Three-component Vector Research Articles

Stokes-vector evolution in a weakly anisotropic inhomogeneous medium

The equation for evolution of the four-component Stokes vector in weakly anisotropic and smoothly inhomogeneous media is derived on the basis of a quasi-isotropic approximation of the geometrical optics method, which provides the consequent asymptotic solution of Maxwell's equations. Our equation generalizes previous results obtained for the normal propagation of electromagnetic waves in stratified media. It is valid for curvilinear rays with torsion and is capable of describing normal mode conversion in inhomogeneous media. Remarkably, evolution of the four-component Stokes vector is described by the Bargmann-Michel-Telegdi equation for relativistic spin precession, whereas the equation for the three-component Stokes vector resembles the Landau-Lifshitz equation describing spin precession in ferromagnetic systems. The general theory is applied for analysis of polarization evolution in a magnetized plasma. We also emphasize fundamental features of the non-Abelian polarization evolution in anisotropic inhomogeneous media and illustrate them by simple examples.

SO(3) symmetry between Néel and ferromagnetic order parameters for graphene in a magnetic field

I consider the Hubbard model of graphene in an external magnetic field and in the Hartree-Fock approximation. In the continuum limit, the ground state energy at half filling becomes nearly symmetric under rotations of the three-component vector (N1,N2,m), with the first two components representing the Neel order parameter orthogonal to and the third component the magnetization parallel with the external magnetic field. When the symmetry breaking effects arising from the lattice, Zeeman coupling, and higher Landau levels are included the system develops a quantum critical point at which the antiferromagnetic order disappears and the magnetization has a kink. The observed incompressible state at filling factor one is argued to arise due to a finite third component of the Neel order parameter at these electron densities. Recent experiments appear consistent with vanishing N1 and N2, and finite N3, at the filling factors zero and one, respectively.

Quaternions and the rotation of a rigid body

The orientation of an arbitrary rigid body is specified in terms of a quaternion based upon a set of four Euler parameters. A corresponding set of four generalized angular momentum variables is derived (another quaternion) and then used to replace the usual three-component angular velocity vector to specify the rate by which the orientation of the body with respect to an inertial frame changes. The use of these two quaternions, coordinates and conjugate moments, naturally leads to a formulation of rigid-body rotational dynamics in terms of a system of eight coupled first-order differential equations involving the four Euler parameters and the four conjugate momenta. The equations are formally simple, easy to handle and free of singularities. Furthermore, integration is fast, since only arithmetic operations are involved.

Tomographic particle image velocimetry

This paper describes the principles of a novel 3D PIV system based on the illumination, recording and reconstruction of tracer particles within a 3D measurement volume. The technique makes use of several simultaneous views of the illuminated particles and their 3D reconstruction as a light intensity distribution by means of optical tomography. The technique is therefore referred to as tomographic particle image velocimetry (tomographic-PIV). The reconstruction is performed with the MART algorithm, yielding a 3D array of light intensity discretized over voxels. The reconstructed tomogram pair is then analyzed by means of 3D cross-correlation with an iterative multigrid volume deformation technique, returning the three-component velocity vector distribution over the measurement volume. The principles and details of the tomographic algorithm are discussed and a parametric study is carried out by means of a computer-simulated tomographic-PIV procedure. The study focuses on the accuracy of the light intensity field reconstruction process. The simulation also identifies the most important parameters governing the experimental method and the tomographic algorithm parameters, showing their effect on the reconstruction accuracy. A computer simulated experiment of a 3D particle motion field describing a vortex ring demonstrates the capability and potential of the proposed system with four cameras. The capability of the technique in real experimental conditions is assessed with the measurement of the turbulent flow in the near wake of a circular cylinder at Reynolds 2,700.

Energy–second-moment map analysis as an approach to quantify the irregularity of Hamiltonian systems

A different approach will be presented that aims to scrutinize the phase-space trajectories of a general class of Hamiltonian systems with regard to their regular or irregular behavior. The approach is based on the "energy-second-moment map" that can be constructed for all Hamiltonian systems of the generic form H=p(2)/2+V(q,t) . With a three-component vector s consisting of the system's energy h and second moments qp, q(2), this map linearly relates the vector s(t) at time t with the vector's initial state s(0) at t=0 . It will turn out that this map is directly obtained from the solution of a linear third-order equation that establishes an extension of the set of canonical equations. The Lyapunov functions of the energy-second-moment map will be shown to have simple analytical representations in terms of the solutions of this linear third-order equation. Applying Lyapunov's regularity analysis for linear systems, we will show that the Lyapunov functions of the energy-second-moment map yields information on the irregularity of the particular phase-space trajectory. Our results will be illustrated by means of numerical examples.

The population RDH index: a novel vector index and graphical method for statistical assessment of antihypertensive treatment reduction, duration, and homogeneity

Current indices used in the evaluation of antihypertensive treatment duration and homogeneity such as the trough-peak, smoothness index, and normalized smoothness index were designed to be applied to ambulatory blood pressure monitoring recordings from individual participants. Evaluation of antihypertensive treatment in populations is often carried out by calculating these individual indices for each of the participants and providing summarizing statistics about the population, such as the mean and median. We describe a new population vector index and graphical method for the statistical assessment of antihypertensive treatment reduction, duration, and homogeneity (RDH) from ambulatory blood pressure monitoring. The population (RDH) was specifically designed as a tool to evaluate and compare blood pressure coverage offered by antihypertensive drugs over 24 h in populations. The population RDH is a three-component vector index that incorporates information about the reduction, duration, and homogeneity of antihypertensive treatment, as well as their statistical significance over the 24 h period. In addition to defining the RDH index, in this paper we also demonstrate its usefulness and advantages as an index and graphical method for antihypertensive treatment duration and homogeneity assessment by using it to analyze two data sets.

The individual RDH index: a novel vector index for statistical assessment of antihypertensive treatment reduction, duration, and homogeneity

We propose a new vector index for the statistical assessment of antihypertensive treatment duration and homogeneity from ambulatory blood pressure monitoring. We termed this approach for evaluating and comparing blood pressure coverage offered by antihypertensive drugs over 24 h as the reduction-duration-homogeneity index. The reduction-duration-homogeneity index is a three-component vector index that incorporates information about the reduction, duration, and homogeneity of antihypertensive treatment, as well as their statistical significance. The advantages of the reduction-duration-homogeneity index are demonstrated by several comparative examples.

Propagation of surface elastic waves in the Cosserat medium

The problem of surface elastic wave propagation in the Cosserat medium (half-space) is considered. The strained state is characterized by two independent vectors: displacement and rotation. Solutions to the equations of motion are sought in the form of wave packets specified by an arbitrary Fourier spectrum. It is shown that, if the solution is sought in the form of a three-component displacement vector and a three-component rotation vector dependent on time, depth, and longitudinal coordinate, the initial system splits into two systems, one of which describes the Rayleigh wave and the other corresponds to a transverse wave decaying with depth. For both waves, analytical solutions in terms of displacements are obtained. It should be particularly noted that, unlike the Rayleigh wave, the solution for the transverse surface wave has no analogues in the classical elasticity theory. The transverse wave solution is numerically compared with the Rayleigh wave solution.

Magnetic field modelling from scalar-only data: Resolving the Backus effect with the equatorial electrojet

Abstract It is well known that models of the global geomagnetic field constructed from only measurements of the field intensity suffer from large errors arising from the Backus or perpendicular error effect. Knowledge of the location of the magnetic dip equator is in principal sufficient to eliminate this error. We investigate constraining the location of the dip equator using observations of the equatorial electrojet in intensity measurements made from the CHAMP satellite. While the models generated are inferior compared with models obtained from oriented three-component vector data, they may be of sufficient quality to allow construction of future global geomagnetic reference models in the absence of vector data.

Representation of Misorientations in Rodrigues-Frank Space: Application to the Bain, Kurdjumov-Sachs, Nishiyama-Wassermann, Pitsch and Greninger-Troiano Orientation Relationships

Orientation relationships between individual crystals can be readily represented in Rodrigues-Frank space because of the one-to-one correspondence between each misorientation and a vector in the fundamental zone of this space. This is done by integrating the rotation angle and axis into a three-component vector. In this study, the three classical orientation relationships describing the γ-to-α transformation, namely the Bain, Kurdjumov-Sachs and Nishiyama- Wassermann, are represented in Rodrigues-Frank space. Also considered are the somewhat less common Pitsch and Greninger-Troiano relationships. The misorientations between these types of transformation variants are displayed in R-F space based on alternative reference systems to highlight the differences. Examples of the various crystallographic relationships between fcc and bcc crystals during the γ-to-α transformation are given to demonstrate the advantages of the use of this space.

Analysis of similarity/dissimilarity of DNA sequences based on a 3-D graphical representation

Based on a 3-D graphical representation of DNA primary sequences by Li and Wang [C. Li, J. Wang, Comb. Chem. High Throughput Screen. 7 (2004) 23], we outline one approach to arrive at quantification of similarities by constructing a three-component vector whose components are the geometrical center and a four-component vector consisting of the graph radius associated with DNA curves. The examination of similarities/dissimilarities among the coding sequences of the first exon of β-globin gene of 11 species illustrates the utility of the approach. Our method has the advantage of greatly reducing the computational complexity, and shows the multiformity of analysis of similarities.

Lattice kinetic simulations in three-dimensional magnetohydrodynamics.

A lattice kinetic algorithm to simulate three-dimensional (3D) incompressible magnetohydrodynamics is presented. The fluid is monitored by a distribution function, which obeys a scalar kinetic equation, subject to an external force due to the imposed magnetic field. Following the work of J. Comput. Phys. 179, 95 (2002)], the magnetic field is represented by a different three-component vector distribution function, which obeys a corresponding vector kinetic equation. Discretization of the 3D phase space is based on a 19-bit scheme for the hydrodynamic part and on a 7-bit scheme for the magnetic part. Numerical results for magnetohydrodynamic (MHD) flow in a rectangular duct with insulating and conducting walls provide excellent agreement with corresponding analytical solutions. The scheme maintains in all cases tested the MHD constraint inverted Delta.B=0 within machine round-off error.

New 2D graphical representation of DNA sequences.

We consider a 2D graphical representations of DNA sequences, which avoids loss of information associated with crossing and overlapping of the corresponding curve. We outline an approach, which is based on the construction of a three-component vector whose components are the normalized leading eigenvalues of the L/L matrices associated with DNA. The examination of similarities/dissimilarities among the coding sequences of the first exon of beta-globin gene of different species illustrates the utility of the approach.

Magnetometer calibration: a joint initiative of Defence and AGSO

The Australian Department of Defence and AGSO aimed to provide a calibration facility for three-component (vector) magnetometers and thereby a cost-effective service for defence and geoscientific purposes. Defence was the main provider of funding and would be spared the expensive and long-winded process of sending its instruments overseas for calibration. Australia would gain from having a world-class facility available to all magnetometer users. Three sets each of four coils are mounted orthogonally and connected to programmable current sources enabling the creation of magnetic fields between 0 mT and 100 mT in any direction. A sequence of field values is generated and compared with the corresponding outputs of the magnetometer under test. The system computer then automatically prints the main parameters describing the instrument?s sensor angles, sensitivities, linearities, and test conditions. Background field variations over the measurement period are subtracted. Our system is capable of generating fields to a magnitude accuracy of 20 ppm and a directional precision of about 10 seconds of arc assured by checking the generated fields with a standard scalar (proton) magnetometer. The Defence Department?s needs and its policy of encouraging general use have enabled AGSO, in close collaboration with its suppliers, to create a unique facility for terrestrial and space exploration, for general scientific research, and a foundation for future client needs. Already AGSO is developing, with funds from the CRC for Satellite Systems, the capacity to magnetically characterise FedSat which will carry a three-component fluxgate magnetometer into polar orbit early next year. FedSat is an Australian Government Centenary of Federation project.

One-dimensional turbulence: vector formulation and application to free shear flows

One-dimensional turbulence is a stochastic simulation method representing the time evolution of the velocity profile along a notional line of sight through a turbulent flow. In this paper, the velocity is treated as a three-component vector, in contrast to previous formulations involving a single velocity component. This generalization allows the incorporation of pressure-scrambling effects and provides a framework for further extensions of the model. Computed results based on two alternative physical pictures of pressure scrambling are compared to direct numerical simulations of two time-developing planar free shear flows: a mixing layer and a wake. Scrambling based on equipartition of turbulent kinetic energy on an eddy-by-eddy basis yields less accurate results than a scheme that maximizes the intercomponent energy transfer during each eddy, subject to invariance constraints. The latter formulation captures many features of free shear flow structure, energetics, and fluctuation properties, including the spatial variation of the probability density function of a passive advected scalar. These results demonstrate the efficacy of the proposed representation of vector velocity evolution on a one-dimensional domain.

Three component spinner magnetometer featuring rapid measurement times

We describe the fabrication and performance of a spinning rock magnetometer for measuring the intensity and direction of the remanent magnetisation of rock specimens collected in palaeomagnetic surveys. The strength of these fields is measured using a high temperature superconductor (HTS) rf SQUID. Samples are rotated around two orthogonal axes to facilitate the calculation of the three-component remanence vector with a minimum of operator intervention. Measurements of remanence values ranging from 10/sup 1/ to 10/sup -5/ A/m, with errors of 1/spl times/10/sup -5/ A/m, have been achieved in measurement times of 20 s, while operating the instrument in a typical geophysical laboratory environment. In this paper we discuss the design requirements and data processing necessary to achieve a user-friendly system.

Polarization state of a biphoton: Quantum ternary logic

Polarization state of biphoton light generated via collinear frequency-degenerate spontaneous parametric down-conversion is considered. A biphoton is described by a three-component polarization vector, its arbitrary transformations relating to the SU(3) group. A subset of such transformations, available with retardation plates, is realized experimentally. In particular, two independent orthogonally polarized beams of type-I biphotons are transformed into a beam of type-II biphotons. Polarized biphotons are suggested as ternary analogs of two-state quantum systems (qubits).

Gravitational and collective effects in an output coupler for a Bose-Einstein condensate in an atomic trap

A three-component vector quantum field theory is developed to describe the electronic-spin-resonance interaction of a magnetically trapped Bose-Einstein condensate with a radio frequency field. The theory is directly applied to simulate the recent experiment of Mewes et al. for an output coupler for a Bose-Einstein condensate in a magnetic trap [Phys. Rev. Lett. 78, 582 (1997)]. By employing the mean-field approach, we study the collective dynamics of the multicomponent matter field via the nonlinear Schr\odinger equation. Including the effect of gravity, we study the spin-resonance Rabi oscillations for the three-component Bose-Einstein condensate in three ground-state hyperfine levels and simulate the dynamic expansion of the condensate components coupled out of the magnetic trap.

Interferometric estimation of three-dimensional ice-flow using ascending and descending passes

Satellite radar interferometry (SRI) provides an important new tool for determining ice-flow velocity. Interferometric measurements made from a single-track direction are sensitive only to a single component of the three-component velocity vector. Observations from along three different track directions would allow the full velocity vector to be determined. A north/south-looking synthetic aperture radar (SAR) could provide these observations over large portions of the globe, but not over large areas of the polar ice sheets. The authors develop and demonstrate a technique that allows the three-component velocity vector to be estimated from data acquired along two track directions (ascending and descending) under a surface-parallel flow assumption. This technique requires that there are accurate estimates of the surface slope, which are also determined interferometrically. To demonstrate the technique, the authors estimate the three-component velocity field for the Ryder Glacier, Greenland. Their results are promising, although they do not have yet ground-truth data with which to determine the accuracy of their estimates.

A three-component force vector cell for in vitro quantification of the force exerted by the papillary muscle on the left ventricular wall

Recent clinical studies indicate that functional mitral regurgitation, which is a common complication in patients who suffer from ischemic heart disease, is related to an increase in the tethering forces acting on the mitral valve leaflets. Alterations in the valvular assembly, displacement of the papillary muscles or dilatation of the mitral valve annulus can disrupt the normal force balance on the mitral leaflets and result in an abnormal coaptation geometry with incomplete mitral leaflet closure. The force balance imposed on the mitral leaflets is created by the coapting forces generated by the transmitral pressure difference and the tethering forces at the leaflet attachments. A unique force vector cell capable of accurately measuring the three-component force vector applied by the papillary muscle on the left-ventricular wall was designed and manufactured to permit quantification of the alteration in the force balance acting on the mitral leaflets, and to allow for the study of the influence of papillary muscle displacement on mitral regurgitation.