Miscellaneum hyperbolicum, et parabolicum : In quo praecipue agitur de centris grauitatis hyperbolae, partium eiusdem, atque nonnullorum solidorum, de quibus nunquam geometria locuta est. Parabola nouiter quadratur dupliciter. Ducuntur infinitarum parabolarum tangentes. Assignantur maxima inscriptibilia, minimaque circumscriptibilia infinitis parabolis, conoidibus, ac semifusis parabolicis. Aliaque geometrica noua exponuntur scitu digna /

Mode of access: Internet.

The elements of the differential calculus, comprehending the general theory of curve surfaces, and of curves of double curvature. Intended for the use of mathematical students in schools and universities.

Available on demand as hard copy or computer file from Cornell University Library.

D. Ioannis Feri, summi templi Moguntinensis concionatoris, de filij prodigi parabola, conciones eruditisimae : tempore Quadragesimali atque Pascha ab eodem recitatae ...

Mode of access: Internet.

A treatise on analytical geometry of three dimensions : containing the theory of curve surfaces and of curves of double curvature /

Mode of access: Internet.

Pore size characterization of mesoporous materials by a thermodynamic approach: A curvature-dependent solid-fluid potential

A thermodynamic analysis of nitrogen adsorption in cylindrical pores of MCM-41 and SBA-15 samples at 77 K is presented within the framework of the Broekhoff and de Boer (BdB) theory. We accounted for the effect of the solid surface curvature on the potential exerted by the pore walls. The developed model is in quantitative agreement with the non-local density functional theory (NLDFT) for pores larger than 2 tun. This modified BdB theory accounting for the Curvature Dependent Potential (CDP-BdB) was applied to determine the pore size distribution (PSD) of a number of MCM-41 and SBA-15 samples on the basis of matching the equilibrium theoretical isotherm against the adsorption branch of the experimental isotherm. In all cases investigated the PSDs determined with the new approach are very similar to those determined with the non-local density functional theory also using the same basis of matching of theoretical isotherm against the experimental adsorption branch. The developed continuum theory is very simple in its utilization, suggesting that CDP-BdB could be used as an alternative tool to obtain PSD for mesoporous solids from the analysis of adsorption branch of adsorption isotherms of any sub-critical fluids.

A Bayesian approach to imposing curvature on distance functions

The estimated parameters of output distance functions frequently violate the monotonicity, quasi-convexity and convexity constraints implied by economic theory, leading to estimated elasticities and shadow prices that are incorrectly signed, and ultimately to perverse conclusions concerning the effects of input and output changes on productivity growth and relative efficiency levels. We show how a Bayesian approach can be used to impose these constraints on the parameters of a translog output distance function. Implementing the approach involves the use of a Gibbs sampler with data augmentation. A Metropolis-Hastings algorithm is also used within the Gibbs to simulate observations from truncated pdfs. Our methods are developed for the case where panel data is available and technical inefficiency effects are assumed to be time-invariant. Two models-a fixed effects model and a random effects model-are developed and applied to panel data on 17 European railways. We observe significant changes in estimated elasticities and shadow price ratios when regularity restrictions are imposed. (c) 2004 Elsevier B.V. All rights reserved.

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.

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.