Radial Curvature Research Articles

Manifolds with asymptotically nonnegative minimal radial curvature

Abstract In this paper we extend results on the geometry of manifolds with asymptotically nonnegative curvature to manifolds with asymptotically nonnegative minimal radial curvature, showing that most of the results obtained by [U. Abresch, Lower curvature bounds, Toponogov's theorem, and bounded topology. Ann. Sci. École Norm. Sup. (4) 18 (1985), 651–670. MR839689 (87j:53058) Zbl 0595.53043], [A. Kasue, A compactification of a manifold with asymptotically nonnegative curvature. Ann. Sci. École Norm. Sup. (4) 21 (1988), 593–622. MR982335 (90d:53049) Zbl 0662.53032] and [S.-H. Zhu, A volume comparison theorem for manifolds with asymptotically nonnegative curvature and its applications. Amer. J. Math. 116 (1994), 669–682. MR1277451 (95c:53049) Zbl 0953.53027] hold in a more general context. Particularly, we show that there exists one and only one tangent cone at infinity to each such manifold, in contrast with the class of manifolds of nonnegative Ricci curvature.

The Gravitropic Response of Poplar Trunks: Key Roles of Prestressed Wood Regulation and the Relative Kinetics of Cambial Growth versus Wood Maturation

In tree trunks, the motor of gravitropism involves radial growth and differentiation of reaction wood (Archer, 1986). The first aim of this study was to quantify the kinematics of gravitropic response in young poplar (Populus nigra x Populus deltoides, 'I4551') by measuring the kinematics of curvature fields along trunks. Three phases were identified, including latency, upward curving, and an anticipative autotropic decurving, which has been overlooked in research on trees. The biological and mechanical bases of these processes were investigated by assessing the biomechanical model of Fournier et al. (1994). Its application at two different time spans of integration made it possible to test hypotheses on maturation, separating the effects of radial growth and cross section size from those of wood prestressing. A significant correlation between trunk curvature and Fournier's model integrated over the growing season was found, but only explained 32% of the total variance. Moreover, over a week's time period, the model failed due to a clear out phasing of the kinetics of radial growth and curvature that the model does not take into account. This demonstrates a key role of the relative kinetics of radial growth and the maturation process during gravitropism. Moreover, the degree of maturation strains appears to differ in the tension woods produced during the upward curving and decurving phases. Cell wall maturation seems to be regulated to achieve control over the degree of prestressing of tension wood, providing effective control of trunk shape.

Behavior of radial curvatures of ends and spectral structure of the Laplacian

Behavior of radial curvatures of ends and spectral structure of the Laplacian

Homologues of Radial Spoke Head Proteins Interact with Ca2+/Calmodulin in Tetrahymena Cilia

Calmodulin (CaM) is an axonemal component. To examine the pathway of Ca(2+)/CaM signaling in cilia, using Ca(2+)/CaM-affinity column, we identified seven Ca(2+)/CaM-associated proteins from a crude dynein fraction and isolated 62 kDa (p62) and 66 kDa (p66) Ca(2+)/CaM-associated proteins in Tetrahymena cilia. The amino acid sequences deduced from the p62 and p66 cDNA sequences suggested that these proteins were similar to Chlamydomonas radial spoke proteins 4 and 6 (RSP4 and RSP6), components of the radial spoke head, and sea urchin sperm p63, which is a homologue of RSP4/6, and isolated as a key component that affect flagellar bending patterns. Although p62 and p66 do not have a conventional CaM-binding site, those have consecutive sequences which showed high normalized scores (>or= 5) from a CaM target database. These consecutive sequences were also found in RSP4, RSP6, and p63. These radial spoke heads proteins have a high similarity region composed of 15 amino acids between the five proteins. Immunoelectron microscopy using anti-CaM antibody showed that CaM was localized along the outer edge of the curved central pair microtubules in axoneme. Therefore, it is possible that the interaction between Ca(2+)/CaM and radial spoke head control axonemal curvature in the ciliary and flagellar waveform.

Spontaneous curvature cancellation in forced thin sheets

In this paper we report numerically observed spontaneous vanishing of mean curvature on a developable cone made by pushing a thin elastic sheet into a circular container [E. Cerda, S. Chaieb, F. Melo, and L. Mahadevan, Nature (London) 401 46 (1999)]. We show that this feature is independent of thickness of the sheet, the supporting radius, and the amount of deflection. Several variants of the developable cone are studied to examine the necessary conditions that lead to the vanishing of mean curvature. It is found that the presence of appropriate amount of radial stress is necessary. The developable cone geometry somehow produces the right amount of radial stress to induce just enough radial curvature to cancel the conical azimuthal curvature. In addition, the circular symmetry of supporting container edge plays an important role. With an elliptical supporting edge, the radial curvature overcompensates the azimuthal curvature near the minor axis and undercompensates near the major axis. Our numerical finding is verified by a crude experiment using a reflective plastic sheet. We expect this finding to have broad importance in describing the general geometrical properties of forced crumpling of thin sheets.

Instabilities in hurricane-like boundary layers

Recent advances in observational technology have led to a more detailed knowledge of the low-level flow in hurricanes. In particular, quasi-streamwise rolls on a variety of scales have been observed. Some of these rolls have radial wavelengths of 4–10 km, which is comparable to rolls associated with instabilities inherent to Ekman-type boundary layers. The evolution and stability of the swirling boundary layer underneath a hurricane-like vortex is studied using both a nonlinear model and linearized stability analysis. The nonlinear model is an axisymmetric model of incompressible fluid flow, which is used to simulate the development of boundary layers underneath vortices with hurricane-like wind profiles. Axisymmetric rolls appear in these boundary layers, which have some similarities to the observed rolls in hurricanes. The axisymmetric flow is also used as the basic-state for a linearized stability analysis. The analysis technique allows for arbitrary variation in the radial and vertical directions for both the basic-state flow and the perturbations. Thus, the strong radial variations and curvature effects common to strong vortices are part of the analysis. The analysis finds both symmetric and asymmetric instabilities that are similar to those in the nonlinear simulations and in observations. The instabilities acquire some of their energy from the vertical shear associated with a reversal of the radial inflow at the top of the boundary layer, and some of their energy from vertical shear of the azimuthal flow. The radial flow energy conversion tends to increase for flows with less inertial stability and for modes oriented across the low-level shear; the azimuthal flow conversion increases for larger inertial stability and for modes aligned with the low-level shear.

How to obtain transience from bounded radial mean curvature

We show that Brownian motion on any unbounded submanifold P in an ambient manifold N with a pole p is transient if the following conditions are satisfied: The p-radial mean curvatures of P are sufficiently small outside a compact set and the p-radial sectional curvatures of N are sufficiently negative. The 'sufficiency' conditions are obtained via comparison with explicit transience criteria for radially drifted Brownian motion in warped product model spaces.

Mechanical Stress Induced Mechanism of Microtubule Catastrophes

Microtubules assembled in vitro from pure tubulin can switch occasionally from growing to shrinking states or resume assembly, an unusual behavior termed "dynamic instability of microtubule growth". Its origin remains unclear and several models have been proposed, including occasional switching of the microtubules into energetically unfavorable configurations during assembly. In this study, we have asked whether the excess energy accumulated in these configurations would be of sufficient magnitude to destabilize the capping region that must exist at the end of growing microtubules. For this purpose, we have analyzed the frequency distribution of microtubules assembled in vitro from pure tubulin, and modeled the different mechanical constraints accumulated in their wall. We find that the maximal excess energy that the microtubule lattice can store is in the order of 11 kBT per dimer. Configurations that require distortions up to approximately 20 kBT are allowed at the expense of a slight conformational change, and larger distortions are not observed. Modeling of the different elastic deformations suggests that the excess energy is essentially induced by protofilament skewing, microtubule radial curvature change and inter-subunit shearing, distortions that must destabilize further the tubulin subunits interactions. These results are consistent with the hypothesis that unfavorable closure events may trigger the catastrophes observed at low tubulin concentration in vitro. In addition, we propose a novel type of representation that describes the stability of microtubule assembly systems, and which might be of considerable interest to study the effects of stabilizing and destabilizing factors on microtubule structure and dynamics.

Mean curvature mapping for detection of corneal shape abnormality

Corneal topography is used to measure the anterior surface of the cornea. It is conventionally represented as radial slope, radial curvature, and elevation. In this paper, we introduce the application of mean curvature mapping as an alternative representation of the corneal topography. The purpose is to improve the detection of keratoconus and other diseases characterized by local increase in corneal curvature. Both simulated keratoconic cornea and real keratoconus data exported from the corneal topography system were analyzed. Four representations of corneal topography were generated and compared. It was found that mean curvature mapping provided the most precise cone location in simulated keratoconus. In both actual and simulated keratoconus cases, the appearance of the cone-like distortion is more consistent on mean curvature maps. Mean curvature mapping may improve the detection and localization of corneal shape abnormalities.

Effect of radial curvature of magnetic media on glide avalanche

Spacing loss due to micro-waviness and curvature effects in the radial direction of a disk was investigated via numerical simulations and measurements. New techniques based on interferometry and laser scanning were employed to measure radius of curvature (ROC) and to reconstruct the profiles at the outer edge of a disk. A good correlation between ROC values and fly height degradation was obtained, and ROC less than 50 m clearly showed the deterioration in fly height by as much as 1.5 nm. The new measurement techniques proved to be an important means to improve substrate manufacturing processes.

GENERALIZED ALEXANDROV-TOPONOGOV THEOREMS FOR RADIALLY CURVED MANIFOLDS AND THEIR APPLICATIONS

We investigate the topology of complete Riemannian manifolds with the radial curvature at the base manifold bounded below by those of model surfaces. Here, model surfaces are generalized surfaces of revolution and warped product models which are diffeomorphic to Sn.

The asymptotic cones of manifolds of roughly non-negative radial curvature

We prove that the asymptotic cone of every complete, connected, non-compact Riemannian manifold of roughly non-negative radial curvature exists, and it is isometric to the Euclidean cone over their Tits ideal boundaries.

Energy landscapes and bistability to finite-amplitude disturbances for the capillary bridge

Axial and radial curvatures compete to destabilize the open catenoidal capillary bridge, and once destabilized, to influence its dynamical trajectory. The fates of finite-amplitude disturbances to the stable capillary shapes are mapped out by analysis of potential energy landscapes in combination with bounds on the total energy (kinetic plus potential) assuming viscous dissipation, following techniques from earlier work on the Rayleigh–Taylor problem. Energy landscapes (floor) and energy bounds (ceiling), in combination, form large trapping regions in the parameter space of bridge length l (scaled by contact radius R) and disturbance amplitude ε2. Regions are identified within which disturbances must always return or can never return to the stable base state. Necessarily, sensitive competition between two or more directions in phase-space cannot occur in these regions. We find l=0.542 to be the location of a change in nature of saddle point in the landscape and discuss its implications for the dynamical behavior.

RELATIVE VOLUME COMPARISON WITH INTEGRAL RADIAL CURVATURE BOUNDS

In this paper, we generalize the Bishop-Gromov volume comparison theorem by considering an integral bound for the part of the radial Ricci curvature which lies below a given smooth function. We also establish a compactness theorem from this result.

Modeling of thermosolutal convection during Bridgman solidification of semiconductor alloys in relation with experiments

Thermosolutal convection during vertical Bridgman directional solidification of Ga 1− x In x Sb alloys has been studied by numerical simulation. The transient analysis of heat, momentum and species transport has been performed by using the finite element code FIDAP ®. In the case of vertical Bridgman configuration, the thermal convection is driven by the radial temperature gradients. The solute (InSb) rejected at the solid–liquid interface, which is heavier than the GaSb component, damps the thermally driven convection. The solutal effect on the melt convection has been analyzed for low ( x=0.01) and high ( x=0.1) doped Ga 1− x In x Sb alloys. It is found that the damping effect is negligible for Ga 0.99In 0.01Sb alloy grown at low pulling rates ( V= 1 μm/s ), but cannot be neglected if the pulling rate is increased. In the case of concentrated alloys, the low level of convection intensity leads to an increase of radial segregation and interface curvature during the whole growth process as also shown by experiments. The effect of solutal buoyancy force on the melt convection is analyzed for the horizontal Bridgman configuration under microgravity conditions. An inverse but lower solutal effect on the melt convection, as compared with vertical Bridgman arrangement, is observed. The results are in good agreement with the experimental data, and show that convective transport can be observed even for low (2×10 −6 g 0) residual gravity levels.

Electrode Area Effect on Breakdown Field Strength of Vacuum Gaps under Non-Uniform Field

This paper has discussed area effect on breakdown field strength in vacuum gaps under non-uniform field. To identify whether the common area effect can be found between uniform and non-uniform field gaps or not, 50 % breakdown field strength (E50) vs. electrode area more than 90 % of maximum fields strength on cathode (S90) was compared among four kinds of plate-plate gaps, three kinds of ring-plate gaps, and two kinds of sphere-plate gaps. As a result, E50 vs. S90 for plate-plate gaps shows different characteristics from the other kinds of gap arrangements when radial curvature of edge is large and gap length is short. We discussed the reason and how to conduct the insulation design for actual high voltage conductors, consist of some curved surfaces and plane surfaces.

Suggestive contours for conveying shape

In this paper, we describe a non-photorealistic rendering system that conveys shape using lines. We go beyond contours and creases by developing a new type of line to draw: the suggestive contour . Suggestive contours are lines drawn on clearly visible parts of the surface, where a true contour would first appear with a minimal change in viewpoint. We provide two methods for calculating suggestive contours, including an algorithm that finds the zero crossings of the radial curvature. We show that suggestive contours can be drawn consistently with true contours, because they anticipate and extend them. We present a variety of results, arguing that these images convey shape more effectively than contour alone.

Disk shape and its effect on flyability

The static spacing loss of slider-disk interface is investigated through careful characterization of disk shape for disks of different form factors with different peak-to-valley flatness values. Measured radial profiles are normalized and found to be confined within a relatively narrow region. The representation of these shapes is reduced to a simple set of quadratic equations. Static flyability models associated with the disk radial slope (crown effect) and radial curvature (camber effect) are presented. Spacing loss is calculated for a negative pressure pico-slider and a positive pressure catamaran nano-slider. Results show that slider geometries and crown/camber sensitivity, as well as disk flatness and shape, all affect static slider-disk spacing loss. The analysis is focused on the effect of disk shape on the static spacing loss and its relationship to slider design.

Radial Graphs of Constant Mean Curvature and Doubly Connected Minimal Surfaces with Prescribed Boundary

In this paper we investigate existence and uniqueness of radial graphs ofconstant mean curvature (cmc) with prescribed boundary. Our main resultestablishes the existence of a minimal radial anullus spanning two givenconvex curves in parallel planes of R3; we also obtain a variant ofa well-known result of James Serrin about the existence of radial cmc graphsover convex domains in the sphere.

Compactification and maximal diameter theorem for noncompact manifolds with radial curvature bounded below

Let M be a complete open n-manifold with a base point p, at which the radial sectional curvature along every minimizing geodesic emanating from p is bounded below by the radial curvature function of a model surface. We discuss the maximal diameter theorem for the compactification of M by attaching the ideal boundary. Under certain conditions we prove that p becomes a pole and that M is isometric to the n-model.