systems Defined by Evolution Operators

Abstract: In the paper we consider the linear, quadratic control and filtering problems for systems defined by integral equations given in terms of evolution operators. We impose very weak conditions on the evolution operators and prove that the solution to both problems leads to an integral Riccati equation which possesses a unique solution. By imposing more structure on the evolution operator we show that the integral Riccati equation can be differentiated, and finally by considering an even smaller class of evolution operators we are able to prove that the differentiated version has a unique solution. The motivation for the study of such systems is that they enable us to consider wide classes of differential delay equations and partial differential equations in the same formulation. We derive new results for such a system and show how all of the existing results can be obtained directly by our methods.

Parallel Addition and Parallel Subtraction of Operators

Abstract: The parallel sum of two invertible nonnegative operators and in a Hilbert space is the operator . This definition was extended to noninvertible operators by Anderson and Duffin for the case and by Fillmore and Williams for the general case.The investigation of parallel addition is continued in this paper; in particular, associativity is proved.Criteria are established for solvability of the equation with an unknown operator when and are given. In the case of solvability, the existence of a minimal solution , called the parallel difference, is proved.Parallel subtraction in a finite-dimensional space is considered in the last section.Bibliography: 11 titles.

Analysis with weak trace ideals and the number of bound states of Schrödinger operators

Abstract: We discuss interpolation theory for the operator ideals Ip p defined on a separable Hilbert space as those operators A whose singular values

The operator pseudoinverse in control and systems identification

Abstract: Publisher Summary This chapter discusses the pseudoinverse of a linear transformation between Hilbert spaces. It presents the basic theory and shows its use for applications to systems identification and the quadratic regulator problem. It also presents an exposition of the basic theory of the pseudoinverse for densely defined linear closed operators with arbitrary range. The theory includes the case of operators that do not have closed range. The chapter discusses the Gauss-Markov theorem on statistical estimation that is shown to be proved under the hypothesis that both the quantity to be estimated and the observations are elements of Hilbert spaces. This theorem applies only to nonsingular covariance operators and reduces to the classical Gauss-Markov theorem when the spaces are finite dimensional. In one classical form of the quadratic regulator problem, it is required to find the minimum energy input that will move a system from some initial state to the origin at a designated time. The chapter presents a reformulation that generalizes this problem to admit a greater variety of linear constraints, possibly including some of which are incompatible and/or unattainable by the system. The solution always exists as a pseudoinverse and reduces to the classical result if the system is capable of meeting the constraint.

Necessary and sufficient conditions for integral representations of Wightman functionals at Schwinger points

Abstract: Given a set of Wightman functions one would like to associate to it a field on Euclidean space admitting a simultaneous diagonalization. We investigate when this can be done in such a way that the Schwinger functions are the expectation values of this commutative field with a bounded metric operator commuting with the field. This requires as a tool the characterization of those linear functionals on the symmetric tensor algebra over a space of test functions which can be represented by complex measures on the corresponding space of distributions.

A learning curve prediction model for operators performing industrial bench assembly operations

Abstract: Learning curves of operators performing industrial operations vary according to the characteristics of the operator and the characteristics of the operation being performed In this paper a method is developed to enable prediction of a specific learning curve of the function Y = c+a(1 −exp −b(t−1)) for a specific operator performing a specific operation This specific curve is based on certain measurements of the operator and certain measurements of the operation

Nuclear spin-lattice relaxation of coupled spin systems in liquids

Abstract: A new method based upon the Zwanzig–Mori projection operator technique has been introduced to treat the spin–lattice relaxation problem of coupled nuclear spin systems in liquids, with emphasis on high resolution NMR spectroscopy. The important quantities required for the use of the method are a set of orthogonal operators which can readily be constructed and given physical meaning in the eigenstate representation for any particular spin system. Some of the expectation values of these orthogonal operators may be related to the measurable quantity in high resolution NMR experiments. The time rate of change of each observable may then be formulated in terms of the trace of various operator functions, allowing the calculation to be carried out in any representation. The computational advantage of using simple spin‐product functions is obvious. The prescription for constructing irreducible orthogonal tensor operators is given, and a variety of such tensor operators are explicitly given for several common spin systems. In the present method each orthogonal operator has a definite symmetry property under spin inversion which can be used to separate the coupled equations into a symmetric and an antisymmetric set. The problem of multiexponential decay in the time evaluation of the longitudinal magnetization, owing to effects of interferences between pairwise dipolar interactions, becomes clear in the present formalism. The present paper is a generalization of earlier work on the relaxation behavior of multiplet spectral structure in high resolution NMR.A new method based upon the Zwanzig–Mori projection operator technique has been introduced to treat the spin–lattice relaxation problem of coupled nuclear spin systems in liquids, with emphasis on high resolution NMR spectroscopy. The important quantities required for the use of the method are a set of orthogonal operators which can readily be constructed and given physical meaning in the eigenstate representation for any particular spin system. Some of the expectation values of these orthogonal operators may be related to the measurable quantity in high resolution NMR experiments. The time rate of change of each observable may then be formulated in terms of the trace of various operator functions, allowing the calculation to be carried out in any representation. The computational advantage of using simple spin‐product functions is obvious. The prescription for constructing irreducible orthogonal tensor operators is given, and a variety of such tensor operators are explicitly given for several common spin...

Conformal covariance in the framework of Wilson's renormalization group approach

Abstract: The author constructs a conformal operator in analogy to the generating operator of Wilson's incomplete-integration renormalization group. The invariance of the partition function with respect to that conformal operation yields identities among the cumulants. Evaluating these identities he finds a generalized and corrected form of the selection rule which determines those two-point cumulants which show a long-range tail. A general equation which governs the asymptotic form of the three-point cumulants is established. It is solved for several examples which involve operators of vector- or tensor-type. It is found that surface effects cannot be excluded a priori. However, the asymptotic expressions for the cumulants are consistent with a neglect of surface effects.

The commutants of certain analytic Toeplitz operators

Abstract: In this paper we characterize the commutants of two classes of analytic Toeplitz operators. We show that if F in H' is univalent and nonvanishing, the (TF2)' = (TZ}'. When g is the product of two Blaschke factors, we characterize { Tp }' in terms of algebraic combinations of Toeplitz and composition operators. Introduction. Let H2 denote the Hilbert space of functions f analytic in the open unit disk D which satisfy sup Ifl(rei)12 dO< x< . 0< r< I Let H denote the algebra of bounded analytic functions on D. For p in H??, T is the analytic Toeplitz operator defined by Tqf = pf. Let { T }' denote the commutant of T , i.e. the algebra of operators which commute with T ,. The study of analytic Toeplitz operators has been extensive and many of their properties are well known [2], [4]. In [6], Nordgren gave a sufficient condition for an analytic Toeplitz operator to have no nontrivial reducing subspaces. Since the projection onto a subspace commutes with an operator if and only if the subspace reduces the operator, the problem of finding reducing subspaces can be generalized to that of determining the commutant of an analytic Toeplitz operator. In a recent paper [3], Deddens and Wong study this latter problem. One of their results is that p univalent implies { T T,}'= {Tz}', the algebra of analytic Toeplitz operators. We extend that result to the case where p is the square of a nonvanishing univalent function. The extension generalizes and simplifies Nordgren's Example 2 in [6]. In certain special cases [1], [3], [8], analytic Toeplitz operators induced by inner functions play a significant role in commutant problems. Since these are unilateral shifts, their commutants can be characterized matricially [3]. On the other hand, the problem of finding more revealing function theoretic characterizations of their commutants is difficult. Our main result is a function theoretic characterization of { T4,}' when p is the product of two Blaschke factors. Received by the editors September 5, 1974 and, in revised form, November 5, 1974. AMS (MOS) subject classifications (1970). Primary 47B35, 47B20, 30A76.

Observer theory for distributed-parameter systems

Abstract: This paper examines the problem of the approximate reconstruction of the unknown state variables in distributed-parameter systems. New results on the observer theory for important classes of linear and non-linear operator, partial differential, and partial differential-integral equations in describing distributed-parameter systems are presented. The specific developments employ the recent results on Lyapunov stability theory, along with the theory of linear and non-linear semigroup operators, and their infinitesimal generators. The questions of observability, stability of the state reconstruction error dynamics associated with the proposed observer structure are discussed. The theoretical results are illustrated with some applications to problems of the kinetic lumping of complex distributed-parameter chemical reaction systems, as well as the observer design for linear and non-linear distributed-parameter diffusion systems.

Weighted norm inequalities for Hilbert transforms and conjugate functions of even and odd functions

Abstract: It is well known that the Hilbert tranformation and the conjugate function operator restricted to even (odd) functions define bounded linear operators on weighted LP spaces under more general conditions than is the case for the unrestricted operators. In analogy with recent results of Hunt, Muckenhoupt and Wheeden for the Hilbert transform and the conjugate function operator, we obtain necessary and sufficient conditions in order that these restricted operators should satisfy weighted weak-type inequalities and hence also necessary and sufficient conditions in order that these operators should be bounded on weighted LP spaces for 1 < p < oo.