Showing papers on "Operator (computer programming) published in 1978"

Fictitious spin 1/2 operator formalism for multiple quantum NMR

Abstract: A formalism is presented that describes the time behavior of the spin density matrix of a nuclear spin system with arbitrary spin in terms of fictitious spin −(1/2) operators. This formalism is an extension of that used earlier for nuclei with spin I=1. For a spin system with n eigenstates we define for every pair of eigenstates ‖i〉 and ‖j〉 three operators Ii−jp, with p=x, y, and z, according to the three 2×2 Pauli matrices σx, σy, and σz. These operators together constitute a complete set of n2−1 independent Hermitian operators, and we can write the n×n density matrix and the spin Hamiltonian of the system in terms of the Ii−jp operators. The commutation relations among the operators make it possible in many cases to solve the equation of motion of the density matrix analytically. Three examples of the use of the Ii−jp operators are presented. Firstly a system of noninteracting spins with I=1 is considered. The Ii−jp operators for this case are compared with the Iq,k operator defined earlier. The cw sign...

Rear viewing system for vehicles

Abstract: A visual aid system for assisting an operator while he maneuvers a vehicle comprises a closed circuit video system having distance measuring apparatus. The system enables the operator to see behind the vehicle and, through the use of the distance indicating apparatus, the operator can tell how far the rear of the vehicle is from objects seen via the video system.

Asymptotic Behavior of Solutions of Evolution Equations

Abstract: Publisher Summary This chapter presents the methods of investigation of the asymptotic behavior of solutions of evolution equations, endowed with a dissipative mechanism, based on the study of the structure of the ω-limit set of trajectories of the evolution operator generated by the equation. The dissipative mechanism usually manifests itself by the presence of a Liapunov functional, which is constant on ω -limit sets; the central idea of the approach presented in the chapter is to use this information in conjunction with properties of ω -limit sets such as invariance and minimality. The chapter discusses two examples of wave equations with weak damping for which the scheme set up by Hale applies. In particular, this requires that the Liapunov functional be continuous on phase space. The chapter explores the case of a hyperbolic conservation law that generates a semigroup on space. It also presents a survey of various applications and extensions of these ideas that may serve as a guide to those interested in learning more about the method.

Unitary operator colligations and their characteristic functions

Abstract: ContentsIntroduction § 1. Unitary colligations § 2. Operations with colligations § 3. Characteristic functions of unitary colligations § 4. Some facts about holomorphic contracting operator functions § 5. Construction of a colligation from its characteristic operator function § 6. Factorization of characteristic functions § 7. The functional model of a unitary colligation § 8. Analytic criteria for regularity of factorizationReferences

Some recent developments in operator theory

Abstract: The spectral picture of an operator Pulling out direct summands The reducing essential matricial spectra of an operator Quasitriangular operators Spectral characterization of nonquasitriangular operators Approximation by nilpotent operators The Lomonosov technique A look at the invariant-subspace problem A model for quasinilpotent operators The Brown-Douglas-Fillmore theorem Bibliography.

An Efficient Method for Reliability Evaluation of a General Network

Abstract: This paper presents a method for finding the terminal-pair reliability expression of a general network. First, the system success function S is found, beginning from the connection matrix for the logic diagram of the network. Second, using the concept of Exclusive operator, S is changed to its equivalent S (disjoint) form, and the reliability expression has been derived. The method has the advantage of not requiring step by step testing for disjointness. Examples illustrate the method.

Approximative methods for nonlinear equations (two approaches to the convergence problem)

Abstract: THIS survey paper gives a functional-analytical treatment of discretization methods such as quadrature formula method for nonlinear integral equations, difference method for nonlinear boundary value problems, etc. Two approaches to the convergence problem have been developed. The first of them (Section 3) is applicable to an equation with differentiable operator and rests on a remark that such an operator is locally almost linear. The second, less traditional approach (Section 4) is based on a topological concept, namely the invariance of the fixed point index under suitable approximations of an operator. As regards the approximation concepts, the paper is built on a relatively novel principle of regular convergence of operators (Section 2). In our fixed opinion, this concept is rather appropriate to applications, and we hope that the reader agrees with us familiarizing himself with the proof ideology of Sections 5-7. Another methodological prop of the paper is the concept of discrete convergence (Section 1). In Sections 5-7 the abstract results of Sections l-4 have been applied to the quadrature formula method for nonlinear integral equations and to the collocation, subregion, Galerkin and difference methods for nonlinear boundary value problems. Only ordinary differential equations are considered. For partial differential equations our approaches are still weakly developed : first works (e.g. [l-3]) concern linear equations. Sections l-3 contain more material than is urgently needed for applications, our significant goal. By stars are labelled the sections, propositions etc. that can be omitted if one wishes to get to applications more quickly. The main text contains only few references. For the reference notes, see the end of the paper.

Measurement of program complexity by the pair: (Cyclomatic Number, Operator Count)

Abstract: In a r e c e n t paper, McCabe [1976] i n t r o d u c e d the cyclomatic number of a program's flow graph as a measure of its complexity. Myers [1977] proposed an improved measure consisting o[ an interval with the original measure as its upperbound. I will argue below that--if two values are to be presented as a measure--it is preferable to couple a variation of the cyclomatic number with a measure o[ the prog~-am's express:ion complexity.

Remarks on the schroedinger operator with singular complex potentials

Abstract: : Schroedinger operators of the form A = delta + V(x), where delta is the Laplacian and V is a scalar potential, arise in quantum mechanics and other areas. Delicate questions concerning what domain should be assigned to A must be settled in order to have a good theory. These questions are answered here for a very general class of potentials V which may even have complex values.

Inequalities between means of positive operators

Abstract: One of the most fruitful – and natural – ways of introducing a partial order in the set of bounded self-adjoint operators in a Hilbert space is through the concept of a positive operator. A bounded self-adjoint operator A denned on is called positive – and one writes A ≥ 0 - if the inner product (ψ, Aψ) ≥ 0 for every ψ ∈ . If, in addition, (ψ, Aψ) = 0 only if ψ = 0, then A is called positive-definite and one writes A > 0. Further, if there exists a real number γ > 0 such that A — γI ≥ 0, I being the unit operator, then A is called strictly positive (in symbols, A ≫ 0). In a finite dimensional space, a positive-definite operator is also strictly positive.

A note on the operator compact implicit method for the wave equation

Abstract: : In a previous paper a fourth order compact implicit scheme was presented for the second order wave equation. A very efficient factorization technique was developed when only second order terms were present. In this note we implement the operator compact implicit spatial discretization method for the second order wave equation when first order terms are present. The resulting algorithm is completely analogous to the compact implicit algorithm when lower order terms were not present. For this more general operator compact implicit spatial approximation the same factorization as in our previous paper is developed. (Author)

Linear response theory revisited. I. The many‐body van Hove limit

Abstract: A critical discussion of linear response theory is given. It is argued that in the formalism as it stands no dissipation is manifest. A physical reinterpretation for the case of a system in weak interaction with a reservoir is given. Mathematically this means that the van Hove limit, as well as the large system limit, is applied to the time‐dependent Heisenberg operators of the Kubo formalism. The reduced operators can be put in a very compact form, viz.,BRα(t) =[exp(−Λdt)]Bα, where Bα is a Schrodinger operator and Λd is the Liouville space superoperator corresponding to the transition operator of the master equation. In this form the relaxation character of the transport expressions, and the approach to equilibrium is at once evident. New expressions for the generalized susceptibility and conductivity in this limit are presented. Also, the Onsager relations and other symmetry properties are confirmed.

Random fixed point theorems

Abstract: Publisher Summary This chapter presents random fixed point theorems. The study of random operator equations was initiated by the Prague school of probabilists around Spacek and Hans in the 1950s. As it seems to be a current trend to use stochastic models rather than deterministic ones, it is not surprising that the interest in random operator equations has been revived in the last years. The basic questions asked about random operator equations contain all problems that are interesting for deterministic operator equations, such as existence, uniqueness, stability, and approximation of solutions. However, the randomization leads to several new questions, such as the measurability of solutions and their statistical properties. The chapter presents the question of single- and multivalued random operators on randomly varying domains of definition.

Choice of operator length for maximum entropy spectral analysis

Abstract: Empirical evidence based on maximum entropy spectra of real seismic data suggests that M = 2N/ln 2N is a reasonable a priori choice of the operator length M for discrete time series of length N. Various examples support this conclusion.

The fuglede commutativity theorem modulo the hilbert-schmidt class and generating functions for matrix operators

Abstract: We prove the following statements about bounded linear operators on a separable, complex Hilbert space: (1) Every normal operator N that is similar to a Hilbert-Schmidt perturbation of a diagonal operator D is unitarily equivalent to a Hilbert-Schmidt perturbation of D; (2) For every normal operator N, diagonal operator D and bounded operator X, the Hilbert-Schmidt norms (finite or infinite) of NX XD and N*X XD* are equal; (3) If NX XN and N*X XN* are Hilbert-Schmidt operators, then their Hilbert-Schmidt norms are equal; (4) If X is a Hilbert-Schmidt operator and N is a normal operator so that NX XN is a trace class operator, then Trace(NX XN) = 0; (5) For every normal operator N that is a Hilbert-Schmidt perturbation of a diagonal operator, and every bounded operator X, the Hilbert-Schmidt norms (finite or infinite) of NX XN and N*X XN* are equal. The main technique employs the use of a new concept which we call 'generating functions for matrices'. Let H denote a separable, complex Hilbert space and let L(H) denote the class of all bounded linear operators acting on H. Let K(H) denote the class of compact operators in L(H) and let Cp denote the Schatten p-class (O 0, there exist a diagonal operator D and a Hilbert-Schmidt operator Ke with lIKe 112 < e for which N _ D + Ke (_ denotes unitary equivalence). (2) For every normal operator N, there exist a diagonal operator D and a K E C2 for which N-D + K. (3) For every normal operator N and bounded operator X, JINX XN112 = IIN*X XN*112. Received by the editors March 3, 1977 and, in revised form, August 29, 1977. AMS (MOS) subject classifications (1970). Primary 47A05, 47A55, 47B10, 47B15, 47B47; Secondary 05A15, 05B20.

On the Operator Identity ∑ AkXBk ≡ 0

Abstract: Let Aj and Bj (1 ≦ j ≦ m) be bounded operators on a Banach space ᚕ and let Φ be the mapping on , the algebra of bounded operators on ᚕ, defined by (1) We give necessary and sufficient conditions for Φ to be identically zero or to be a compact map or (in the Hilbert space case) for the induced mapping on the Calkin algebra to be identically zero. These results are then used to obtain some results about inner derivations and, more generally, about mappings of the form For example, it is shown that the commutant of the range of C(S, T) is “small” unless S and T are scalars.

Construction of triangular finite element universal matrices

Abstract: The various ‘universal’ matrices from which finite element matrices for triangular elements are assembled in many electromagnetics and acoustics problems, can all be derived from a basic set of three fundamental matrices. These represent, respectively, the metric of the linear manifold spanned by the triangle interpolation polynominals, the finite differentiation operator on that same manifold, and a product-embedding operator for the corresponding manifold for interpolation polynomials one order higher. Two of these have already been tabulated and published; the required method for computing the third is given in this paper, along with tables of low-order matrices.

Self-Adjoint Variational Formulation of Problems Having Non-Self-Adjoint Operators

Abstract: A systematic approach is given for deriving a variational formulation, previously stated by Stakgold, of non-self-adjoint operators from the standard quadratic functional for self-adjoint operators given by Mikhlin. If the same set of basis functions is used to approximate the solution of the operator equation and its adjoint equation, the resulting system of equations is identical to that derived from the Galerkin method. By using two differing sets of basis functions, one obtains a system of equations which corresponds to that derived from the moment method in general. As a particular and important example, the integral equation for the interface problem between differing media is considered. Compared to the method used by McDonald, Friedman, and Wexler, the present formulation involves no danger of finding a false solution, results in a simpler set of equations, requires fewer integrations, and is seen--in the case of integral equations--to correspond to the Galerkin method. It is also shown that for wave propagation through a lossy medium, which involves the solution of the non-self-adjoint complex Helmholtz equation, the resulting system of linear equations takes the same form as those for the real self-adjoint case but for the addition of complex arithmetic.

Cluster expansion of the wavefunction. Structure of the closed‐shell orbital theory

Abstract: A new approach to the closed‐shell orbital theory is presented with the formalism of the cluster expansion of the wavefunction. The four independent excitation operators are used to represent the general determinantal wavefunction and also to discuss the stability of the Hartree–Fock solution. This leads to a new concept called the ’’stability dilemma,’’ which is the key not only in understanding the structure of the orbital theory but also for the extension of the orbital model. Only when the stability dilemma is removed, the correlation effect is taken into account within the framework of the orbital approximation. The closed‐shell orbital theory including the electron correlation is defined as ‖Φ〉=P exp[iF]‖0〉, where the F is the excitation operator to generate the variational space and the P is the projection operator to remove the stability dilemma. The various orbital theories (some are known but some are new) can be obtained by appropriate choices of the F and P. It is shown that the above cluster ...

Simplified irrigation controller

Abstract: The present invention discloses a simplified solid state controller for use in controlling automatic irrigation systems. The operator input panel provides no means for the inputting of numerical irrigation information such as station start times and station run times. Rather, the input panel provides switch means whereby the operator can initiate a number of parameterization logic sequences wherein the current information data in memory is displayed to the operator. Through the input panel, the operator can cause the information displayed to be advanced to a new desired value on a step-by-step basis. Upon the displayed value being equal to the value desired as the new value by the operator, the operator signals to the logic of the controller through the input panel and the current value is updated to the presently displayed value.