Flat Curvature Research Articles

2-Dimensional flat curvature flow of crystals

In the impressive and seminal paper [5], Fred Almgren, Jean Taylor, and Lihe Wang introduced flat curvature flow in Rn, a variational time-discretization scheme for (isotropic or anisotropic) mean curvature flow. Their main result asserts the Holder continuity, in time, of these flows. This essential estimate requires a boundary regularity result, a uniform lower density ratio bound condition, which they proved for each n > 3. Similar estimates for Brownian flows, from important work by N. K. Yip on stochastic mean curvature flow [30], also rely on this pivotal regularity result. In this paper, we complete this analysis for the case n = 2 by establishing the necessary uniform lower density ratio bounds.

SYMMETRIES, LADDER OPERATORS AND QUANTUM INTEGRABLE SYSTEMS

We consider a class of deformations of Laplace-Beltrami operators on flat and constant curvature spaces, which possess a family of commuting operators. These are built as deformations of the symmetries of the underlying geometric space. In flat spaces it is also possible to extend some symmetries into ladder operators. In all cases it is possible to choose sub-classes which are super-integrable.

On higher spin theory: strings, BRST, dimensional reductions

We briefly review some modern developments in higher spin field theory and their links with superstring theory. The analysis is based on various BRST constructions allowing us to derive the Lagrangians for massive and massless higher spin fields on flat or constant curvature backgrounds of arbitrary dimensions.

BAYESIAN ANALYSIS OF THE CHAPLYGIN GAS AND COSMOLOGICAL CONSTANT MODELS USING THE SNe Ia DATA

The type Ia supernovae observational data are used to estimate the parameters of a cosmological model with cold dark matter and the Chaplygin gas. This exotic gas, which is characterized by a negative pressure varying with the inverse of density, represents in this model the dark energy responsible for the acceleration of the Universe. The Chaplygin gas model depends essentially on four parameters: the Hubble constant, the velocity of the sound of the Chaplygin gas, the curvature of the Universe and the fraction density of the Chaplygin gas and the cold dark matter. The Bayesian parameter estimation yields [Formula: see text] and [Formula: see text]. These and other results indicate that a Universe completely dominated by the Chaplygin gas is favoured, what reinforces the idea that the Chaplygin gas may unify the description for dark matter and dark energy, at least as the type Ia supernovae data are concerned. A closed and accelerating Universe is also favoured. The Bayesian statistics indicates that the Chaplygin gas model is more likely than the standard cosmological constant (ΛCDM) model at 55.3% confidence level when an integration on all free parameters is performed. Assuming the spatially flat curvature, this percentage mounts to 65.3%. On the other hand, if the density of dark matter is fixed at zero value, the Chaplygin gas model becomes more preferred than the ΛCDM model at 91.8% confidence level. Finally, the hypothesis of flat Universe and baryonic matter (Ωb0=0.04) implies a Chaplygin gas model preferred over the ΛCDM at a confidence level of 99.4%.

Impact of graft diameter on corneal power and the regularity of postkeratoplasty astigmatism before and after suture removal

Objective To assess the impact of graft diameter on corneal curvature before and after removal of a double-running suture after nonmechanical penetrating keratoplasty (PK). Design Prospective, nonrandomized, comparative (self-controlled) single-center clinical trial. Patients Four hundred eighty-nine eyes with “two sutures in” and 308 eyes with “all sutures out” (mean age, 52±19 years) were included. The diagnoses were keratoconus (48%), Fuchs' and stromal dystrophies (31%), aphakic or pseudophakic bullous keratopathy (11%), and scars (10%). Interventions In all eyes, a central trephination was performed (donor trephination from the epithelial side) using the 193-nm Meditec excimer laser (Carl Zeiss Meditec, Jena, Germany) along metal masks with eight “orientation teeth/notches.” Diameters were 8.0 mm, 7.5 mm, and 7.0 mm with a graft oversize of 0.1 mm. In 29% of eyes, additional cataract, intraocular lens surgery, or both were performed simultaneously. In all eyes, a double-running 10-0 nylon suture was applied. Zeiss keratometry and TMS-1 topography analysis were performed before removal of the first suture (14±4 months) and at least 6 weeks after removal of the second suture (20±4 months), but before any additional surgery, such as cataract extraction or refractive keratotomies. Main outcome measures Topographic central corneal power (CP; keratometric diopters), keratometric astigmatism (KA), surface regularity index (SRI), and surface asymmetry index (SAI). The regularity of keratometry mires was recorded semiquantitatively from 0 = regular to 3 = not measurable (as published earlier). Results With both sutures in, median CP in 7.0-mm (42.0 diopters [D]; P = 0.04) and in 7.5-mm grafts (42.3 D; P = 0.007) was significantly lower than in 8.0-mm grafts (43.0 D). Keratometric astigmatism did not differ between groups (3.0 D vs. 3.0 D vs. 2.7 D). The SRI (1.66 vs. 1.43 vs. 1.11) and SAI (1.55 vs. 1.24 vs. 0.85) decreased significantly with increasing diameter. The proportion of regular keratometry mires (13% vs. 17% vs. 29%) increased, and the proportion of not measurable keratometries (45% vs. 18% vs. 9%) decreased with increasing diameter. With all sutures out, CP in 7.0-mm grafts (40.4 D) was significantly smaller than in 7.5-mm (43.6 D; P = 0.04) and 8.0-mm grafts (43.3 D; P = 0.04). Again, KA did not differ between groups (3.0 D vs. 3.2 D vs. 3.0 D). The SRI (1.40 vs. 1.09 vs. 0.84) and SAI (1.24 vs. 0.83 vs. 0.62) decreased significantly with increasing diameter. The proportion of regular keratometry mires (5% vs. 31% vs. 52%) increased, and the proportion of not measurable keratometries (42% vs. 11% vs. 4%) decreased with increasing diameter. Conclusions After PK, a smaller graft diameter results in a flatter curvature and a higher degree of topographic irregularity, but not in higher net astigmatism. After suture removal, graft topography tends to regularize, whereas the principal differences between diameters do persist.

Regular cosmological bouncing solutions in low energy effective action from string theories

The possibility of obtaining singularity free cosmological solutions in four dimensional effective actions motivated by string theory is investigated. In these effective actions, in addition to the Einstein-Hilbert term, the dilatonic and the axionic fields are also considered as well as terms coming from the Ramond-Ramond sector. A radiation fluid is coupled to the field equations, which appears as a consequence of the Maxwellian terms in the Ramond-Ramond sector. Singularity free bouncing solutions in which the dilaton is finite and strictly positive are obtained for models with flat or negative curvature spatial sections when the dilatonic coupling constant is such that $\omega < - 3/2$, which may appear in the so called $F$ theory in 12 dimensions. These bouncing phases are smoothly connected to the radiation dominated expansion phase of the standard cosmological model, and the asymptotic pasts correspond to very large flat spacetimes.

Molecular Organization of Gemini Surfactants in Cylindrical Micelles: An Infrared Dichroism Spectroscopy and Molecular Dynamics Study

The organization of cationic dimeric (gemini) surfactants assembled in cylindrical micelles was investigated using polarization modulation infrared linear dichroism (PM-IRLD) and molecular dynamics simulations. These molecules consist of two quaternary ammonium amphiphiles connected at the headgroup by an ethylene spacer, resulting in an elongated polar head. PM-IRLD measurements and molecular dynamics simulations indicate that the long axis of the surfactant headgroup may adopt a preferred orientation of ca. 30-50° with respect to the micelle cylinder axis. This may be interpreted as the result of the competition between the charge repulsion between headgroups which favors an orientation of the molecules perpendicular to the cylinder axis (high curvature) and the steric repulsion between hydrophobic tails which favors an orientation of the molecules parallel with the cylinder axis (flat curvature). Experiment and calculation both show a water layer a few angstroms thick with a specific orientation at the micelle interface.

Noncommutative Riemannian geometry of the alternating group [formula omitted

We study the noncommutative Riemannian geometry of the alternating group A 4=( Z 2× Z 2)⋊ Z 3 using the recent formulation for finite groups. We find a unique ‘Levi-Civita’ connection for the invariant metric, and find that it has Ricci flat but nonzero Riemann curvature. We show that it is the unique Ricci flat connection on A 4 with the standard framing (we solve the vacuum Einstein’s equation). We also propose a natural Dirac operator for the associated spin connection and solve the Dirac equation. Some of our results hold for any finite group equipped with a cyclic conjugacy class of four elements. In this case the exterior algebra Ω( A 4) has dimensions 1:4:8:11:12:12:11:8:4:1 with top-form nine-dimensional. We also find the noncommutative cohomology H 1( A 4)= C .

A curvature-based bound on the number of frictionless fingers required to immobilize three-dimensional objects

This paper presents a curvature-based bound on the number of frictionless fingers or fixtures required to immobilize 3D objects. A recently developed second-order mobility theory has shown that in addition to first-order geometrical effects, second-order or curvature effects play an important role in the kinematics of contact. We show that when second-order effects are included, four convex fingers or fixtures with sufficiently flat curvature can immobilize any generic smooth or polyhedral 3D object. The derivation of the improved bound proceeds by first constructing a suitable equilibrium grasp of the given object, called a pre-immobilizing grasp. Depending on the object's geometry, a pre-immobilizing grasp may have two, three, or four contacts. The conversion of a pre-immobilizing grasp to a four-finger immobilizing grasp depends on the object's curvature at the contacts. This curvature can be convex, concave, or saddle-like. Since a pre-immobilizing grasp can have k = 2, 3, 4 contacts, there are 31 cases to consider. We construct immobilizing grasps for all of these cases, and present simulation results showing the immobilization of selected object types.

Gaussian superpositions in scalar-tensor quantum cosmological models

A free scalar field minimally coupled to gravity model is quantized and the Wheeler-DeWitt equation in minisuperspace is solved analytically, exhibiting positive and negative frequency modes. The analysis is performed for positive, negative and zero values of the curvature of the spatial section. Gaussian superpositions of the modes are constructed, and the quantum Bohmian trajectories are determined in the framework of the Bohm--de Broglie interpretation of quantum cosmology. Oscillating universes appear in all cases, but with a characteristic scale of the order of the Planck scale. Bouncing regular solutions emerge for the flat curvature case. They contract classically from infinity until a minimum size, where quantum effects become important, acting as repulsive forces avoiding the singularity and creating an inflationary phase, expanding afterwards to an infinite size, approaching the classical expansion as long as the scale factor increases. These are nonsingular solutions which are viable models to describe the early Universe.

Meridional ray contributions to scattering by tilted cylindrical shells above the coincidence frequency: ray theory and computations

High-frequency scattering enhancements from a submerged infinite cylindrical shell are studied by means of partial wave series (PWS) calculations and a ray approximation. For certain oblique angles of incidence it is possible to launch a generalized leaky Lamb wave which propagates strictly in the axial direction on the shell. This meridional leaky wave radiates with a locally flat wavefront curvature and produces large far-field amplitudes over a range of tilt angles. When combined with a reflection from a discontinuity in the shell, this process can give rise to large backscattering enhancements at large tilt angles for thick finite cylindrical shells [S. F. Morse et al., J. Acoust. Soc. Am. 103, 785–794 (1998)] and is present in sonar images. A convolution formulation ray theory for the far-field meridional ray amplitude is compared with the exact PWS solution in the region where enhancements are observed for both the a0 and s0 Lamb waves. The ray theory correctly predicts both the peak enhancement amplitude and the general shape of the enhancement curve.

On forms with non-Abelian charges and their dualities

We describe forms with non-Abelian charges. We avoid the use of theories with flat curvatures by working in the context of topological field theory. We obtain TQFTs for a form and its dual. We leave open the question of getting gauges in which the form, or its dual, can be gauged away, in such way that the model has two dual formulations. We give the example of charged two-forms in six dimensions.

Euclidean supergravity in terms of Dirac eigenvalues

It has been recently shown that the eigenvalues of the Dirac operator can be considered as dynamical variables of Euclidean gravity. The purpose of this paper is to explore the possiblity that the eigenvalues of the Dirac operator might play the same role in the case of supergravity. It is shown that for this purpose some primary constraints on covariant phase space as well as secondary constraints on the eigenspinors must be imposed. The validity of primary constraints under covariant transport is further analyzed. It is show that in the this case restrictions on the tanget bundle and on the spinor bundle of spacetime arise. The form of these restrictions is determined under some simplifying assumptions. It is also shown that manifolds with flat curvature of tangent bundle and spinor bundle and spinor bundle satisfy these restrictons and thus they support the Dirac eigenvalues as global observables.

The use of computerized videokeratography as an aid in fitting rigid gas permeable contact lenses

To assess the efficacy of computerized corneal topography in (1) fitting cosmetic rigid gas permeable contact lenses on normal corneas and (2) patients who have keratoconus. In study 1, 26 patients were fitted with cosmetic contact lenses. In 13 patients, the first rigid gas permeable lens was chosen on the basis of corneal topography using the EyeSys videokeratoscope and contact lens software version 2.11w. Thirteen control patients were fitted with rigid gas permeable contact lenses by means of diagnostic contact lenses. The number of parameter changes required in each group to reach the best fit were compared. In study 2, the records of 28 patients (41 eyes) with keratoconus who had been successfully fit with rigid gas permeable lenses were reviewed. Topographic analysis using the EyeSys videokeratoscope was conducted. The following 8 data points were analyzed: maximum dioptric curvature with respect to the central visual axis, the power at the visual axis (VK), the average keratometry readings within the 3-mm zone, the average flat K at 3-mm central and 5-mm paracentral regions, and the K reading at 1.5, 2.0, and 2.5 mm superior to the visual axis. Study 1: In 10 eyes (38%) fitted with cosmetic contact lenses utilizing corneal topography and the EyeSys contact lens program, the first lens chosen by the computer program was correct with no additional changes in lens parameters required. In the other 16 eyes, 21 parameter changes were required. In the control group, 14 eyes (53%) required no additional change in parameters from the original contact lens that was fitted using diagnostic lenses. In the other 12 eyes, 12 parameter changes were required. Study 2: In the keratoconus group, of the 9 measurements analyzed, the final base curve of patients' rigid gas permeable lenses was correlated most closely with the average flat corneal curvature at the 3-mm central zone. Study 1: In fitting cosmetic rigid gas permeable contact lenses, the videokeratoscope with the EyeSys software 2.11w contact lens program proved to be inferior to diagnostic lenses in selecting the final correct contact lens parameters. However, it may be useful in selecting an initial diagnostic lens with which to begin rigid gas permeable fitting. Study 2: In patients with irregular corneas such as keratoconus, corneal topography is helpful in determining the initial base curve of the contact lens. In patients with keratoconus, it appears to be the average flat corneal curvature at the 3.0-mm zone.

On curvature homogeneous and locally homogeneous affine connections

This paper deals with curvature homogeneous affine connections on 2-dimensional manifolds. We give a sufficient condition for a protectively flat curvature homogeneous connection to be locally homogeneous and show how to construct curvature homogeneous connections that are not locally homogeneous. The concept of a curvature homogeneous Riemannian connection was introduced by I. M. Singer in [5] and then studied in many papers. A similar notion can be defined in affine geometry. Curvature homogeneous affine connections do not have, in general, properties similar to Riemannian ones. For instance, a curvature homogeneous Riemannian connection on a 2-dimensional manifold is automatically locally symmetric and consequently locally homogeneous. As regards the affine case, it is easy to find curvature homogeneous affine connections on 2-dimensional manifolds which are not locally homogeneous. For instance, connections with symmetric Ricci tensor of rank 1 on connected manifolds are always curvature homogeneous and only in exceptional cases locally homogeneous. In this note we begin to study curvature homogeneous affine connections on 2-dimensional manifolds. The differences between the curvature homogeneity, locally homogeneity and local symmetry conditions are illustrated by means of protectively flat connections, which, in some sense, are the closest to locally symmetric ones. 1 Manifolds are assumed to be connected and connections torsion-free. For a given connection its curvature and Ricci tensors will be denoted by R and Ric respectively. Definition 1.1. Let V be a connection on a manifold M. It is called curvature homogeneous of order r if and only if for every x, y E M there exists a linear isomorphism F: TsM TyM such that F*(ViR)y = (V'R), for every i = 0, ..., r. By a curvature homogeneous connection we mean a connection which is curvature homogeneous of order 0. Received by the editors November 15, 1994. 1991 Mathematics Subject Classification. Primary 53B05, 53C30.

New bounds on the number of frictionless fingers requied to immobilize

AbstractUsing a novel second‐order mobility theory that was recently developed by the authors, this article develops new lower bounds on the number of frictionless fingers required to immobilize generic planar objects. We show that any generic smooth or polygonal planar object can be immobilized with three convex fingers or fixtures that have sufficiently flat curvature. Further, if it is possible to specify the fingertip curvature, then any generic smooth or polygonal object can be immobilized with two, possibly concave, fingers. © 1995 John Wiley &amp; Sons, Inc.

A COVARIANT HAMILTONIAN FOR GRAVITY THEORIES

Differential form methods are used to obtain a covariant Hamiltonian formulation for any metric-compatible gravitational theory. The Hamiltonian includes a covariant expression for the conserved quantities of an asymptotically flat or constant curvature space. Several applications are noted.

Covariant form for the conserved currents of the sine-Gordon and Liouville theories

A conserved covariant fourth rank tensor current J ; αβγ is constructed for these models both in flat and constant curvature space. For flat space, ∫ d x + J ++++ and its parity conjugate agree with well known results for the lowest grade sine-Gordon conserved charges. However potentially new charges such as ∫ d x + J +++− and ∫ d x + J +++ α ϵ αβ x β either vanish or fail to be conserved because J μαβγ is not symmetric in μ↔ γ. There is one curious exception for sine-Gordon models in anti-de Sitter space.

Differential geometry in the large and compactification of higher-dimensional gravity

Some well-known results from differential geometry are applied to some of the major issues of compactification of higher-dimensional gravity. The results apply both to the theories generally known as Kaluza–Klein and the recently more promising super string theories. These results are primarily due to Yano [K. Yano, Integral Formulas in Differential Geometry (Marcel Dekker, New York, 1970); Differential Geometry on Complex and Almost Complex Manifolds (Macmillian, New York, 1965)] and have profound implications for the Kaluza–Klein scenario with respect to the cosmological constant problem and the massless sector of the theory. While the results are well known in the mathematical literature, the present author has only seen a fragmentary account presented by a few physicists. The necessary introduction to complex manifolds is also provided including Kähler manifolds and their possible relevance to the problem of compactification. The Ricci tensor provides the central role in the discussion of metric isometries, holomorphy, and holonomy. The incumbent role of Calabi–Yau manifolds with Ricci flat curvature and SU(n) holonomy, which have been recently conjectured in regard to super string compactification, is also mentioned.

Aggregates of amphiphilic molecules in lyotropic liquid crystals

Aamphiphilic molecules are made of two parts with different chemical affinities. They build interfaces which limit the contact between these two parts. We shall focus our attention on the case of soap molecules in the presence of water, in which the interfaces separate the paraffinic chains from the water and delimit aggregates of various shapes. Classically the aggregates can be infinite, with flat or cylindrical interfacial curvatures, and packed with long-range translational order in the well-known lamellar, cubic and hexagonal phases; or they can be finite, quasi-spherical and packed without any long-range order in the micellar phase. However, recent investigations show that a long-range ordering of finite aggregates is possible. There is one example of translational order, in which the micelles build a cubic structure, and much more examples now of orientational order. In the latter case the micelles, oblate or prolate spheroids, are dispersed in the solution with their axes nearly parallel to each other, a nematic phase is formed. We shall discuss the possible factors controlling the shapes of the aggregates and their interactions.