Vector Description Research Articles

Representation of orientation and disorientation data for cubic, hexagonal, tetragonal and orthorhombic crystals

A convenient method for the description of orientation data for cubic, hexagonal, tetragonal and orthorhombic crystals is given. The method can also be used for the representation of disorientation data, where disorientations between any two crystals of the specified symmetry lattices are considered. It is based on the quaternion formalism introduced into the discussion of orientations and disorientations by Grimmer [Acta Cryst. (1974), A30, 685–688], Frank [(1987). Proc. Int. Conf. on Texture of Materials 8 (INCOTOM 8), Santa Fe, USA, pp. 3–13] and others. Since orientations and disorientations can be interpreted as rotations which in turn can be represented by only three parameters a vector description is used. These vectors span a rotation space corresponding to the usual space of Eulerian angles. It is called Rodrigues vector space [Rodrigues (1840). J. Math. Pure Appl. 5, 380–440; Becker & Panchanadeeswaran (1989). Text. Microstruct. 10, 167]. The direction of a Rodrigues vector is parallel to the rotation axis and its length is tan (θ/2), where θ describes the rotation angle. A method for selecting a unique representative out of the numerous symmetrically equivalent Rodrigues vectors is given. Since these selection rules depend on the symmetry of the crystal lattices considered they yield compact domains in the Rodrigues vector space which are typical for each type of lattice or lattice pair. These domains are always bounded by planes. Frank (1987) called them fundamental zones and described them for the orientations of cubic, hexagonal and orthorhombic crystals.

Quantum kinetic energy operator for a group of particles in terms of scalar basic rotational invariant coordinates derived from a generalized Jacobi vector (GJV) description. III. A metric tensors approach

A more compact and standard formulation for the quantum kinetic energy operator of a system of nuclei exercising arbitrary amplitude motion is presented, deriving from a direct transformation from Cartesian centre of mass source coordinates of a set of GJV describing the system to internal and rotational curvilinear coordinates.

The quantum kinetic energy operator for a group of particles in terms of scalar basic rotational invariant coordinates derived from a generalized Jacobi vector (GJV) description. I. Frames derived from two GJVs

Previous work defining the quantum kinetic energy operator for relative motion of a group of particles in terms of Cartesian components of generalized Jacobi vectors (GJVs) is logically extended. The first such extension examines alternate definitions of the noninertial reference frame which follow from the minimum specification by two GJVs, the second invokes the simplest scalar internal coordinates which are the basic rotational invariants of the GJV. KE operators are explicitly evaluated and compared to assess the relative value (in terms of separability) of alternate approaches.

The quantum kinetic energy operator for a group of particles in terms of scalar basic rotational invariant coordinates derived from a generalized Jacobi vector (GJV) description. II. Frames derived from three GJVs

Previous work defining the quantum kinetic energy operator for relative motion of a group of particles in terms of Cartesian components of generalized Jacobi vectors (GJVs) is logically extended. The first such extension examines definitions of the noninertial reference frame which follow from its definition by three GJVs, the second invokes the simplest scalar internal coordinates which are the basic rotational invariants of the GJVs. KE operators are explicitly evaluated and assessed in terms of separability.

Vector and tensor form of intrinsic shell-theory relations

Vector description of the local shell shape, strain and equilibrium is complemented by discussion of the interconnection of the vector and tensor formulations of the theory. The vector parameters are shown to constitute dyadic components of the second order tensors of the classical theory. These relations are reflected in those between the vector and the component equations of compatibility and equilibrium. Some new light shed on the physical meaning of the components of tensors and vectors involved in the theory should be helpful in dealing with specific problems, particularly, non-linear ones.

Landau-Lifshitz equation and a generalized two-level system

The connection between the vector description of a generalized two-level system and magnetic momentum motion equations in the form of Landau-Lifshitz and Bloch equations is analyzed. It is shown that the Schrödinger equation for a two-level system with an effective non-hermitian hamiltonian is equivalent to that of the Landau-Lifshitz type and analogous to the quadric Dirac equation.

Induced Weak Currents in Nuclei

The vector--axial vector description of weak charged currents in nuclear theory is reviewed. Tests of conserved vector current, PCAC, and second--class currents are discussed. (AIP)

Phase effects for a sine wave masked by reproducible noise

A tone‐in‐noise detection task was used to assess the filtering properties of the auditory system. In the first experiment, ten 150‐ms samples of reproducible noise were used as maskers. The signal was a 500‐Hz tone either 20 or 100 ms in duration. The 100‐ms signal was centered temporally in the noise; the 20‐ms signal occurred either at the beginning, center, or end of the noise. The starting phase of the signal was varied from 0°–315° in steps of 45°. Signal thresholds, collected by the method of adjustment, were a cyclical function of starting phase and could be described by an energy model [Green and Swets, Signal Detection Theory and Psychophysics (Krieger, Huntington, NY, 1966/1974)]. A vector description of these data revealed an invariant property of each sample of noise, which we call the ‘‘noise vector.’’ The relationship among the parameters of the noise vectors over the various signal conditions suggest the presence of temporal interactions due to narrow‐band filtering. These relationships are consistent with the bandwidth estimates from the energy model which were 36, 36, and 52 Hz for the three subjects. In the second experiment, a 20‐ms signal was used and 32‐ms maskers were derived from the original ten maskers by deleting portions of the noise that had not occurred simultaneously with the signal. The noise vectors measured for these stimuli differed from those obtained in the first experiment. Moreover, the vectors were better predicted by the energy model than by the vectors of the first experiment, supporting such a model and adding further evidence of temporal interactions due to narrow‐band filtering by the auditory system.

A vector description of multiple-quantum coherence in AX n spin systems

Many NMR pulse sequences include a time period during which multiple-quantum coherence evolves. This paper describes a vector description of the time evolution of multiple-quantum coherences. Each vector corresponds to a transition on the energy level diagram, and evolves in the appropriate two-dimensional space depending on whether it corresponds to a single-, double-, or other multiple-quantum transition. Pulses cause transitions between these different spaces in a way that can best be predicted using a description in terms of spin angular momentum coherences. The transformation between these descriptions is shown to be straightforward. These methods are illustrated by applying a pulse sequence with a variable multiple-quantum evolution time to AX, AX 2, and AX 3 spin systems.

Asymptotic Behavior of the Closed Loop Poles of Linear Optimal Multivariable Systems

A new theory to describe the asymptotic optimal closed-loop pole positions and their related eigenvectors is presented. This is accomplished by utilizing the optimal return difference relation in a matrix/input vector description of the asymptotic eigenstructure. The paper contains an analysis for the finite closed-loop positions showing that for nonsquare systems, there can be finite closed loop-poles not associated with the system invariant zeros. Also presented is an elementary analysis for the asymptotically infinite closed-loop pole positions. Introduced here are the concepts of Butterworth patterns and some simple subspace characterisations of the input vector directions.

First-gradient balancing of higher-order gradiometers

A formalism for the vector and tensor description of a balancing procedure is developed and a first-gradient and field-balancing method is outlined. The experimental apparatus and measurements of the first-gradient common-mode vector,C G 1, are discussed.

Zonal Security Technique for Use in Strategic and Tactical Response Planning

This work describes a statistical technique for the determination of the penetration probability of each physical point in a protected facility as a function of time. The technique is based on a state vector description of the facility in conjunction with a transition matrix containing the transition rates between the various states. The state vector denotes the probability density that the intruders are in state i at time t. The time evolution of this quantity is determined by utilization of a Markov assumption for the system. While the basic solution utilizes a time independent transition matrix, a means for extending the method to a time dependent transition matrix is described.

Polarization transfer pulse sequences for two-dimensional NMR by heisenberg vector analysis

There are 24 possible polarization transfer pulse sequences having two or three precessional periods in which the effects of heteronuclear coupling and chemical shift are separated by refocusing pulses. Six of the most useful of these pulse sequences are described in detail using the Heisenberg vector approach. The vector description enables a clear analysis of the outcome of each sequence and of the modifications which are necessary to optimize each sequence. Each sequence is analyzed for the production of artifacts and a simple general method for the phase correction of the two-dimensional spectra is described.

Polarized light scattering from biological systems: A technique for cell differentiation

Measurements of polarized light scattered from spores, bacterial cells, and red and white blood cells in various conditions yield optical data that can differentiate between species, strains, surface characteristics, killing methods, antibody binding, red blood cell types, and two types of human white blood cells. The light scattering instrument used, which operates in the context of the Mueller matrix/Stokes vector description of the scattering process, is described. One particular matrix element, S34, is especially sensitive to small changes and cellular differences.

On vector description of arbitrary deformation of shells

The purpose of the present note is to remove the restriction of linearity of deformation analysis from the vectorial kinetostatics relations of shell theory [1,2]. The resulting nonlinear vector compatibility equations have a prototype in the moment stress elasticity [3].

Crystal subgrain misorientations observed by X-ray topography in reflection

Based on the principles of conservation of momentum and energy for X-ray diffraction, a vector description is obtained for the displacement of adjacent subgrain images in reflection topographs. The analysis includes, in addition to those crystal parameters defining the misorientation at a subgrain boundary, the combined effects of (horizontal and vertical) divergence in the incident X-ray beam and of the position where the X-ray images are recorded. The vector description is matched with a stereographic projection method of analysis for describing the subgrain misorientations. These total considerations are applied to the characterization of subgrain boundaries grown into a nickel single-crystal solidified along [010], including specification of the dislocation structure within the boundaries.

Measurement of the combination tones f2-f1 and 2f1-f2.

A three‐alternative forced‐choice technique was employed to estimate the phase and amplitude of the combination tone f2−f1 and 2f1−f2. The measurement procedure is based on a simple vector description of combination tone interactions. The task of the observer was to discriminate a change in intensity of the combination tone produced by a change of either the phase or amplitude of the cancellation tone. Measurements of f2−f1 essentially constitute a replication of previously reported data obtained with adjustment techniques. Considerable difficulty was encountered, however, in the measurement of the phase and amplitude of the cubic difference tone. Masking of additional low‐frequency combination tones with low‐pass noise did not eliminate the problem. It is concluded that phase induced changes in the envelope of the acoustic waveform as well as the magntiudes of higher‐order combination tones contribute to a monaural phase effect and hamper the forced‐choice measurement of 2f1−f2.

Coding of information about rapid eye movements in the pontine reticular formation of alert monkeys

Neurons in the rostral paramedian zone of the pontine reticular formation (PRF) have distinct frequency changes prior to and during quick eye movements, but generally little or no tonic activity assocaited with eye position. Evidence indicates that horizontal saccades and quick phases of nystagmus are generated in this region. Firing of units activated with eye movements (burst units) and of units which are inhibited (pause units) was analyzed. Eye movements were described by a vector having an amplitude (A) and an angle (α). These parameters were related to position changes in certain planes by the equation Δ;pos=A·cosα. In each of 80 cells in the PRF which were encountered, the activity could be related to some parameter of the above equation: change of position (Δ pos), amplitude (A), or the cosine of the angle between the direction of movement and a reference direction (cos α). Units coding amplitude of eye movement or change in position in a particular plane conveyed the information by number of spikes. Units coding direction of movement did so by frequency. In many units, the information coding was precise so that the direction or amplitude of single eye movements could be predicted from the frequency changes of single units. In other units, this could be determined from averages of the frequency changes. Cells coding direction or change in position had frequency maxima only in planes corresponding to the pulling direction of the eye muscles. The results suggest that a vector description is not only a convenient mathematical tool, but is the way eye movements are coded in the PRF and possibly elsewhere in the central nervous system.

Non-diagonal neutral currents and the scattering of neutrinos and antineutrinos by electrons

Non-diagonal currents arise if the initial and final neutrinos in neutral current interactions are not identical. Implications of non-diagonal neutral currents for the scattering of neutrinos and antineutrinos by electrons are considered within a general vector and axial vector description. Several observable relations can test for the presence of the non-diagonal pieces.

A vector description of parallel search in an associative memory

A vector description of parallel search in an associative memory