Showing papers on "Operator (computer programming) published in 1984"

A boundedness criterion for generalized Caldéron-Zygmund operators

Abstract: In 1965, A. P. Calderon showed the L2-boundedness of the so-called first Calderon commutator. This is one of the first examples of a non-convolution operator associated to a kernel which satisfies certain size and smoothness properties comparable to those of the kernel of the Hilbert transform. These properties, together with the L2-boundedness, imply the LP-boundedness for all p's in ]1, + oo[. Many operators in analysis, such as certain classes of pseudo-differential operators and the Cauchy integral operator on a curve, are associated with kernels having these properties. For these operators, one of the major questions is if they are bounded on L2. We are going to give necessary and sufficient conditions for such an operator to be bounded on L2. They are essentially that the images of the function 1 under the actions of the operator and its adjoint both lie in BMO. In the case of the aforementioned first commutator this can be checked by an integration by parts. In the first section we present some basic notions and state the theorem, which is proved in Sections 2 and 3. In Section 4 we show how to recover some classical results. In Sections 5 and 6 we construct a functional calculus for small perturbations of A, and in Section 7 we show a connection between the theory of Calderon-Zygmund operators and Kato's conjecture. It is a pleasure to express our thanks to R. R. Coifman and Y. Meyer for suggesting many elegant simplifications in our proofs and most of the applications. We also wish to thank Stephen Semmes for several pertinent remarks.

Calculations in external fields in quantum chromodynamics. Technical review

Abstract: We review the technique of calculation of operator expansion coefficients. The main emphasis is put on gluon operators which appear in expansion of n-point functions induced by colourless quark currents. Two convenient schemes are discussed in detail: the abstract operator method and the method based on the Fock-Schwinger gauge for the vacuum gluon field. We consider a large number of instructive examples important from the point of view of physical applications.

The t expansion: A nonperturbative analytic tool for Hamiltonian systems

Abstract: A systematic nonperturbative scheme is developed to calculate the ground-state expectation values of arbitrary operators for any Hamiltonian system. Quantities computed in this way converge rapidly to their true expectation values. The method is based upon the use of the operator e/sup -t/H to contract any trial state onto the true ground state of the Hamiltonian H. We express all expectation values in the contracted state as a power series in t, and reconstruct t..-->..infinity behavior by means of Pade approximants. The problem associated with factors of spatial volume is taken care of by developing a connected graph expansion for matrix elements of arbitrary operators taken between arbitrary states. We investigate Pade methods for the t series and discuss the merits of various procedures. As examples of the power of this technique we present results obtained for the Heisenberg and Ising models in 1+1 dimensions starting from simple mean-field wave functions. The improvement upon mean-field results is remarkable for the amount of effort required. The connection between our method and conventional perturbation theory is established, and a generalization of the technique which allows us to exploit off-diagonal matrix elements is introduced. The bistate procedure is used to develop a tmore » expansion for the ground-state energy of the Ising model which is, term by term, self-dual.« less

On the rate of convergence to equilibrium in one-dimensional systems

Abstract: We determine the essential spectral radius of the Perron-Frobenius-operator for piecewise expanding transformations considered as an operator on the space of functions of bounded variation and relate the speed of convergence to equilibrium in such one-dimensional systems to the greatest eigenvalues of generalized Perron-Frobenius-operators of the transformations (operators which yield singular invariant measures).

Foundations of the relativistic theory of many‐electron bound states

Abstract: Most of the existing calculations of relativistic effects in many-electron atoms or molecules are based on the Dirac–Coulomb Hamiltonian HDC. However, because the electron–electron interaction mixes positive- and negative-energy states, the operator HDC has no normalizable eigenfunctions. This fact undermines the quantum-theoretic rationale for the Dirac–Hartree–Fock (DHF) equations and therefore that of the relativistic configuration-interaction (RCI) and multiconfiguration Dirac–Fock (MCDF) methods. An approach to this problem based on quantum electrodynamics is reviewed. It leads to a configuration-space Hamilton H which involves positive-energy projection operators dependent on an external potential U; identification of U with the nuclear potential Vext corresponds to use of the Furry bound-state interaction picture. It is shown that the RCI method can be reinterpreted as an approximation scheme for finding eigenvalues of a Hamiltonian H, with U identified as the DHF potential; the theoretical interpretation of the MCDF method needs further clarification. It is emphasized that if U differs from Vext one must consider the effects of virtual-pair creation by the difference potential δU = Vext − U; an approximate formula for the level-shift arising from δU is derived. Some ideas for dealing with the technical problems introduced by the projection operators are discussed and relativistic virial theorems are given. Finally, a possible scheme for adapting current MCDF methods to Hamiltonians involving projection operators is described.

Rates of convergence of variational calculations and of expectation values

Abstract: We present a mathematical and numerical analysis of the rates of convergence of variational calculations and their impact on the issue of the convergence or divergence of expectation values obtained from variational wave functions. The rate of convergence of a variational calculation is critically dependent on the ability of finite linear combinations of basis functions to simulate the nonanalyticities (cusps) in the exact wave function being approximated. A slow rate of convergence of the variational energy can imply that the corresponding variational wave functions will yield divergent expectation values of physical operators not relatively bounded by the Hamiltonian. We illustrate the sorts of problems which can arise by examining Gauss‐type approximations to hydrogenic orbitals. Since all many‐electron wave functions have cusps similar to those in hydrogenic wave functions, this simple example is relevant to variational calculations performed on atoms and molecules. Finally, we offer suggestions on what types of variational wave functions are likely to yield rapid rates of convergence for the energy and reasonable rates of convergence for physical operators such as the dipole moment operator.

Some results on operator performance in emergency events

Abstract: This paper synthesizes results from several studies of operator performance in simulated power plant emergencies and from retrospective analyses of operator performance in five actual power plant critical incidents. This synthesis is feasible because all of these studies used process tracing techniques and a common perspective on decision making as the basis for their analysis of operator performance. The paper focuses on two areas: operator's ability to detect and correct errors and how operators utilize procedures. The results in these areas are assessed in terms of their implications for concepts and models relevant to operator performance and in terms of their implications for man-machine system improvements.

Effective operators in the atomic hyperfine interaction

Abstract: The many-body perturbation theory (MBPT) is reviewed and applied to the atomic hyperfine interaction. Graphical methods are introduced by without mathematical details. The results are interpreted in terms of effective operators. For systems with a single valence electron-such as the alkali atoms-this operator has the same form as the ordinary hyperfine operator and is identical to the operator commonly used in the analysis of experimental hyperfine data. The origin of different contributions to this operator is discussed. Numerical results are given for the 22S and 22P states of the lithium atom, where accurate MBPT calculations have recently been performed. For systems with several valence electrons additional parameters are needed or, alternatively, the parameters of the one-body effective operator are allowed to be term-dependent. Recent experiments and corresponding theoretical investigations on alkaline-earth elements, with two valence electrons, are reviewed and, in particular, MBPT calculations on the calcium atom are discussed.

Quantum Mechanical Path Integrals With Wiener Measures for All Polynomial Hamiltonians

Abstract: We construct arbitrary matrix elements of the quantum evolution operator for a wide class of self-adjoint canonical Hamiltonians, including those which are polynomial in the Heisenberg operators, as the limit of well defined path integrals involving Wiener measure on phase space, as the diffusion constant diverges. A related construction achieves a similar result for an arbitrary spin Hamiltonian.

Exponentially bounded positive definite functions

Abstract: Equivalent conditions for scalar (or operator valued) positive definite functions, on a commutative semigroup $S$ with identity $e$, to admit a disintegration with respect to a regular positive (operator valued) measure supported by an arbitrary compact subset of semicharacters are given. The theory links to the theory of $\tau$-positive functions presented previously by the second author and comparisons between the two are given. Old and new theorems to classical and modern moment problems are obtained as a consequence.

OR Practice-A Queueing Model for Telephone Operator Staffing

Abstract: This paper describes a queueing model of telephone operator staffing and the application of the model. The Bell System has used the model to reduce the cost of meeting its service criteria, for planning purposes, and to help explain to regulatory agencies changes in service measurement criteria. The model deals with large server team sizes, bimodal service time distributions, nonstationary arrivals, customer abandonments and reattempts, and certain priority queueing structures.

Formulas for the eigenvalues of the Laplacian on tensor harmonics on symmetric coset spaces

Abstract: On a symmetric coset space G/H the eigenvalues of the Laplacian and the Lichnerowicz operator acting on arbitrary tensor harmonics are given in terms of the eigenvalues of the quadratic Casimir operators of G and H. Explicit examples for Sn, CPn, and real (complex) Grassmann manifolds are analyzed.

The Coupling Method for Solving Integral Equations

Abstract: This paper presents a new method to reduce integral operators of various classes to simpler operators, which often are just finite matrices. By this method the problem to find the inverse, generalized inverses, kernel and image of an integral operator is reduced for several cases to the corresponding problem for a finite matrix. The classes of integral operators dealt with include integral operators of the second kind with a finite rank or semi-separable kernel and also, which is more surprising, systems of Wiener-Hopf integral operators and singular integral operators with rational matrix symbols.

Expansions over the ‘‘squared’’ solutions and difference evolution equations

Abstract: The completeness relation for the system of ‘‘squared’’ solutions of the discrete analog of the Zakharov–Shabat problem is derived It allows one to rederive the known statements concerning the class of difference evolution equations related to this linear problem and to obtain additional results These include: (i) the expansion of the potential and its variations over the system of ‘‘squared’’ solutions, the expansion coefficients being the scattering data and their variations, respectively; thus the interpretation of the inverse scattering transform (IST) as a generalized Fourier transform becomes obvious; (ii) compact expressions for the trace identities through the operator Λ, for which the ‘‘squared’’ solutions are eigenfunctions; (iii) brief exposition of the spectral theory of the operator Λ; (iv) direct calculation of the action‐angle variables based on the symplectic form of the completeness relation; (v) the generating functional of the M operators in the Lax representation; (vi) the quantum ve

Microlocal regularity theorems for nonsmooth pseudodifferential operators and applications to nonlinear problems

Abstract: On developpe un calcul d'operateurs pseudodifferentiels a coefficients non lisses afin d'etudier la regularite des solutions des equations lineaires P(x,D)u=f

A Full Coupled-Cluster Singles, Doubles, and Triples Model for the Description of Electron Correlation

Abstract: Equations for the determination of the cluster coefficients in a full coupled cluster theory involving single, double and triple cluster operators with respect to an independent particle reference, expressible as a single determinant of spin-orbitals, are derived The resulting wave operator is full, or untruncated, consistant with the choice of cluster operator truncation and the requirements of the connected cluster theorem A time-independent diagrammatic approach, based on second quantization and the Wick theorem, is employed Final equations are presented that avoid the construction of rank three intermediary tensors The model is seen to be a computationally viable, size-extensive, high-level description of electron correlation in small polyatomic molecules

Quantum mechanics in the Gaussian wave-packet phase space representation

Abstract: We present here some results related to an alternative mapping formulation (to the well-known Wigner-Weyl transform) which makes use of the Gaussian Wave Packet or coherent states representation | pq ↩. The pair p-q which labels this state defines a phase-space in which the abstract operators P, Q , of momentum and position, are represented as differential operators. In the mapped expression of an operator A ( P, Q ), the quantum effects appear when they are absent in the corresponding Wigner-Weyl transform.

Granger-Causality and Policy Effectiveness

Abstract: It is generally recognized that, if a set of monetary and fiscal policy variables Granger-cause1 real economic variables, this does not imply that alternative deterministic rules for determining the values of these policy instruments will alter the joint density function of the real variables.2 It has, however, also been asserted that (letting X denote a (suitably restricted) vector of "real" economic aggregates, g a list of monetary and fiscal policy variables and E the mathematical expectation operator),

Monotone explicit iterations of the finite element approximations for the nonlinear boundary value problem

Abstract: In this paper, we consider monotone explicit iterations of the finite element schemes for the nonlinear equations associated with the boundary value problem Δu=bu 2, based on piecewise linear polynomials and the lumping operator. These iterations construct the monotonically decreasing and increasing sequences, and convergence proofs are given. Finally, we present some numerical examples verifying the effectiveness of the theory.

Isovector electromagnetic exchange currents in the chiral approach

Abstract: The problem studied in the paper concerns the structure of the isovector e.m. MEC operators of the A1−, ϱ- and π ranges in the interval of intermediate energies. The two main dynamic principles which we invoke in our considerations are the current conservation and the gauge chiral invariance. Respecting them consistently allows us to describe correctly the interaction of NA1 ϱπ system with the external e.m. field. Our main results are as follows: (i) We verified that our Lagrangian approach is consistent with the prediction of the low energy theorem at threshold. (ii) We showed explicitly the continuity equation which the longitudinal parts of our currents obey. (iii) We proved the equivalence relation for the MEC operator of the pion range and demonstrated the existence of the seagull current in the MEC operator built up using PS πN coupling. This new term influences strongly the exchange charge density.