Operator Notation Research Articles

Tables of crystallographic properties of magnetic space groups

Tables of crystallographic properties of the reduced magnetic superfamilies of space groups, i.e. the 7 one-dimensional, 80 two-dimensional and 1651 three-dimensional group types, commonly referred to as magnetic space groups, are presented. The content and format are similar to that of non-magnetic space groups and subperiodic groups given in International Tables for Crystallography. Additional content for each representative group of each magnetic space-group type includes a diagram of general positions with corresponding general magnetic moments, Seitz notation used as a second notation for symmetry operations, and general and special positions listed with the components of the corresponding magnetic moments allowed by symmetry.

The power and the limits of Wittgenstein's N operator

The power of Wittgenstein's N operator described in the Tractatus is that every proposition which can be expressed in the Russellian variant of the predicate calculus familiar to him has an equivalent proposition in an extended variant of his N operator notation. This remains true if the bound variables are understood in the usual inclusive sense or in Wittgenstein's restrictive exclusive sense. The problematic limit of Wittgenstein's N operator comes from his claim that symbols alone reveal the logical status of propositions. This would require knowledge of the size of the Tractarian domain of unchangeable, simple objects, and this, Wittgenstein maintained, cannot be known.

Operator representation as a new differential optical absorption spectroscopy formalism

UV-visible absorption spectroscopy with extraterrestrial light sources is a widely used technique for the measurement of stratospheric and tropospheric trace gases. We focus on differential optical absorption spectroscopy (DOAS) and present an operator notation as a new formalism to describe the different processes in the atmosphere and the simplifying assumptions that compose the advantage of DOAS. This formalism provides tools to classify and reduce possible error sources of DOAS applications.

Tables of properties of magnetic subperiodic groups

Tables of crystallographic properties of the magnetic subperiodic groups, i.e. the 31 magnetic frieze-group types, the 394 magnetic rod-group types and the 528 magnetic layer-group types are presented. The content and format is similar to that of non-magnetic subperiodic groups and space groups given in International Tables for Crystallography. Additional content for each group type includes a diagram of general positions with corresponding general magnetic moments, Seitz notation used as a second notation for symmetry operations, and general and special positions listed with the components of the corresponding magnetic moments allowed by symmetry.

A generalized integral equation formulation for mixed dielectric‐PEC scatterers

The known integral representation of electromagnetic fields is reviewed. Based on the review, a new formulation allows us to model any general case. The formulation of PEC, dielectric, dielectric with metallic surfaces, and dielectric scatterers with dielectric or PEC inclusions is derived. Finally, a new operator notation allows a concise and accurate representation of the integral equation formulations. The electric field integral equation and the magnetic field integral equation are contained in this formulation.

VIBRATIONS OF CONTINUOUS SYSTEMS WITH A GENERAL OPERATOR NOTATION SUITABLE FOR PERTURBATIVE CALCULATIONS

The operator notation previously developed to analyze vibrations of continuous systems has been further generalized to model a system with an arbitrary number of coupled differential equations. Linear parts of the equations are expressed with an arbitrary linear differential and/or integral operators, and non-linear parts are expressed with arbitrary quadratic and cubic operators. Equations of motion are solved in their general form using the method of multiple scales, a perturbation technique. The case of primary resonances of the external excitation and one-to-one internal resonances between the natural frequencies of the equations is considered. The algorithm developed is applied to a non-linear cable vibration problem having small sag-to-span ratios.

An overview and generalization of implicit Navier–Stokes algorithms and approximate factorization

A theme of linearization and approximate factorization provides the context for a retrospective overview of the development and evolution of implicit numerical methods for the compressible and incompressible Euler and Navier–Stokes algorithms. This topic was chosen for this special volume commemorating the recent retirements of R.M. Beam and R.F. Warming. A generalized treatment of approximate factorization schemes is given, based on an operator notation for the spatial approximation. The generalization focuses on the implicit structure of Euler and Navier–Stokes algorithms as nonlinear systems of partial differential equations, with details of the spatial approximation left to operator definitions. This provides a unified context for discussing noniterative and iterative time-linearized schemes, and Newton iteration for unsteady nonlinear schemes. The factorizations include alternating direction implicit, LU and line relaxation schemes with either upwind or centered spatial approximations for both compressible and incompressible flows. The noniterative schemes are best suited for steady flows, while the iterative schemes are well suited for either steady or unsteady flows. This generalization serves to unify a large number of schemes developed over the past 30 years.

Quantum noise of a mode-locked laser

The master equation of mode locking written in operator notation is supplemented with noise-source terms that conserve commutator brackets. The noise sources are associated with the reservoirs responsible for loss and gain. The output of a mode-locked laser with the least possible quantum noise is determined.

On the operator representation of the Green's functions of the dynamic theory of elasticity

Starting from the operator notation for the dyadic Green's function for elastic displacement of the nonstationary theory of elasticity, we propose a method of factoring the mutually orthogonal components of the Green's function in the form of tensor products of operators acting respectively on the coordinates of the points of observation and the source.

Multipulse NMR. Part I. The simplest case: Quantum mechanical treatment of a single-spin system

This article is addressed mostly to those students who have just begun to study modern NMR. Starting with the fundamental physical principles, the quantum-mechanical formalism is introduced to describe a single-spin system. Short operator and superoperator notation is used for the derivation of formulas and rules. Equilibrium state, pulse, and chemical shift effects are treated. The close relationship between the quantum-mechanical approach and the classical vector model is discussed. © 1996 John Wiley & Sons, Inc.

Background subtraction of multiple-fold gamma-ray coincidence data: an operator approach

An expression for the background subtraction of multiple gamma-ray coincidence data sets may be generated in a simple operator notation. The operator and its physical justification are discussed. Approximations which simplify the expressions for common reduced data sets are evaluated and demonstrated.

Zeeman and zero-field splitting of 3d 4 and 3d 6 ions with orbital singlet ground state at orthorhombic and tetragonal symmetry sites

The spin Hamiltonian (SH) parameters for 3 d 4 and 3 d 6 ions with the spin S = 2 in the first- and second-kind orthorhombic crystal field (CF) are studied within the 5 D approximation. Four possible energy level schemes with a distinct orbital singlet ground state with S = 2 have been established due to our previous CF studies. For each scheme the microscopic spin Hamiltonian (MSH) formulas for the zero-field splitting (ZFS) parameters in the extended Stevens operator notation B q k as well as the electronic Zeeman parameters g i are derived by computer using the ALTRAN algebraic program. The microscopic approach enables one to express the parameters B q k and g i in terms of the spin-orbit (λ) and spin-spin (ρ) coupling parameters, the orthorhombic energy level splittings (Δ i) and mixing coefficients (q,s). The present perturbation calculations extend to higher orders of perturbation theory as well as being more comprehensive than the earlier ones, since two new energy level schemes not considered before are now dealt with. Due to the ascent in symmetry method, the MSH results are also applicable to tetragonal symmetry cases involving the real ZFS terms. Some inconsistencies in the previous MSH derivations carried out by hand are clarified. Conversion relations between the ZFS parameters B 0 2, B 2 2, B 0 4, B 2 4, B 4 4 and those expressed in several other notations used in the EPR literature are provided. Specific applications of the MSH formulas derived here provide guidelines for the millimeter/submillimeter and high magnetic field EPR studies of 3 d 4 and 3 d 6 ions at orthorhombic and tetragonal symmetry sites.

Normal-Ordering Renormalization and Gaussian Wave-Functional Method

Through the explicit calculation to the Gaussian effective potential in the sine-Gordon model in (1+1) dimensions with the normal-ordering prescription of renormalization, we find that the "m" in Coleman's formalism in operator notation can be regarded as the renormalized mass.

Linear algebraic transformations of the bidomain equations: Implications for numerical methods

A mathematical framework is presented for the treatment of the bidomain equations used to model propagation in cardiac tissue. This framework is independent of the model used to represent membrane ionic currents and incorporates boundary conditions and other constraints. By representing the bidomain equations in the operator notation Lφ = F ̃ , various algebraic transformations can be expressed as PLQ -1ψ = P F ̃ , where P and Q are linear operators. The authors show how previous work fits into this framework and discuss the implications of various transformation for numerical methods of solution. Although such transformations allow many choices of independent variable, these results emphasize the fundamental importance of the transmembrane potential.

Imaging transforms for visualizing surfaces and volumes.

Three-dimensional (3D) visualization in biomedical and other imaging areas is a rapidly emerging discipline. The major developments in this field are described in a unified and concise way. To this end, we introduce an operator notation to describe the basic imaging transforms commonly used in 3D visualization and to identify a comprehensive set of basic transforms. We also introduce several new basic transforms for filtering and interpolating scenes and structures and for rendering surfaces and volumes. We demonstrate not only how the existing visualization methodologies can be described concisely, but we also show how a great variety of new methodologies can be generated using both the existing imaging transforms and the new transforms introduced in this paper. A comprehensive evaluation method to compare objectively rendering methods used in visualization based on task-specific mathematical phantoms is described. We examine in detail separate transform sequences that are best suited for rendering robust and frail structures (ie, structures with well- and poorly defined boundaries).

Superposition model and crystal-field analysis of the 4A2 and a2E states of Cr3+ ions at C3 sites in LiNbO3

The energy levels and wavefunctions including the two lowest-lying levels, namely 4A2 and a2E, for which reliable experimental data exist for Cr3+ ions at C3 symmetry sites in LiNbO3, are calculated using the complete matrix diagonalization method within the 3d3 configuration. The Hamiltonian considered includes the electrostatic term, the Trees correction, the spin-orbit interaction and the crystal-field interaction. The role of the additional low-symmetry crystal-field term B4-3O4-3 (in the Stevens operator notation), neglected in the C3v approximation used so far in the literature, is studied. The superposition model is developed for 3d3 ions at C3 symmetry sites and applied to study the site occupancy of Cr3+ in LiNbO3. Analysis of the optical data indicates that Cr3+ ions substitute at Nb sites and Li sites simultaneously. The present considerations offer an improvement over the earlier approximations using C3v symmetry only. The zero-field splitting predicted by the crystal-field calculations for Cr3+ at the Nb site matches the experimental value from EPR studies very well. This is contrary to the earlier prediction by the superposition model analysis of the spin-Hamiltonian parameters indicating that the zero-field splitting for Cr3+ ions at Li sites matches the experimental zero-field splitting better than that for Cr3+ at Nb sites. Since the present calculations involve fitting not only the zero-field splitting but also the energies of the a2E state, the present predictions may be more reliable than the previous predictions.

Modern equations of diffractometry. Diffraction geometry

The various geometries of area-detector diffractometers and cameras are best described using a coordinate-free abstract operator notation. Modem methods of geometry, including especially the combined application of vectors and covectors, are used; they confer the simultaneous advantages of simplifying, virtualizing and unifying the analysis, which becomes applicable to all methods and machines. A second, and most valuable, prize arising from this approach, itself a major theme of this paper, is the complete avoidance of computationally expensive and analytically inconvenient trigonometric functions in area diffractometry. The very few occasions when they are unavoidable have already been discussed fully in a previous paper on goniometry. Basic diffraction geometry is presented first, giving all the equations necessary to identify diffraction spots and to calculate a useful generalization of the Lorentz factor. These are a formalized and extended version of those presented to the EEC Cooperative Workshop on Position-Sensitive Detector Software held at LURE in Paris in 1986.Then, various previously unpublished formulae describing beam divergence, dispersion and polarization, crystal mosaicity and angular widths of diffraction spots are presented. Finally, three specific calculations appropriate to the use of an area diffractometer are given, including a calculation of window sizes, a model of the backstop shadow and a method of surveying a diffraction pattern for assessment and prealignment.

Elastic wave propagation using fully vectorized high order finite‐difference algorithms

Two finite‐difference schemes for solving the elastic wave equation in heterogeneous two‐dimensional media are implemented on a vector computer. A modified Lax‐Wendroff scheme that is second‐order accurate both in time and space and is a version of the MacCormack scheme that is second‐order accurate in time and fourth‐order in space. The algorithms are based on the matrix times vector by diagonals technique that is fully vectorized and is described using a novel notation for vector supercomputer operations. The technique described can be implemented on a vector processor of modest dimensions and increase the applicability of finite differences. The two difference operators are compared and the programs are tested for a simple case of standing sinusoidal waves for which the exact solution is known and also for a two‐layer model with a line source. A comparison of the results for an actual well‐to‐well experiment verifies the usefulness of the two‐dimensional approach in modeling the results.

A theoretical solution for the buckling of sandwich panels with laminated face plates using a computer algebra system

Abstract The governing equations for both symmetric and asymmetric wrinkling of sandwich panels are derived using operator notation. In each case differential systems of three equations in three unknowns are obtained and two of the three variables are eliminated by taking combinations of the equations. Whilst the mathematical techniques employed are straightforward, the expressions involved are lengthy and recourse is made to the computer algebra system MACSYMA, without which a solution for the asymmetric case would have been difficult to achieve. Explicit algebraic expressions are given for buckling loads which can be used by structural engineers to calculate allowable design loads for sandwich panels.

R-parity violation at hadron colliders

Using a new operator notation we present a systematic study of all possible R-parity violating signals at a hadron collider.