Vacuum spacetimes with constant Weyl eigenvalues

Einstein spacetimes (that is vacuum spacetimes possibly with a non-zero cosmological constant A) with constant non-zero Weyl eigenvalues are considered. For type Petrov II & D this assumption allows one to prove that the non-repeated eigenvalue necessarily has the value 2A/3 and it turns out that the only possible spacetimes are some Kundt-waves considered by Lewandowski which are type II and a Robinson-Bertotti solution of type D. For Petrov type I the only solution turns out to be a homogeneous pure vacuum solution found long ago by Petrov using group theoretic methods. These results can be summarised by the statement that the only vacuum spacetimes with constant Weyl eigenvalues are either homogeneous or are Kundt spacetimes. This result is similar to that of Coley et al. who proved their result for general spacetimes under the assumption that all scalar invariants constructed from the curvature tensor and all its derivatives were constant.

Curvature-driven smoothing: a learning algorithm for feed-forward networks

The performance of feed-forward neural networks in real applications can be often be improved significantly if use is made of a-priori information. For interpolation problems this prior knowledge frequently includes smoothness requirements on the network mapping, and can be imposed by the addition to the error function of suitable regularization terms. The new error function, however, now depends on the derivatives of the network mapping, and so the standard back-propagation algorithm cannot be applied. In this paper, we derive a computationally efficient learning algorithm, for a feed-forward network of arbitrary topology, which can be used to minimize the new error function. Networks having a single hidden layer, for which the learning algorithm simplifies, are treated as a special case.

Incorporating curvature information into on-line learning

We analyse the dynamics of a number of second order on-line learning algorithms training multi-layer neural networks, using the methods of statistical mechanics. We first consider on-line Newton's method, which is known to provide optimal asymptotic performance. We determine the asymptotic generalization error decay for a soft committee machine, which is shown to compare favourably with the result for standard gradient descent. Matrix momentum provides a practical approximation to this method by allowing an efficient inversion of the Hessian. We consider an idealized matrix momentum algorithm which requires access to the Hessian and find close correspondence with the dynamics of on-line Newton's method. In practice, the Hessian will not be known on-line and we therefore consider matrix momentum using a single example approximation to the Hessian. In this case good asymptotic performance may still be achieved, but the algorithm is now sensitive to parameter choice because of noise in the Hessian estimate. On-line Newton's method is not appropriate during the transient learning phase, since a suboptimal unstable fixed point of the gradient descent dynamics becomes stable for this algorithm. A principled alternative is to use Amari's natural gradient learning algorithm and we show how this method provides a significant reduction in learning time when compared to gradient descent, while retaining the asymptotic performance of on-line Newton's method.

Phakometric measurement of ocular surface radii of curvature, axial separations and alignment in relaxed and accommodated human eyes

Measurements (autokeratometry, A-scan ultrasonography and video ophthalmophakometry) of ocular surface radii, axial separations and alignment were made in the horizontal meridian of nine emmetropes (aged 20-38 years) with relaxed (cycloplegia) and active accommodation (mean ± 95% confidence interval: 3.7 ± 1.1 D). The anterior chamber depth (-1.5 ± 0.3 D) and both crystalline lens surfaces (front 3.1 ± 0.8 D; rear 2.1 ± 0.6 D) contributed to dioptric vergence changes that accompany accommodation. Accommodation did not alter ocular surface alignment. Ocular misalignment in relaxed eyes is mainly because of eye rotation (5.7 ± 1.6° temporally) with small amounts of lens tilt (0.2 ± 0.8° temporally) and decentration (0.1 ± 0.1 mm nasally) but these results must be viewed with caution as we did not account for corneal asymmetry. Comparison of calculated and empirically derived coefficients (upon which ocular surface alignment calculations depend) revealed that negligible inherent errors arose from neglect of ocular surface asphericity, lens gradient refractive index properties, surface astigmatism, effects of pupil size and centration, assumed eye rotation axis position and use of linear equations for analysing Purkinje image shifts. © 2004 The College of Optometrists.

Phakometric measurement of ocular surface radius of curvature and alignment: evaluation of method with physical model eyes

Ophthalmophakometric measurements of ocular surface radius of curvature and alignment were evaluated on physical model eyes encompassing a wide range of human ocular dimensions. The results indicated that defocus errors arising from imperfections in the ophthalmophakometer camera telecentricity and light source collimation were smaller than experimental errors. Reasonable estimates emerged for anterior lens surface radius of curvature (accuracy: 0.02–0.10 mm; precision 0.05–0.09 mm), posterior lens surface radius of curvature (accuracy: 0.10–0.55 mm; precision 0.06–0.20 mm), eye rotation (accuracy: 0.00–0.32°; precision 0.06–0.25°), lens tilt (accuracy: 0.00–0.33°; precision 0.05–0.98°) and lens decentration (accuracy: 0.00–0.07 mm; precision 0.00–0.07 mm).

The effect of corneal thickness and corneal curvature on pneumatonometer measurements

Purpose. The purpose of this study was to investigate the influence of corneal topography and thickness on intraocular pressure (IOP) and pulse amplitude (PA) as measured using the Ocular Blood Flow Analyzer (OBFA) pneumatonometer (Paradigm Medical Industries, Utah, USA). Methods. 47 university students volunteered for this cross-sectional study: mean age 20.4 yrs, range 18 to 28 yrs; 23 male, 24 female. Only the measurements from the right eye of each participant were used. Central corneal thickness and mean corneal radius were measured using Scheimpflug biometry and corneal topographic imaging respectively. IOP and PA measurements were made with the OBFA pneumatonometer. Axial length was measured using A-scan ultrasound, due to its known correlation with these corneal parameters. Stepwise multiple regression analysis was used to identify those components that contributed significant variance to the independent variables of IOP and PA. Results. The mean IOP and PA measurements were 13.1 (SD 3.3) mmHg and 3.0 (SD 1.2) mmHg respectively. IOP measurements made with the OBFA pneumatonometer correlated significantly with central corneal thickness (r = +0.374, p = 0.010), such that a 10 mm change in CCT was equivalent to a 0.30 mmHg change in measured IOP. PA measurements correlated significantly with axial length (part correlate = -0.651, p < 0.001) and mean corneal radius (part correlate = +0.459, p < 0.001) but not corneal thickness. Conclusions. IOP measurements taken with the OBFA pneumatonometer are correlated with corneal thickness, but not axial length or corneal curvature. Conversely, PA measurements are unaffected by corneal thickness, but correlated with axial length and corneal radius. These parameters should be taken into consideration when interpreting IOP and PA measurements made with the OBFA pneumatonometer.

The interrogation and multiplexing of long period grating curvature sensors using a Bragg grating based, derivative spectroscopy technique

A long period grating is interrogated with a fibre Bragg grating using a derivative spectroscopy technique. A quasi-linear relationship between the output of the sensing scheme and the curvature experienced by the long period grating is demonstrated, with a sensitivity of 5.05 m and with an average curvature resolution of 2.9 × 10-2 m-1. In addition, the feasibility of multiplexing an in-line series of long period gratings with this interrogation scheme is demonstrated with two pairs of fibre Bragg gratings and long period gratings. With this arrangement the cross-talk error between channels was less than ± 2.4 × 10-3 m-1.

Constant shear along a boundary of a flume

'I'he accurate rreasurement of bed shear stress has been extremely difficult due to its changing values until white propunded a theory which would give constant shear along the bed of a flume. In this investigation a flume has been designed according to White's theory and by two separate methods proven to give constant shearing force along the bed. The first method applied the Hydrogen Bubble Technique to obtain accurate values of velocity thus allowing the velocity profile to be plotted and the momentum at the various test sections to be calculated. The use of a 16 mm Beaulieu movie camera allowed the exact velocity profiles created by the hydrogen bubbles to be recorded whilst an analysing projector gave the means of calculating the exact velocities at the various test sections. Simultaneously Preston's technique of measuring skin friction using Pitot tubes was applied. Twc banks of open ended water manometer were used for recording the static and velocity head pressure drop along the flume. This tvpe of manometer eliminated air locks in the tubes and was found to be sufficiently accurate. Readings of pressure and velocity were taken for various types and diameters of bed material both natural sands and glass spheres and the results tabulated. Graphs of particle Reynolds Number against bed shear stress were plotted and gave a linear relationship which dropped off at high values of Reynolds number. It was found that bed movement occurred instantaneously along the bed of the flume once critical velocity had been reached. On completion of this test a roof curve inappropriate to the bed material was used and then the test repeated. The bed shearing stress was now no longer constant and yet bed movement started instantaneously along the bed of the flume, showing that there are more parameters than critical shear stress to bed movement. It is concluded from the two separate methods applied that the bed shear stress is constant along the bed of the flume.

Generalized methods and solvers for noise removal from piecewise constant signals. I. Background theory

Removing noise from piecewise constant (PWC) signals is a challenging signal processing problem arising in many practical contexts. For example, in exploration geosciences, noisy drill hole records need to be separated into stratigraphic zones, and in biophysics, jumps between molecular dwell states have to be extracted from noisy fluorescence microscopy signals. Many PWC denoising methods exist, including total variation regularization, mean shift clustering, stepwise jump placement, running medians, convex clustering shrinkage and bilateral filtering; conventional linear signal processing methods are fundamentally unsuited. This paper (part I, the first of two) shows that most of these methods are associated with a special case of a generalized functional, minimized to achieve PWC denoising. The minimizer can be obtained by diverse solver algorithms, including stepwise jump placement, convex programming, finite differences, iterated running medians, least angle regression, regularization path following and coordinate descent. In the second paper, part II, we introduce novel PWC denoising methods, and comparisons between these methods performed on synthetic and real signals, showing that the new understanding of the problem gained in part I leads to new methods that have a useful role to play.

Generalized methods and solvers for noise removal from piecewise constant signals. II. New methods

Removing noise from signals which are piecewise constant (PWC) is a challenging signal processing problem that arises in many practical scientific and engineering contexts. In the first paper (part I) of this series of two, we presented background theory building on results from the image processing community to show that the majority of these algorithms, and more proposed in the wider literature, are each associated with a special case of a generalized functional, that, when minimized, solves the PWC denoising problem. It shows how the minimizer can be obtained by a range of computational solver algorithms. In this second paper (part II), using this understanding developed in part I, we introduce several novel PWC denoising methods, which, for example, combine the global behaviour of mean shift clustering with the local smoothing of total variation diffusion, and show example solver algorithms for these new methods. Comparisons between these methods are performed on synthetic and real signals, revealing that our new methods have a useful role to play. Finally, overlaps between the generalized methods of these two papers and others such as wavelet shrinkage, hidden Markov models, and piecewise smooth filtering are touched on.

A priori prediction of aggregation efficiency and rate constant for fluidized bed melt granulation

This paper presents a predictive aggregation rate model for spray fluidized bed melt granulation. The aggregation rate constant was derived from probability analysis of particle–droplet contact combined with time scale analysis of droplet solidification and granule–granule collision rates. The latter was obtained using the principles of kinetic theory of granular flow (KTGF). The predicted aggregation rate constants were validated by comparison with reported experimental data for a range of binder spray rate, binder droplet size and operating granulator temperature. The developed model is particularly useful for predicting particle size distributions and growth using population balance equations (PBEs).

Build-up of auditory stream segregation induced by tone sequences of constant or alternating frequency and the resetting effects of single deviants

A sequence of constant-frequency tones can promote streaming in a subsequent sequence of alternating-frequency tones, but why this effect occurs is not fully understood and its time course has not been investigated. Experiment 1 used a 2.0-s-long constant-frequency inducer (10 repetitions of a low-frequency pure tone) to promote segregation in a subsequent, 1.2-s test sequence of alternating low- and high-frequency tones. Replacing the final inducer tone with silence substantially reduced reported test-sequence segregation. This reduction did not occur when either the 4th or 7th inducer was replaced with silence. This suggests that a change at the induction/test-sequence boundary actively resets build-up, rather than less segregation occurring simply because fewer inducer tones were presented. Furthermore, Experiment 2 found that a constant-frequency inducer produced its maximum segregation-promoting effect after only three tones—this contrasts with the more gradual build-up typically observed for alternating-frequency sequences. Experiment 3 required listeners to judge continuously the grouping of 20-s test sequences. Constant-frequency inducers were considerably more effective at promoting segregation than alternating ones; this difference persisted for ~10 s. In addition, resetting arising from a single deviant (longer tone) was associated only with constant-frequency inducers. Overall, the results suggest that constant-frequency inducers promote segregation by capturing one subset of test-sequence tones into an ongoing, preestablished stream, and that a deviant tone may reduce segregation by disrupting this capture. These findings offer new insight into the dynamics of stream segregation, and have implications for the neural basis of streaming and the role of attention in stream formation. (PsycINFO Database Record (c) 2013 APA, all rights reserved)

Optical bend sensor for vector curvature measurement based on Bragg grating in eccentric core polymer optical fibre

We present an optical bend sensor based on a Bragg grating written in an eccentric core polymer optical fibre. The grating wavelength shifts are studied as a function of bend curvature and fibre orientation and the device exhibits strong fibre orientation dependence, wide bend curvature range of ± 22.7 m-1 and high bend sensitivity of 63 pm/m-1, which is 80 times higher than the reported sensor based on an offset-FBG in standard single mode silica fibre.

The interrogation and multiplexing of long period grating curvature sensors using a Bragg grating based, derivative spectroscopy technique

A long period grating is interrogated with a fibre Bragg grating using a derivative spectroscopy technique. A quasi-linear relationship between the output of the sensing scheme and the curvature experienced by the long period grating is demonstrated, with a sensitivity of 5.05 m and with an average curvature resolution of 2.9 × 10-2 m-1. In addition, the feasibility of multiplexing an in-line series of long period gratings with this interrogation scheme is demonstrated with two pairs of fibre Bragg gratings and long period gratings. With this arrangement the cross-talk error between channels was less than ± 2.4 × 10-3 m-1.

Bifurcation study for a vertical channel with constant flux and large aspect ratio

Using suitable coupled Navier-Stokes Equations for an incompressible Newtonian fluid we investigate the linear and non-linear steady state solutions for both a homogeneously and a laterally heated fluid with finite Prandtl Number (Pr=7) in the vertical orientation of the channel. Both models are studied within the Large Aspect Ratio narrow-gap and under constant flux conditions with the channel closed. We use direct numerics to identify the linear stability criterion in parametric terms as a function of Grashof Number (Gr) and streamwise infinitesimal perturbation wavenumber (making use of the generalised Squire’s Theorem). We find higher harmonic solutions at lower wavenumbers with a resonance of 1:3exist, for both of the heating models considered. We proceed to identify 2D secondary steady state solutions, which bifurcate from the laminar state. Our studies show that 2D solutions are found not to exist in certain regions of the pure manifold, where we find that 1:3 resonant mode 2D solutions exist, for low wavenumber perturbations. For the homogeneously heated fluid, we notice a jump phenomenon existing between the pure and resonant mode secondary solutions for very specific wavenumbers .We attempt to verify whether mixed mode solutions are present for this model by considering the laterally heated model with the same geometry. We find mixed mode solutions for the laterally heated model showing that a bridge exists between the pure and 1:3 resonant mode 2D solutions, of which some are stationary and some travelling. Further, we show that for the homogeneously heated fluid that the 2D solutions bifurcate in hopf bifurcations and there exists a manifold where the 2D solutions are stable to Eckhaus criterion, within this manifold we proceed to identify 3D tertiary solutions and find that the stability for said 3D bifurcations is not phase locked to the 2D state. For the homogeneously heated model we identify a closed loop within the neutral stability curve for higher perturbation wavenumubers and analyse the nature of the multiple 2D bifurcations around this loop for identical wavenumber and find that a temperature inversion occurs within this loop. We conclude that for a homogeneously heated fluid it is possible to have abrup ttransitions between the pure and resonant 2D solutions, and that for the laterally heated model there exist a transient bifurcation via mixed mode solutions.