Showing papers on "Operator (computer programming) published in 1989"

Phase properties of the quantized single-mode electromagnetic field.

Abstract: The usual mathematical model of the single-mode electromagnetic field is the harmonic oscillator with an infinite-dimensional state space, which unfortunately cannot accommodate the existence of a Hermitian phase operator. Recently we indicated that this difficulty may be circumvented by using an alternative, and physically indistinguishable, mathematical model of the single-mode field involving a finite but arbitrarily large state space, the dimension of which is allowed to tend to infinity after physically measurable results, such as expectation values, are calculated. In this paper we investigate the properties of a Hermitian phase operator which follows directly and uniquely from the form of the phase states in this space and find them to be well behaved. The phase-number commutator is not subject to the difficulties inherent in Dirac's original commutator, but still preserves the commutator--Poisson-bracket correspondence for physical field states. In the quantum regime of small field strengths, the phase operator predicts phase properties substantially different from those obtained using the conventional Susskind-Glogower operators. In particular, our results are consistent with the vacuum being a state of random phase and the phases of two vacuum fields being uncorrelated. For higher-intensity fields, the quantum phase properties agree with those previously obtained by phenomenological and semiclassical approaches, where such approximations are valid. We illustrate the properties of the phase with a discussion of partial phase states. The Hermitian phase operator also allows us to construct a unitary number-shift operator and phase-moment generating functions. We conclude that the alternative mathematical description of the single-mode field presented here provides a valid, and potentially useful, quantum-mechanical approach for calculating the phase properties of the electromagnetic field.

Adapting operator probabilities in genetic algorithms

Abstract: In the vast majority of genetic algorithm implementations, the operator probabilities are xed throughout a given run. However, it can be convincingly argued that these probabilities should vary over the course of a genetic algorithm run | so as to account for changes in the ability of the operators to produce children of increased tness. This dissertation describes an empirical investigation into this question. The e ect upon genetic algorithm performance of adaptation methods upon both well-studied theoretical problems, and a hard problem from Operations Research | the owshop sequencing problem, is examined. Acknowledgements I would rst of all like to thank my supervisor, Dr Peter Ross, for his valuable guidance of the research reported here. Thanks also to Dave Corne and the members of the DAI EC group for their occasional advice. Thanks to the other MScs who have provided a very enjoyable working atmosphere. Special thanks to Michael L, Emma C, Mike F, Richard W and Larry F (for proving that heartless Canadians are the exception rather than the rule!). Mention and gratitude must also go to Pete, Daren, Jon, Evonne and Rob for their entertaining and supportive email. I also have to thank Dave and Elaine for providing the same support over more mundane communication methods! I am grateful to Keith and Frankie Tuson for their encouragement and nancial support during my academic career. Extra special thanks must go to Dr Hugh Cartwright of the Physical and Theoretical Chemistry Laboratory at Oxford University. His supervision of my chemistry Part II research project proved to be a turning point for me, and inspired my interest in AI no amount of malt whisky can express my gratitude enough. Finally, I thank the EPSRC who provided nancial support during my year of study via studentship no: 94415692. Dedication I would like to dedicate this dissertation to my family: Laurence, Margaret and Karen Tuson who have supported and encouraged me throughout my academic career. i Table of

Quaternionic analysis and elliptic boundary value problems

Abstract: 1. Quaternionic Analysis.- 1.1. Algebra of Real Quaternions.- 1.2. H-regular Functions.- 1.3. A Generalized LEIBNIZ Rule.- 1.4. BOREL-POMPEIU's Formula.- 1.5. Basic Statements of H-regular Functions.- 2. Operators.- 2.3. Properties of the T-Operator.- 2.4. VEKUA's Theorems.- 2.5. Some Integral Operators on the Manifold.- 3. Orthogonal Decomposition of the Space L2,H(G).- 4. Some Boundary Value Problems of DIRICHLET's Type.- 4.1. LAPLACE Equation.- 4.2. HELMHOLTZ Equation.- 4.3. Equations of Linear Elasticity.- 4.4. Time-independent MAXWELL Equations.- 4.5. STOKES Equations.- 4.6. NAVIER-STOKES Equations.- 4.7. Stream Problems with Free Convection.- 4.8. Approximation of STOKES Equations by Boundary Value Problems of Linear Elasticity.- 5. H-regular Boundary Collocation Methods.- 5.1. Complete Systems of H-regular Functions.- 5.2. Numerical Properties of H-complete Systems of H-regular Functions.- 5.3. Foundation of a Collocation Method with H-regular Functions for Several Elliptic Boundary Value Problems.- 5.4. Numerical Examples.- 6. Discrete Quaternionic Function Theory.- 6.1. Fundamental Solutions of the Discrete Laplacian.- 6.2. Fundamental Solutions of a Discrete Generalized CAUCHY-RIEMANN Operator.- 6.3. Elements of a Discrete Quaternionic Function Theory.- 6.4. Main Properties of Discrete Operators.- 6.5. Numerical Solution of Boundary Value Problems of NAVIER-STOKES Equations.- 6.6. Concluding Remarks.- References.- Notations.

Admissible observation operators for linear semigroups

Abstract: Consider a semigroupT on a Banach spaceX and a (possibly unbounded) operatorC densely defined inX, with values in another Banach space. We give some necessary as well as some sufficient conditions forC to be an admissible observation operator forT, i.e., any finite segment of the output functiony(t)=CTtx,t≧0, should be inLp and should depend continuously on the initial statex. Our approach is to start from a description of the map which takes initial states into output functions in terms of a functional equation. We also introduce an extension ofC which permits a pointwise interpretation ofy(t)=CTtx, even if the trajectory ofx is not in the domain ofC.

The weighted particle method for convection-diffusion equations. I. The case of an isotropic viscosity

Abstract: A particle method for convection-diffusion equations based on the approximation of diffusion operators by integral operators and the use of a particle method to solve integro-differential equations previously described is presented and studied. The isotropic diffusion operators are dealt with first. Two approximation possibilities are obtained, depending on whether or not the integral operator is positive. An extension of the method to anisotropic diffusion operators follows. The consistency and the accuracy of the method require much more complex conditions on the cutoff functions than in the isotropic case. After detailing these conditions, several examples of cutoff functions which can be used for practical computations are given. A detailed error analysis is then performed. 24 refs.

Hilbert Modules over Function Algebras

Abstract: Much of the early motivation for the study of operator theory came from integral equations although early in this century both operator theory and functional analysis took on a life of their own. Self-adjoint operators, both bounded and unbounded, occupied center stage for several decades either singly or in algebras. During the last two or three decades various approaches to the non-selfadjoint theory have been introduced with considerable success at least in the case of a single operator. The generalization to several operators, whether commuting or non-commuting, has largely eluded us. In this note we want to outline a different point of view which may assist in guiding developments in this area.

Heuristic search approach to distribution system restoration

Abstract: Service restoration, service reconfiguration, and other related problems are formulated and solved by heuristic search, which is a search strategy (e.g. depth-first search) armed with practical rules (e.g. based on operator experience) to guide the search. The method is based on operator procedures and usually generates the same solutions as operators would do. However, it is also possible to investigate alternatives that normally would not be considered by system operators, which can be very helpful under certain critical operating conditions. Moreover, the proposed framework makes it possible to investigate the effect of practical rules on the optimality of the final solution and so can be a useful tool in designing new algorithms. Test results are presented, and an illustrative example is given. The preliminary results indicate that the proposed technique is very promising. >

Lectures on Hyponormal Operators

Abstract: I: Subnormal operators.- 1. Elementary properties and examples.- 2. Characterization of subnormality.- 3. The minimal normal extension.- 4. Putnam's inequality.- 5. Supplement: Positive definite kernels.- Notes.- Exercises.- II: Hyponormal operators and related objects.- 1. Pure hyponormal operators.- 2. Examples of hyponormal operators.- 3. Contractions associated to hyponormal operators.- 4. Unitary invariants.- Notes.- Exercises.- III: Spectrum, resolvent and analytic functional calculus.- 1. The spectrum.- 2. Estimates of the resolvent function.- 3. A sharpened analytic functional calculus.- 4. Generalized scalar extensions.- 5. Local spectral properties.- 6. Supplement: Pseudo-analytic extensions of smooth functions.- Notes.- Exercises.- IV: Some invariant subspaces for hyponormal operators.- 1. Preliminaries.- 2. Scott Brown's theorem.- 3. Hyperinvariant subspaces for subnormal operators.- 4. The lattice of invariant subspaces.- Notes.- Exercises.- V: Operations with hyponormal operators.- 1. Operations.- 2. Spectral mapping results.- Notes.- Exercises.- VI: The basic inequalities.- 1. Berger and Shaw's inequality.- 2. Putnam's inequality.- 3. Commutators and absolute continuity of self-adjoint operators.- 4. Kato's inequality.- 5. Supplement: The structure of absolutely continuous self-adjoint operators.- Notes.- Exercises.- VII: Functional models.- 1. The Hilbert transform of vector valued functions.- 2. The singular integral model.- 3. The two-dimensional singular integral model.- 4. The Toeplitz model.- 5. Supplement: One dimensional singular integral operators.- a) The Cauchy integral.- b) The Sohotskii - Plemelj - Privalov formulae.- c) The Hilbert transform on L2(R).- d) Singular integral operators with continuous symbol.- e) The Riemann - Hilbert problem.- Notes.- Exercises.- VIII: Methods of perturbation theory.- 1. The phase shift.- 2. Abstract symbol and Friedrichs operations.- 3. The Birman - Kato - Rosenblum scattering theory.- 4. Boundary behaviour of compressed resolvents.- 5. Supplement: Integral representations for a class of analytic functions defined in the upper half-plane.- Notes.- Exercises.- IX: Mosaics.- 1. The phase operator.- 2. Determining functions.- 3. The principal function.- 4. Symbol homomorphisms and mosaics.- 5. Properties of the mosaic.- 6. Supplement: A spectral mapping theorem for the numerical range.- Notes.- Exercises.- X: The principal function.- 1. Bilinear forms with the collapsing property.- 2. Smooth functional calculus modulo trace-class operators and the trace formula.- 3. The properties of the principal function.- 4. Berger's estimates.- Notes.- Exercises.- XI: Operators with one dimensional self-commutator.- 1. The global local resolvent.- 2. The kernel function.- 3. A functional model.- 4. The spectrum and the principal function.- Notes.- Exercises.- XII: Applications.- 1. Pairs of unbounded self-adjoint operators.- 2. The Szego limit theorem.- 3. A two dimensional moment problem.- Notes.- Exercises.- References.- Notation and symbols.

Spectral theory of two-dimensional periodic operators and its applications

Abstract: CONTENTS Introduction Chapter I. The spectral theory of the non-stationary Schrodinger operator § 1. The perturbation theory for formal Bloch solutions § 2. The structure of the Riemann surface of Bloch functions § 3. The approximation theorem § 4. The spectral theory of finite-gap non-stationary Schrodinger operators § 5. The completeness theorem for products of Bloch functions Chapter II. The periodic problem for equations of Kadomtsev-Petviashvili type § 1. Necessary information on finite-gap solutions § 2. The perturbation theory for finite-gap solutions of the Kadomtsev-Petviashvili –2 equation § 3. Whitham equations for space two-dimensional "integrable systems" § 4. The construction of exact solutions of Whitham equations § 5. The quasi-classical limit of two-dimensional integrable equations. The Khokhlov-Zabolotskaya equationChapter III. The spectral theory of the two-dimensional periodic Schrodinger operator for one energy level § 1. The perturbation theory for formal Bloch solutions § 2. The structure of complex "Fermi-curves" § 3. The spectral theory of "finite-gap operators with respect to the level E0" and two-dimensional periodic Schrodinger operators References

Factorisation of energy dependent Schrödinger operators: Miura maps and modified systems

Abstract: We consider the energy dependent Schrodinger operator\(\mathbb{L} = \sum\limits_{i = 0}^N {\lambda ^i (\varepsilon _i \partial ^2 + u_i )} \), which we have previously shown to be associated with multi-Hamiltonian structures [2]. In this paper we use an unusual form of the Lax approach to derive by asingle construction the time evolutions of the eigenfunctions of\(\mathbb{L}\), the associated Hamiltonian operators and the Hamiltonian functionals. We then generalise the well known factorisation of standard Lax operators to the case of energy-dependent operators. The simple product of linear factors is replaced by a λ-dependent quadratic form. We thus generalise the resulting construction of Miura maps and modified equations. We show that for some of our systems there exists a sequence ofN such modifications, therth modification possessing (N−r+1) Hamiltonian structures.

Document processing system having integrated expert module

Abstract: A bank document processing system for processing checks, deposit slips and other financial documents which advantageously combines an automatic expert reconciliation system with conventional multi-tasking document processing apparatus so as to enable the expert reconciliation system to function as an integral part of the overall document processing system. In particular, the combined system allows expert and conventional data processing systems to efficiently and automatically communicate with one another and an operator so that expert solutions as well as conventional prebalancing results can be displayed to the operator to permit him to expeditiously perform transaction balancing functions without the need for the operator to have special expertise in finding these solutions.

Evolution equations with dynamic boundary conditions

Abstract: In this paper, we study boundary problems with dynamic boundary conditions, that is, with boundary operators containing time derivatives. The equations under consideration are transformed into abstract Cauchy problems x – Cx = f and x(0) = x0. Abstract theoretical results concerning the operators C are obtained by the study of a naturally arising pseudodifferential operator. For existence and uniqueness theorems concerning solutions of parabolic and hyperbolic equations, we then apply the theory of semigroups in Banach spaces. Some examples of semilinear and quasilinear problems, to which our results apply, are given.

Self-consistent perturbation approach to rough surface scattering in stratified elastic media

Abstract: A unified, self‐consistent perturbation approach to rough surface scattering in stratified media is presented By introducing a boundary‐condition operator formulation, the effect of the scattering on the mean field is accounted for by replacing the boundary conditions for the smooth interface with a set of effective boundary conditions involving relatively simple matrix operations The formulation is valid for any type of interface between fluid or elastic layers, with the only change involved being the actual boundary operators The use of boundary operator makes the formulation compatible with existing propagation models for stratified media, allowing simulation of scattering loss of the coherent component of the field due to the generation of a scattered field in the stratified fluid–solid media with an arbitrary number of rough interfaces The scattered field, a by‐product of the simulation is, in effect, the reverberation field in the stratified waveguide The approach is verified by agreement with

An Introduction to Operator Polynomials

Abstract: 1. Linearizations.- 1.1 Definitions and examples.- 1.2 Uniqueness of linearization.- 1.3 Existence of linearizations.- 1.4 Operator polynomials that are multiples of identity modulo compacts.- 1.5 Inverse linearization of operator polynomials..- 1.6 Exercises.- 1.7 Notes.- 2. Representations and Divisors of Monic Operator Polynomials.- 2.1 Spectral pairs.- 2.2 Representations in terms of spectral pairs.- 2.3 Linearizations.- 2.4 Generalizations of canonical forms.- 2.5 Spectral triples.- 2.6 Multiplication and division theorems.- 2.7 Characterization of divisors in terms of subspaces.- 2.8 Factorable indexless polynomials.- 2.9 Description of the left quotients.- 2.10 Spectral divisors.- 2.11 Differential and difference equations.- 2.12 Exercises.- 2.13 Notes.- 3. Vandermonde Operators and Common Multiples.- 3.1 Definition and basic properties of the Vandermonde operator.- 3.2 Existence of common multiples.- 3.3 Common multiples of minimal degree.- 3.4 Fredholm Vandermonde operators.- 3.5 Vandermonde operators of divisors.- 3.6 Divisors with disjoint spectra.- Appendix: Hulls of operators.- 3.7 Application to differential equations.- 3.8 Interpolation problem.- 3.9 Exercises.- 3.10 Notes.- 4. Stable Factorizations of Monic Operator Polynomials.- 4.1 The metric space of subspaces in a Banach space.- 4.2 Spherical gap and direct sums.- 4.3 Stable invariant subspaces.- 4.4 Proof of Theorems 4.3.3 and 4.3.4.- 4.5 Lipschitz stable invariant subspaces and one-sided resolvents.- 4.6 Lipschitz continuous dependence of supporting subspaces and factorizations.- 4.7 Stability of factorizations of monic operator polynomials.- 4.8 Stable sets of invariant subspaces.- 4.9 Exercises.- 4.10 Notes.- 5. Self-Adjoint Operator Polynomials.- 5.1 Indefinite scalar products and subspaces..- 5.2 J-self-adjoint and J-positizable operators.- 5.3 Factorizations and invariant semidefinite subspaces.- 5.4 Classes of polynomials with special factorizations.- 5.5 Positive semidefinite operator polynomials.- 5.6 Strongly hyperbolic operator polynomials.- 5.7 Proof of Theorem 5.6.4.- 5.8 Invariant subspaces for unitary and self-adjoint operators in indefinite scalar products.- 5.9 Self-adjoint operator polynomials of second degree.- 5.10 Exercises.- 5.11 Notes.- 6. Spectral Triples and Divisibility of Non-Monic Operator Polynomials.- 6.1 Spectral triples: definition and uniqueness.- 6.2 Calculus of spectral triples.- 6.3 Construction of spectral triples.- 6.4 Spectral triples and linearization.- 6.5 Spectral triples and divisibility.- 6.6 Characterization of spectral pairs.- 6.7 Reduction to monic polynomials.- 6.8 Exercises.- 6.9 Notes.- 7. Polynomials with Given Spectral Pairs and Exactly Controllable Systems.- 7.1 Exactly controllable systems.- 7.2 Spectrum assignment theorems.- 7.3 Analytic dependence of the feedback.- 7.4 Polynomials with given spectral pairs.- 7.5 Invariant subspaces and divisors.- 7.6 Exercises.- 7.7 Notes.- 8. Common Divisors and Common Multiples.- 8.1 Common divisors.- 8.2 Common multiples.- 8.3 Coprimeness and Bezout equation.- 8.4 Analytic behavior of common multiples.- 8.5 Notes.- 9. Resultant and Bezoutian Operators.- 9.1 Resultant operators and their kernel.- 9.2 Proof of Theorem 9.1.4.- 9.3 Bezoutian operator.- 9.4 The kernel of a Bezoutian operator.- 9.5 Inertia theorems.- 9.6 Spectrum separation.- 9.7 Spectrum separation problem: deductions and special cases.- 9.8 Applications to difference equations.- 9.9 Notes.- 10. Wiener-Hopf Factorization.- 10.1 Definition and the main result.- 10.2 Pairs of finite type and proof of Theorem 10.1.1.- 10.3 Finite-dimensional perturbations.- 10.4 Notes.- References.- Notation.

Automatic program generation from teaching data for the hybrid control of robots

Abstract: An efficient method is developed to generate programs for the hybrid position/force control of robots from teaching data. An operator's motion is measured in terms of the force exerted by the operator and the position of the end-effector. The acquired data are then analyzed in order to understand what the operator intended to do, and necessary information is obtained to generate the hybrid control program. Determinations are made of which control mode, position or force, should be taken in each direction, how much force should be exerted, and what trajectory the end-effector should follow. The interpreted motion is then translated into a robot language, which explicitly describes the motion strategy that the human operator conceived. The method was implemented on a direct-drive arm and a personal computer, and the efficiency of the method was demonstrated through experiments. >

Dirac equation in external vector fields: Separation of variables

Abstract: The method of separation of variables in the Dirac equation in the external vector fields is developed through the search for exact solutions. The essence of the method consists of the separation of the first‐order matricial differential operators that define the dependence of the Dirac bispinor on the related variables, but commutation of such operators with the operator of the equations or between them is not assumed. This approach, which is perfectly justified in the presence of gravitational fields, permits one to prove rigorous theorems about necessary and sufficient conditions on the field functions that allow one to separate variables in the Dirac equation. In analogous investigations by other authors [Bagrov et al., Exact solutions of Relativistic Wave Equations (Nauka, Novosibirst, 1982)] for electromagnetic fields an essential demand related to the operators that define the dependence of the bispinor on the separated variables is the demand for the commutation of a complete set of operators betw...

3D table-driven migration

Abstract: An efficient full 3D wavefield extrapolation technique is presented. The method can be used for any type of subsurface structure and the degree of accuracy and dip-angle performance are user-defined. The extrapolation is performed in the space-frequency domain as a space-dependent spatial convolution with recursive Kirchhoff extrapolation operators. To get a high level of efficiency the operators are optimized such that they have the smallest possible size for a specified accuracy and dip-angle performance. As both accuracy and maximum dip-angle are input parameters for the operator calculation, the method offers the possibility of a trade-off between these quantities and efficiency. The operators are calculated in advance and stored in a table for a range of wavenumbers. Once they have been calculated they can be used many times. At the basis of the operator design is the well-known phase-shift operator. Although this operator is exact for homogeneous media only, it is assumed that it may be applied locally in case of inhomogeneities. Lateral velocity variations can then be handled by choosing the extrapolation operator according to the local value of the velocity. Optionally the operators can be designed such that they act as spatially variant high-cut filters. This means that the evanescent field can be suppressed in one pass with the extrapolation. The extrapolation method can be used both in prestack and post-stack applications. In this paper we use it in zero-offset migration. Tests on 2D and 3D synthetic and 2D real data show the excellent quality of the method. The full 3D result is much better then the result of two-pass migration, which has been applied to the same data. The implementation yields a code that is fully vectorizable, which makes the method very suitable for vector computers.

Vehicle use-locking and unlocking system

Abstract: The specification describes a wireless, signal controlled locking and unlocking system for use in vehicles. A control unit mounted on the vehicle door is operated remotely by a control unit held by the operator.

Anderson localization for the 1-D discrete Schrödinger operator with two-frequency potential

Abstract: We prove the complete exponential localization of eigenfunctions for the 1-D discrete Schrodinger operators with quasi-periodic potentials having two basic frequencies. It is shown also that for such operators there is no forbidden zones in the spectrum, unlike the operators with one basic frequency.

On Asymptotic Toeplitz and Hankel Operators

Abstract: An operator T is asymptotic Toeplitz if for S the unilateral shift, the sequence {S ✹n n TS n } converges. It is asymptotic Hankel if for J n the permutation insometry on the subspace determined by the first n coordinate vectors, the sequence {J n TS n+1 } converges. The relationship between these notions is studied and operator analogues of the A.A.K. distance formulae in terms of the s numbers of a Hankel operator are obtained.

Regular solutions of transmission and interaction problems for wave equations

Abstract: Consider n bounded domains Ω ⊆ ℝ and elliptic formally symmetric differential operators A1 of second order on Ωi Choose any closed subspace V in , and extend (Ai)i=1,…,n by Friedrich's theorem to a self-adjoint operator A with D(A1/2) = V (interaction operator). We give asymptotic estimates for the eigenvalues of A and consider wave equations with interaction. With this concept, we solve a large class of problems including interface problems and transmission problems on ramified spaces.25,32 We also treat non-linear interaction, using a theorem of Minty29.