Showing papers on "Operator (computer programming) published in 1973"

On the theory of brownian motion

Abstract: The spectrum of the Fokker-Planck operator for weakly coupled gases is considered. The operator is decomposed into operators acting on functions whose angular dependence is given by spherical harmonics. It is shown that the operator corresponding to l = 0 has zero for a point eigenvalue (the eigenfunction is the Maxwell distribution). There are no other point eigenvalues and the continuous spectrum of all of the operators is the entire negative real axis. Some consequences are briefly discussed.

A space‐time calculus based on pairs of null directions

Abstract: A formalism is presented for the treatment of space‐times, which is intermediate between a fully covariant approach and the spin‐coefficient method of Newman and Penrose. With the present formalism, a pair of null directions only, rather than an entire null tetrad, is singled out at each point. The concept of a spin‐ and boost‐weighted quantity is defined, the formalism operating entirely with such quantities. This entails the introduction of modified differentiation operators, one of which represents a natural extension of the definition of the operator ð which had been introduced earlier by Newman and Penrose. For suitable problems, the present formalism should lead to considerable simplifications over that achieved by the standard spin‐coefficient method.

A Local Visual Operator Which Recognizes Edges and Lines

Abstract: An earlier published edge operator is generalized so as to include line recognition. The linear projection space as well as the nonlinear pattern space is extended. The recognition principle of the old operator is further investigated and put to use for "edge-line" recognition. A new Solution Theorem presolves the recognition problem in generality and thus leaves only final evaluations to the computer. The speed of the operator is 23 arithmetic operations per picture point. A description of the edge or line and a reliability assessment accompany every recognition process. The operator program and a computer experiment are presented.

A generalization of the concept of equivalent linearization

Abstract: A method is presented whereby a non-linear second order dynamical system is replaced by a linear system in such a way that an average of the difference between the two systems is minimized. Provided the averaging operator possesses certain properties, it is shown that the replacement is unique and can be accomplished in a straightforward manner. The parameters of the replacement linear system are expressed in terms of averages of functions of the linearized solution.

Projection Operators in Hartree-Fock Theory

Abstract: Publisher Summary This chapter reviews some selected facets of the use of the projection operator technique in Hartree–Fock type theories. In the approximate treatment of many-electron systems, one often applies the variational procedure where the wave function and accordingly the energy of the system are expressed as functionals of electron orbital functions. The virtual orbital energy is usually positive, suggesting that they correspond to a state in the continuum. With the use of the modified Hartree-Fock operator, iterative self-consistent field (SCF) calculation is not required. If the solution to the usual Hartree-Fock equations is known, all that is required is the construction of the modified operator. The new orbital energy has a very simple physical interpretation and is appropriate for use in discussions of excitation energy. This is a definite conceptual improvement as it is difficult to visualize the relationship between the excitation energy and the old orbital energy. The chapter explains the concept of separability that gives an idea of separating the electrons of an atomic or molecular system into sets describing relevant regions (or groups) and sets describing unimportant regions.

Resolution space; operators and systems

Abstract: 1. Causality.- A. Resolution Space.- B. Causal Operators.- C. Closure Theorems.- D. The Integrals of Triangular Truncation.- E. Strictly Causal Operators.- F. Operator Decomposition.- G. Problems and Discussion.- 2. Feedback Systems.- A. Well-Posedness.- B. Stability.- C. Sensitivity.- D. Optimal Controllers.- E. Problems and Discussion.- 3. Dynamical Systems.- A. State Decomposition.- B. Controllability, Observability and Stability.- C. The Regulator Problem.- D. Problems and Discussion.- 4. Time-Invariance.- A. Uniform Resolution Space.- B. Spaces of Time-Invariant Operators.- C. The Fourier Transform.- D. The Laplace Transform.- E. Problems and Discussion.- Appendices.- A. Topological Groups.- A. Elementary Group Concepts.- B. Character Groups.- C. Ordered Groups.- D. Integration on (LCA) Groups.- E. Differentiation on (LCA) Groups.- B. Operator Valued Integration.- A. Operator Valued Measures.- B. The Lebesgue Integral.- C. The Cauchy Integrals.- D. Integration over Spectral Measures.- C. Spectral Theory.- A. Spectral Theory for Unitary Groups.- B. Spectral Multiplicity Theory.- C. Spectral Theory for Contractive Semigroups.- D. Representation Theory.- A. Resolution Space Representation Theory.- B. Uniform Resolution Space Representation Theory.- References.

Oscillator phase states, thermal equilibrium and group representations

Abstract: Eigenstates of the annihilation type operator U = C + iS, where C and S are the ``cosine'' and ``sine'' operators for harmonic oscillator phase, are shown to be closely related to thermal equilibrium states of the oscillator and to provide a new interpretation of the thermal equilibrium density operator. The problem of creating such states is considered and a general theorem is established leading to the construction of interaction Hamiltonians which transform the eigenstates of U among themselves and, in particular, create them from the oscillator ground state. These Hamiltonians lead to representations of the Lie algebras of O(2,1) and O(3). It is suggested that the mathematical technique used, in which generalized U‐type operators provide the link between a group and its representations, has its own intrinsic interest for the study of Lie groups.

Cramér-Rao inequalities for operator-valued measures in quantum mechanics

Abstract: The theory of estimation of parameters of quantum-mechanical density operators is expressed in terms of the measurement of operator-valued measures. Lower bounds on mean-square errors of parameter estimates are set by two quantum-mechanical forms of the Cramer-Rao inequality of classical statistics, derived here in terms of such measures. The results are exemplified by the simultaneous estimation of the real and imaginary parts of the complex amplitude of a coherent oscillation in the presence of thermal noise.

Duffin‐Kemmer‐Petiau subalgebras: Representations and applications

Abstract: A reduction of the Duffin‐Kemmer‐Petiau algebra to a direct sum of irreducible subalgebras for spin‐0 and spin‐1 bosons is presented. The subalgebras are defined by multiplication rules for the linearly independent basis elements. In the representations discussed the spin projection operators are independent basis elements of the subalgebras. The formal utility of these representations is demonstrated by obtaining the reduction of arbitrary operator products and trace theorems. The practical utility is demonstrated by application to the analysis of free and interacting boson field currents. Most importantly, one can understand the differences between DKP nonconserved currents and those obtained from second‐order wave equations.

A decomposition for some operators

Abstract: A decomposition of some operators is obtained that represents a generalization of Brown's (1953) work on quasi-normal operators. It is shown that the decomposition obtained is of most interest when the operator considered is far from being finite-dimensional. An application of the results obtained to the study of quasi-triangular operators is presented for illustration.

Mental set and mental arithmetic

Abstract: State University of New York, Buffalo, New York 14226 Ss performed mental arithmetic problems in which they added, subtracted, or multiplied two one-digit numbers. The presentation order of the operator symbol and the digits was varied. With three possible operators, presentation of the operator prior to the digits (OD) led to faster RTs. With two possible operators, the opposite order (digits prior to operator, DO) led to faster RTs, because RTs in the OD condition were unaffected by the number of possible operators. These results are discussed in terms of the trade off between accessing active memory for a small number of items in the DO condition vs retrieving information from relatively large tables in long-term memory in the OD condition.

On Chisholm's paradox

Abstract: It has been maintained that we are quite able to express (1*)–(4*) without the introduction of a dyadic deontic operator, provided only that we supply our standard deontic logic with a stronger conditional than material implication. The lesson learned from Chisholm's paradox has been the eminently convincing, indeed obvious, one: that what we ought to do is not determined by what is the case in some perfect world, but by what is the case in the best world we can ‘get to’ from this world. What we ought to do depends upon how we are circumstanced.

Some essentially self-adjoint Dirac operators with spherically symmetric potentials

Abstract: It is shown that the one electron Dirac operator in a stationary electric field is essentially self-adjoint, on the domain of infinitely differentiable functions of compact support, for a class of spherically symmetric potentials including the Coulomb potential, for atomic numbers less than or equal to 118. In addition, the domain of the closure of the perturbed operator is the same as the domain of the closure of the unperturbed operator. We also give an abstract theorem on domain-preserving essential self-adjointness for perturbed operators, which is perhaps of independent interest.

Strap applying tool

Abstract: A tool is disclosed for applying and tightening a strap around a bundle of articles such as electrical wires or cables. When the strap has been tensioned to a predetermined value, the tool is operative to either release or sever the strap at the discretion of the operator.

The Symmetric Groups and Calculation of Energies of n-Electron Systems in Pure Spin States

Abstract: Publisher Summary This chapter discusses certain aspects of the theory of symmetric group representations and algebras in an essentially spin-free way with the use of certain Young operators. Specifically, a t basis based upon the Young operator θ N α P α N α has been used. The determinantal form of this operator provides a solution to the problem of summing over all n! permutations in such expressions. The chapter also examines some of the problems of doing quantum mechanical calculations of electronic structures of atoms and molecules for a spin-free Hamiltonian. For a spin-free Hamiltonian the spin functions are essentially determined by symmetry and the dynamical problem of obtaining the energy may be confined to the spatial part of the wave function. There is a unique space-spin function corresponding to the symmetry adapted space functions that are constructed in the chapter.

On convex power series of a conservative Markov operator

Abstract: A. Brunel proved that a conservative Markov operator, P, has a finite invariant measure if and only if every operator Q= Zn=o Ix Pn where xc_O and Cn = I is conservative. In this note we give a different proof and study related problems. Introduction. We shall use the notation and definitions of [3]. Let us quote some basic results: The operator P is conservative if and only if for every 0_f E Lo" the sum oj?=o Pnf assumes the values 0 or oo only. The operator P is conservative if and only if whenever 0 0 and Z on=1. Such operators will be called convex power series of P, and denoted by A(P) where A(z)= >n=oCZn. 1. Conditions for Q to be conservative. THEOREM 1.1. Let P be a conservative operator and Q =A (P) a convex power series of P. If Z12l na'< oo then Q is conservative too. PROOF. Note first that

Time Operators, Partial Stationarity, and the Energy-Time Uncertainty Relation

Abstract: The reciprocal time operator which is suggested by the notion of partial stationarity is shown to permit an unambiguous and nonsingular statement of the energy-time uncertainty relation.

Eigenfield expansion technique for efficient computation of field‐swept fixed‐frequency spectra from relaxation master equations

Abstract: If the Hamiltonian and Liouville operators of a spectral intensity problem are functions of a field parameter x computation of the intensity as a function of x requires, in effect, inversion of a different large matrix for each value of x. Here we show that when the Liouville operator is a polynomial in x, with operator coefficients, solution of one generalized eigenvalue problem followed by a single solution of a system of linear equations yields the intensity for all x. This formulation promises to save large amounts of computational time, particularly for electron paramagnetic resonance problems involving large zero‐field splittings.

On a class of finite dimensional contractive perturbations of restricted shifts of finite multiplicity

Abstract: We study a class of finite-dimensional contractive perturbations of shift operators of finite multiplicity restricted to left invariant subspaces of vectorialH 2 spaces. We determine their spectra in terms of the characteristic function of the unperturbed operator and the perturbation.

Prediction of Human Operator Performance

Abstract: The Kalman Filter technique is applied to the problem of predicting human operator performance in the execution of a wide variety of tasks described by an exponential improvement model. Reliable predictions can be used as a guide by management on the efficiency of task design, operator selection, and operator training functions. Results of industrial case studies involving mechanical and electrical assemblies show that realistic predictions can be made even when the model parameters are nonstationary. Steady-state detection is also included in the paper to permit the isolation of the ``improvement plateau'' phenomenon which indicates a false performance ceiling. In such instances both the initial improvement phase and the recovery phase are described by exponential models.

Elliptic pseudodifferential boundary value problems with a small parameter in the coefficient of the leading operator

Abstract: Solvability theorems are proved and a priori estimates uniform with respect to a parameter are established. Bibliography: 17 items.