On a Class Almost Contact Manifolds with Norden Metric

Certain curvature properties and scalar invariants of the mani- folds belonging to one of the main classes almost contact manifolds with Norden metric are considered. An example illustrating the obtained results is given and studied.

Linear Connections on Normal Almost Contact Manifolds with Norden Metric

Families of linear connections are constructed on almost con- tact manifolds with Norden metric. An analogous connection to the symmetric Yano connection is obtained on a normal almost contact manifold with Norden metric and closed structural 1-form. The curvature properties of this connec- tion are studied on two basic classes of normal almost contact manifolds with Norden metric.

Spectral analysis and the Aharonov-Bohm\~ effect on certain almost-Riemannian manifold

We study spectral properties of the Laplace-Beltrami operator on two relevant almost-Riemannian manifolds, namely the Grushin structures on the cylinder and on the sphere. This operator contains first order diverging terms caused by the divergence of the volume. We get explicit descriptions of the spectrum and the eigenfunctions. In particular in both cases we get a Weyl's law with leading term Elog E. We then study the drastic effect of Aharonov-Bohm magnetic potentials on the spectral properties. Other generalised Riemannian structures including conic and anti-conic type manifolds are also studied. In this case, the Aharonov-Bohm magnetic potential may affect the self-adjointness of the Laplace-Beltrami operator.

Natural Connections with Totally Skew-Symmetric Torsion on Manifolds with Norden-Type Metrics

This paper is a survey of results obtained by the authors on the geometry of connections with totally skew-symmetric torsion on the following manifolds: almost complex manifolds with Norden metric, almost contact manifolds with B-metric and almost hypercomplex manifolds with Hermitian and anti-Hermitian metric.

Contact CR-Submanifolds of Kenmotsu Manifolds

2000 Mathematics Subject Classification: 53C15, 53C42.

Lie Groups as Four-Dimensional Special Complex Manifolds with Norden Metric

Марта Теофилова - Конструиран е пример на четиримерно специално комплексно многообразие с норденова метрика и постоянна холоморфна секционна кривина чрез двупара-метрично семейство от разрешими алгебри на Ли. Изследвани са кривинните свойства на полученото многообразие. Дадени са необходими и достатъчни усло-вия за разглежданото многообразие да бъде изотропно келерово.

Augmentations and Rulings of Legendrian Links

For any Legendrian knot in R^3 with the standard contact structure, we show that the existence of an augmentation to any field of the Chekanov-Eliashberg differential graded algebra over Z[t,t^{-1}] is equivalent to the existence of a normal ruling of the front diagram, generalizing results of Fuchs, Ishkhanov, and Sabloff. We also show that any even graded augmentation must send t to -1.

We extend the definition of a normal ruling from J^1(S^1) given by Lavrov and Rutherford to a normal ruling for Legendrian links in #^k(S^1\times S^2). We then show that for Legendrian links in J^1(S^1) and #^k(S^1\times S^2), the existence of an augmentation to any field of the Chekanov-Eliashberg differential graded algebra over Z[t,t^{-1}] is equivalent to the existence of a normal ruling of the front diagram. For Legendrian knots, we also show that any even graded augmentation must send t to -1. We use the correspondence to give nonvanishing results for the symplectic homology of certain Weinstein 4-manifolds.

A ‘cost-per-wear’ model based on contact lens replacement frequency

Purpose: The aim was to construct and advise on the use of a cost-per-wear model based on contact lens replacement frequency, to form an equitable basis for cost comparison. ---------- Methods: The annual cost of professional fees, contact lenses and solutions when wearing daily, two-weekly and monthly replacement contact lenses is determined in the context of the Australian market for spherical, toric and multifocal prescription types. This annual cost is divided by the number of times lenses are worn per year, resulting in a ‘cost-per-wear’. The model is presented graphically as the cost-per-wear versus the number of times lenses are worn each week for daily replacement and reusable (two-weekly and monthly replacement) lenses.---------- Results: The cost-per-wear for two-weekly and monthly replacement spherical lenses is almost identical but decreases with increasing frequency of wear. The cost-per-wear of daily replacement spherical lenses is lower than for reusable spherical lenses, when worn from one to four days per week but higher when worn six or seven days per week. The point at which the cost-per-wear is virtually the same for all three spherical lens replacement frequencies (approximately AUD$3.00) is five days of lens wear per week. A similar but upwardly displaced (higher cost) pattern is observed for toric lenses, with the cross-over point occurring between three and four days of wear per week (AUD$4.80). Multifocal lenses have the highest price, with cross-over points for daily versus two-weekly replacement lenses at between four and five days of wear per week (AUD$5.00) and for daily versus monthly replacement lenses at three days per week (AUD$5.50).---------- Conclusions: This cost-per-wear model can be used to assist practitioners and patients in making an informed decision in relation to the cost of contact lens wear as one of many considerations that must be taken into account when deciding on the most suitable lens replacement modality.

Power profiles of multifocal contact lenses and their interpretation

Purpose Many contact lens (CL) manufacturers produce simultaneous-image lenses in which power varies either smoothly or discontinuously with zonal radius. We present in vitro measurements of some recent CLs and discuss how power profiles might be approximated in terms of nominal distance corrections, near additions, and on-eye visual performance. Methods Fully hydrated soft, simultaneous-image CLs from four manufacturers (Air Optix AQUA, Alcon; PureVision multifocal, Bausch & Lomb; Acuvue OASYS for Presbyopia, Vistakon; Biofinity multifocal- ‘‘D’’ design, Cooper Vision) were measured with a Phase focus Lens Profiler (Phase Focus Ltd., Sheffield,UK) in a wet cell and powerswere corrected to powers in air. All lenses had zero labeled power for distance. Results Sagittal power profiles revealed that the ‘‘low’’ add PureVision and Air Optix lenses exhibit smooth (parabolic) profiles, corresponding to negative spherical aberration. The ‘‘mid’’ and ‘‘high’’ add PureVision and Air Optix lenses have biaspheric designs, leading to different rates of power change for the central and peripheral portions. All OASYS lenses display a series of concentric zones, separated by abrupt discontinuities; individual profiles can be constrained between two parabolically decreasing curves, each giving a valid description of the power changes over alternate annular zones. Biofinity lenses have constant power over the central circular region of radius 1.5 mm, followed by an annular zone where the power increases approximately linearly, the gradient increasing with the add power, and finally an outer zone showing a slow, linear increase in power with a gradient being almost independent of the add power. Conclusions The variation in power across the simultaneous-image lenses produces enhanced depth of focus. The throughfocusnature of the image, which influences the ‘‘best focus’’ (distance correction) and the reading addition, will vary with several factors, including lens centration, the wearer’s pupil diameter, and ocular aberrations, particularly spherical aberration; visual performance with some designs may show greater sensitivity to these factors.

International contact lens prescribing in 2015

We have been collecting data on worldwide contact lens prescribing habits for almost 20 years. Over this time period, we have amassed prospective information about 315,000 contact lens fits in 59 countries. This article marks our 15th report in Contact Lens Spectrum and features a breakdown of more than 23,000 contact lens fits in 34 markets. As in previous years, our international network of coordinators distributed survey forms to eyecare practitioners in their market who then recorded generic information about the first 10 patients fit with contact lenses after receipt. Information is gathered about patient age and gender; whether the contact lenses are prescribed as a new fit or a refit; contact lens material, design, and replacement frequency; number of intended days per week of use; wearing modality; and care system. Contact lens fits are weighted to reflect the number of fits undertaken by each eyecare practitioner. The study data were entered and processed at the University of Manchester and at the University of Waterloo.

Systematic study on crystal-contact engineering of diphthine synthase: influence of mutations at crystal-packing regions on X-ray diffraction quality

It is well known that protein crystallizability can be influenced by site-directed mutagenesis of residues on the molecular surface of proteins, indicating that the intermolecular interactions in crystal-packing regions may play a crucial role in the structural regularity at atomic resolution of protein crystals. Here, a systematic examination was made of the improvement in the diffraction resolution of protein crystals on introducing a single mutation of a crystal-packing residue in order to provide more favourable packing interactions, using diphthine synthase from Pyrococcus horikoshii OT3 as a model system. All of a total of 21 designed mutants at 13 different crystal-packing residues yielded almost isomorphous crystals from the same crystallization conditions as those used for the wild-type crystals, which diffracted X-rays to 2.1 angstrom resolution. Of the 21 mutants, eight provided crystals with an improved resolution of 1.8 angstrom or better. Thus, it has been clarified that crystal quality can be improved by introducing a suitable single mutation of a crystal-packing residue. In the improved crystals, more intimate crystal-packing interactions than those in the wild-type crystal are observed. Notably, the mutants K49R and T146R yielded crystals with outstandingly improved resolutions of 1.5 and 1.6 angstrom, respectively, in which a large-scale rearrangement of packing interactions was unexpectedly observed despite the retention of the same isomorphous crystal form. In contrast, the mutants that provided results that were in good agreement with the designed putative structures tended to achieve only moderate improvements in resolution of up to 1.75 angstrom. These results suggest a difficulty in the rational prediction of highly effective mutations in crystal engineering.

Note on the locally product and almost locally product structures

This paper is concerned with a study of some of the properties of locally product and almost locally product structures on a differentiable manifold X n of class C k . Every locally product space has certain almost locally product structures which transform the local tangent space to X n at an arbitrary point P in a set fashion: this is studied in Theorem (2.2). Theorem (2.3) considers the nature of transformations that exist between two co-ordinate systems at a point whenever an almost locally product structure has the same local representation in each of these co-ordinate systems. A necessary and sufficient condition for X n to be a locally product manifold is obtained in terms of the pseudo-group of co-ordinate transformations on X n and the subpseudo-groups [cf., Theoren (2.1)]. Section 3 is entirely devoted to the study of integrable almost locally product structures.

Almost quarter-pinched Kähler metrics and Chern numbers

Given n is an element of Z(+) and epsilon > 0, we prove that there exists delta = delta(epsilon, n) > 0 such that the following holds: If (M(n),g) is a compact Kahler n-manifold whose sectional curvatures K satisfy -1 -delta <= K <= -1/4 and c(I)(M), c(J)(M) are any two Chern numbers of M, then |c(I)(M)/c(J)(M) - c(I)(0)/c(J)(0)| < epsilon, where c(I)(0), c(J)(0) are the corresponding characteristic numbers of a complex hyperbolic space form. It follows that the Mostow-Siu surfaces and the threefolds of Deraux do not admit Kahler metrics with pinching close to 1/4.

Dynamic contact angle beating from drops impacting onto solid surfaces exhibiting anisotropic wetting

This paper reports an experimental investigation of low Weber number water drops impacting onto solid surfaces exhibiting anisotropic wetting. The wetting anisotropy is created by patterning the solid surfaces with unidirectional parallel grooves. Temporal measurements of impacting drop parameters such as drop base contact diameter, apparent contact angle of drop, and drop height at the center are obtained from high-speed video recordings of drop impacts. The study shows that the impact of low Weber number water drops on the grooved surface exhibits beating phenomenon in the temporal variations of the dynamic contact angle anisotropy and drop height at the center of the impacting drop. It is observed that the beating phenomenon of impacting drop parameters is caused by the frequency difference between the dynamic contact angle oscillations of impacting drop liquid oriented perpendicular and parallel to the direction of grooves on the grooved surface. The primary trigger for the phenomenon is the existence of non-axisymmetric drop flow on the grooved surface featuring pinned and free motions of drop liquid in the directions perpendicular and parallel to the grooves, respectively. The beat frequency is almost independent of the impact drop Weber number. Further experimental measurements with solid surfaces of different groove textures show that the grooved surface with larger wetting anisotropy may be expected to show a dominant beating phenomenon. The phenomenon is gradually damped out with time and is fully unrecognizable at higher drop impact Weber numbers. (C) 2011 Elsevier B.V. All rights reserved.

On the dynamic contact angle in simulation of impinging droplets with sharp interface methods

Effects of dynamic contact angle models on the flow dynamics of an impinging droplet in sharp interface simulations are presented in this article. In the considered finite element scheme, the free surface is tracked using the arbitrary Lagrangian-Eulerian approach. The contact angle is incorporated into the model by replacing the curvature with the Laplace-Beltrami operator and integration by parts. Further, the Navier-slip with friction boundary condition is used to avoid stress singularities at the contact line. Our study demonstrates that the contact angle models have almost no influence on the flow dynamics of the non-wetting droplets. In computations of the wetting and partially wetting droplets, different contact angle models induce different flow dynamics, especially during recoiling. It is shown that a large value for the slip number has to be used in computations of the wetting and partially wetting droplets in order to reduce the effects of the contact angle models. Among all models, the equilibrium model is simple and easy to implement. Further, the equilibrium model also incorporates the contact angle hysteresis. Thus, the equilibrium contact angle model is preferred in sharp interface numerical schemes.