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To explore the curvature dependence of solid–fluid interfacial thermodynamics, we calculate, using Grand Canonical Monte Carlo simulation, the surface free energy for a 2d hard-disk fluid confined in a circular hard container of radius R as a function of the bulk packing fraction η and wall curvature C ̄ = − 1 / R. (The curvature is negative because the surface is concave.) Combining this with our previous data [Martin et al., J. Phys. Chem. B 124, 7938–7947 (2020)] for the positive curvature case (a hard-disk fluid at a circular wall, C ̄ = + 1 / R), we obtain a complete picture of surface thermodynamics in this system over the full range of positive and negative wall curvatures. Our results show that γ is linear in C ̄ with a slope that is the same for both positive and negative wall curvatures, with deviations seen only at high negative curvatures (strong confinement) and high density. This observation indicates that the surface thermodynamics of this system is consistent with the predictions of so-called morphometric thermodynamics at both positive and negative curvatures. In addition, we show that classical density functional theory and a generalized scaled particle theory can be constructed that give excellent agreement with the simulation data over most of the range of curvatures and densities. For extremely high curvatures, where only one or two disks can occupy the container at maximum packing, it is possible to calculate γ exactly. In this limit, the simulations and density functional theory calculations are in remarkable agreement with the exact results. [ABSTRACT FROM AUTHOR]

Bendway weirs are low-level transverse structures constructed on the outer bank of a channel bend to improve navigation, create aquatic habitat and control bend migration. This experimental study was aimed at investigating the erosion and sedimentation patterns in a 90° bend by varying the relative curvature radius (i.e. normalized to the channel width), in the absence and in the presence of bendway weirs, under different flow conditions. A series of flat weirs with relative heights of 30% and 70% were considered. The results indicate that as the relative curvature increases, the erosion near the weirs decreases while the point-bar at the inner bend wall extends. Regardless of the weir height, the increase in the relative curvature leads to a deeper scour depth at the tip of the weirs. Bendway weirs with submergence ratio of 70% are found to be the most efficient in protecting the outer bank from erosion. [ABSTRACT FROM AUTHOR]

The dispersive and Strichartz estimates are essential for establishing well posedness results for nonlinear equations as well as long time behaviour of solutions to the equation. While in the boundary-less case these estimates are well understood, the case of boundary the situation can become much more difficult. In this work, we establish local in time dispersive estimates for solutions of the model case Dirichlet wave equation inside cylindrical convex domains ⊂ Ω R³ with smooth boundary ∂Ω ≠ ø. In this paper, we provide detailed proofs of the results established in [16, 17]. Let us recall that dispersive estimates are key ingredients to prove Strichartz estimates. Strichartz estimates for waves inside an arbitrary domain Ω have been proved by Blair, Smith, Sogge [4, 5]. Optimal estimates in strictly convex domains have been obtained in [12]. Our case of cylindrical domains is an extension of the result of [12] in the case when the nonnegative curvature radius depends on the incident angle and vanishes in some directions. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Chongqing University of Technology (Natural Science) is the property of Chongqing University of Technology and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

A ball cage flexible drill pipe is a new type of ultra-short-radius drilling tool, which consists of multiple flexible joints hinged together. During the drilling process, the flexible members will come into contact and wear, which reduces the efficiency of load transfer. The multi-body collision contact dynamics model was proposed to study the performance of the ball cage flexible drill pipe. The method considered the influence of the borehole curvature. The kinematic equations of the ball cage flexible drill pipe were established. The Lankarani–Nikravesh collisional contact model was used to characterize the normal contact force, and the Coulomb friction model was used to describe the tangential contact force. The multi-body motion state of the flexible drill pipe was simulated, the contact force distribution of the flexible drill pipe during the motion cycle was analyzed, and the influence of the borehole curvature radius on the size of the flexible joints and the contact force was studied. The results show that the running form of the ball cage flexible drill pipe shows a "folded" shape compared with the initial form; the contact force of different flexible joints is in a state of fluctuation; the normal contact force is much larger than the tangential contact force; the matching relationship between the borehole curvature and the length and radius of the flexible joints is derived, which provides criteria for the design of the flexible joints to ensure the reliability of the flexible drill pipe in large curvature borehole; the borehole curvature has an important influence on the collision contact force and load transfer efficiency of flexible drill pipe. [ABSTRACT FROM AUTHOR]

For a general radially symmetric, non-increasing, non-negative kernel h ∈ L l o c 1 (R d) , we study the rigidity of measurable sets in R d with constant nonlocal h -mean curvature. Under a suitable "improved integrability" assumption on h , we prove that these sets are finite unions of equal balls, as soon as they satisfy a natural nondegeneracy condition. Both the radius of the balls and their mutual distance can be controlled from below in terms of suitable parameters depending explicitly on the measure of the level sets of h. In the simplest, common case, in which h is positive, bounded and decreasing, our result implies that any bounded open set or any bounded measurable set with finite perimeter which has constant nonlocal h -mean curvature has to be a ball. [ABSTRACT FROM AUTHOR]

Consider two coaxial round circles with the same radius. A mathematical model of a tiny liquid drop trapped between them are constant mean curvature (CMC) surfaces because a CMC surface is a critical point of the area for all variations that preserve the enclosed volume and satisfy given boundary condition. A CMC surface is said to be stable if the second variation of the area for any such variation is nonnegative. In this paper, we judge the stability of one period of the so-called unduloid between two consecutive bulges rigorously for the first time. [ABSTRACT FROM AUTHOR]

To examine the flow characteristics of particles in the vertical tube corresponding to different curvature radius bends, the particle velocity is measured at the Minimum Pressure Drop (MPD) using a high-speed Particle Image Velocimeter (PIV). This experiment explores the effects of particle flow characteristics at different curvature radius bends and their corresponding vertical tubes with pressure drop, power consumption, the intensity of particle fluctuation velocity, power spectrum and time-frequency characteristics. It is observed that the pressure drop and power consumption can be reduced with the help of a large curvature radius bend. Besides, the reduction of particle velocity in the large curvature radius bend and its corresponding vertical tube is less, and the particle possesses a larger intensity of fluctuation velocity in its corresponding vertical tube. The particles in the vertical tube corresponding to the large curvature radius bend lead to large peaks of the power spectrum in the low-frequency region, which is closely linked to the pressure drop. Eventually, the dynamics of particles in a vertical tube are revealed from the perspective of time-frequency analysis by using a continuous wavelet transform. [ABSTRACT FROM AUTHOR]

In this work, we establish precise local in time dispersive estimates for solutions of the model case Dirichlet wave equation inside cylindrical convex domains \Omega \subset \mathbb {R}^3 with smooth boundary \partial \Omega \neq \emptyset. This result is the improved estimates established by Len Meas [C. R. Math. Acad. Sci. Paris 355 (2017), pp. 161–165]. Let us recall that dispersive estimates are key ingredients to prove Strichartz estimates. Strichartz estimates for waves inside an arbitrary domain \Omega have been proved by Blair, Smith, Sogge [Proc. Amer. Math. Soc. 136 (2008), pp. 247–256; Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), pp.1817–1829]. Optimal estimates in strictly convex domains have been obtained by Ivanovici, Lebeau, and Planchon [Ann. of Math. 180 (2014), pp. 323–380]. Our case of cylindrical domains is an extension of the result of Ivanovici, Lebeau, and Planchon [Ann. of Math. 180 (2014), pp. 323–380] in the case when the nonnegative curvature radius depends on the incident angle and vanishes in some directions. [ABSTRACT FROM AUTHOR]

We introduce a notion of Ricci curvature for Cayley graphs that can be thought of as "medium-scale" because it is neither infinitesimal nor asymptotic, but based on a chosen finite radius parameter. We argue that it gives the foundation for a definition of Ricci curvature well adapted to geometric group theory, beginning by observing that the sign can easily be characterized in terms of conjugation in the group. With this conjugation curvature  κ , abelian groups are identically flat, and in the other direction we show that κ ≡ 0 implies the group is virtually abelian. Beyond that, κ captures known curvature phenomena in right-angled Artin groups (including free groups) and nilpotent groups, and has a strong relationship to other group-theoretic notions like growth rate and dead ends. We study dependence on generators and behavior under embeddings, and close with directions for further development and study. [ABSTRACT FROM AUTHOR]

We investigate criteria for circle packing (CP) types of disk triangulation graphs embedded into simply connected domains in C. In particular, by studying combinatorial curvature and the combinatorial Gauss-Bonnet theorem involving boundary turns, we show that a disk triangulation graph is CP parabolic if ∑n=1∞ 1/∑j=0n-1 (kj + 6) = ∞, where kn is the degree excess sequence defined by kn = ∑v∈Bn (deg v − 6) for combinatorial balls Bn of radius n and centered at a fixed vertex. It is also shown that the simple random walk on a disk triangulation graph is recurrent if ∑n=1∞ 1/∑j=0n-1 (kj + 6) + ∑j=0n (kj + 6) = ∞. These criteria are sharp, and generalize a conjecture by He and Schramm in their paper from 1995, which was later proved by Repp in 2001. We also give several criteria for CP hyperbolicity, one of which generalizes a theorem of He and Schramm, and present a necessary and sufficient condition for CP types of layered circle packings generalizing and confirming a criterion given by Siders in 1998. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the density dependence of the nuclear symmetry energy S (ρ) in the KIDS (Korea-IBS-Daegu-SKKU) framework for the nuclear equation of state (EoS) and energy-density functional (EDF). The aim is to constrain the value of the curvature parameter ( K sym ) based on recent astronomical data. First, assuming a standard saturation point, we calculate bulk nuclear properties within KIDS-EDF for different values of the compression modulus of symmetric nuclear matter ( K 0 ) and of the leading-order symmetry energy parameters, i.e., the symmetry energy (J) and slope (L) at saturation density, each within a broadly accepted range, as well as K sym . All of the above EoS parameters are varied independently of each other. The skewness parameter ( Q sym ) is presently kept fixed at 650 MeV. For all EoS parameter sets which describe the selected nuclear data within better than 0. 3 % , we calculate the neutron star EoS and mass–radius relation and analyze the results in terms of Pearson correlation coefficients r. We find that the value of K sym is strongly correlated with the radius of both a canonical and a massive star (| r | > 0. 9). If we impose that all known astronomical constraints on the neutron star radii must be satisfied, we deduce − 1 5 0 < K sym < 0. As a result, the symmetry energy as a function of the density is consistently found to have an inflection point at ρ 0 < ρ < 2 ρ 0 . We take the opportunity to report that the neutron skin thickness of 2 0 8 Pb shows no correlation at all with the neutron star radii (| r | < 0. 1) , in contrast with studies which focus on the role of L only. [ABSTRACT FROM AUTHOR]

According to the existing practical engineering data, the settlement curve of shield tunnel with small curvature radius is obviously different from that of straight tunnel, so using Peck formula to forecast the surface settlement is not applicable. This paper describes results from a series of the laboratory test carried out in sandy soil, which based on similarity theory. According to the test results, the characteristics of surface settlement caused by small radius tunnel excavation are summarized, and the effects of turning radius and buried depth on the surface settlement are demonstrated. Based on Gaussian formula, a method of forecasting the surface settlement caused by construction of shield tunnel with small curvature radius is developed. [ABSTRACT FROM AUTHOR]

In the analytic modeling of shell buckling, much attention has been paid to closed cylindrical shells or cylindrical panels with at least two opposite edges simply supported whose solutions are known to be Lévy-type solutions. However, there have been very few reports on analytic solutions of commonly used non-Lévy-type cylindrical panels, which is mainly attributed to the widely acknowledged difficulty in solving the governing higher-order partial differential equations under prescribed boundary conditions. To address this gap, the present study provides some new analytic buckling solutions of non-Lévy-type cylindrical panels within the Hamiltonian-system-based symplectic framework. The buckling of a cylindrical panel is first formulated in the Hamiltonian system to realize a new matrix-form governing equation. An original problem with non-Lévy-type boundary conditions is then treated as the superposition of two elaborated subproblems that are solved by the rigorous symplectic approach. The final solution is obtained according to the equivalence between the superposition of the subproblems and the original problem. For benchmark use, comprehensive buckling loads and buckling modes are presented for typical non-Lévy-type panels with different ratios of in-plane dimensions, different ratios of in-plane dimension to radius of curvature, and different ratios of thickness to in-plane dimension. The present study is the first successful extension of the symplectic superposition method to the buckling analysis of cylindrical panels, which may enable access to more new analytic solutions for similar issues. [ABSTRACT FROM AUTHOR]

An annulus is a configuration characterizing the cross section of a rotating detonation engine (RDE), and the dimensionless radius is a unique geometric feature of the annulus. The present study examines the effects of the dimensionless radius of the annulus on the detonation wave propagation characteristics in an RDE. The concept of a helical computational domain is introduced for a two-dimensional analogy of the annular cross section of the RDE, and the moving window technique is used to accelerate computation. The results suggest the existence of a critical dimensionless radius ( R critical ) condition, above which stable propagation is feasible. Unstable overdriven detonation is observed for values of the dimensionless radius below the critical value. Based on these investigations, the propagation characteristics of detonation waves in three non-circular cross sections are studied for RDE design flexibility. A track-shaped channel was studied in depth for the transitional phenomena between circular to linear and linear to circular sections. A rounded-square configuration and a star-shaped configuration were also examined, in which the simultaneously curved and linear effects exist and dictate the stability of detonation propagation. In the rounded-square channel, this instability is stabilized only at a high radius of curvature. In the star-shaped channel, the detonation propagates in the critical mode for most cases owing to the small local radius of curvature. [ABSTRACT FROM AUTHOR]

The differential equation characterizing a spherical curve in R³ expresses the radius of curvature of the curve in terms of its torsion. In this paper, we have given a generalization of this equation for a curve lying in an arbitrary surface in R³. Moreover, we have established the analogue of the Frenet equations for a curve lying in a surface of R³. We have also revisited some formulas for the geodesic torsion of a curve lying in a surface of R³. [ABSTRACT FROM AUTHOR]

It is shown that in core-radius cutoff regularized simplified elasticity (where the elastic energy depends quadratically on the full displacement gradient rather than its symmetrized version), the force on a dislocation curve by the negative gradient of the elastic energy asymptotically approaches the mean curvature of the curve as the cutoff radius converges to zero. Rigorous error bounds in Holder spaces are provided. As an application, convergence of dislocations moving by the gradient flow of the elastic energy to dislocations moving by the gradient flow of the arclength functional, when the motion law is given by an H1-type dissipation, and convergence to curve shortening flow in co-dimension 2 for the usual L2-dissipation is established. In the second scenario, existence and regularity are assumed while the H1-gradient flow is treated in full generality (for short time). The methods developed here are a blueprint for the more physical setting of linearized isotropic elasticity. [ABSTRACT FROM AUTHOR]

The governing equations of electro-elastic bending problem are derived for size-dependent electro-elastic analysis of three-layered piezoelectric doubly curved nano shells. The three-layered nano shell is composed of an elastic homogeneous core and two piezoelectric nano layers at top and bottom. The doubly curved piezoelectric nano shell is subjected to applied electric potential and transverse loads. A comparative study is presented for validation of formulation and solution. The influence of nonlocal parameter, applied electric potential, and geometric parameters such as radii of curvature and ratio of thickness is studied on the electro-elastic results of the simply-supported doubly curved nano shell. [ABSTRACT FROM AUTHOR]

This paper studies a smooth one-parameter family of standard Lorentzian circles with fixed radius. Such a surface is called a timelike circular surface with constant radius. We call each circle a generating circle. A new type of timelike circular surfaces was identified and coined as the timelike tangent circular surface. The new timelike tangent circular surface has the property of all generating circles being lines of curvature and its Gaussian and mean curvatures being independent of the geodesic curvature of the spherical indicatrix. [ABSTRACT FROM AUTHOR]

The high stresses on plaque wall experienced by stent implantation into a stenotic artery can go beyond the material strength of plaque tissue, potentially leading to plaque rupture. Two non-commercial stents with different link structures called S-type and N-type were taken into account, respectively. A non-linear finite element model was developed to investigate the influence of the stenosis level (i.e., 24%, 40%, and 50%) and arterial curvature radius (i.e., 6, 10, and 20 mm) on the stress induced within the plaque tissue during stent expansion. The numerical results indicated that the severer stenosis level and more tortuous artery caused the higher stress on plaque wall. The maximal stresses on the plaque wall were in the fracture level of 1.79 MPa for N-type stent, and 1.82 MPa for S-type stent under the 50% stenosis rate and 6 mm curvature radius. Due to smaller compliance mismatch to the curved vessel, the N-type stent was founded to induce less stress gradient on the plaque wall than that by the S-type stent. This suggests a lower risk of the plaque rupture for the N-type stent. This study showed how the arterial curvature radius and stenosis level were correlated with the plaque vulnerability. Therefore, it is possible to choose a suitable stent in terms of arterial stenosis geometry and thereby optimize the outcome of stenting procedure. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Drainage & Irrigation Machinery Engineering / Paiguan Jixie Gongcheng Xuebao is the property of Editorial Department of Drainage & Irrigation Machinery Engineering and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, we establish a unilateral global bifurcation result for half-linear perturbation problems with mean curvature operator in Minkowski space. As applications of the abovementioned result, we shall prove the existence of nodal solutions for the following problem − div ∇ v / 1 − ∇ v 2 = α x v + + β x v − + λ a x f v , in B R 0 , v x = 0 , on ∂ B R 0 , where λ ≠ 0 is a parameter, R is a positive constant, and B R 0 = x ∈ ℝ N : x < R is the standard open ball in the Euclidean space ℝ N N ≥ 1 which is centered at the origin and has radius R. a(|x|) ∈ C[0, R] is positive, v + = max{ v , 0}, v − = −min{ v , 0}, α(|x|), β(|x|) ∈ C[0, R]; f ∈ C ℝ , ℝ , s f (s) > 0 for s ≠ 0, and f0 ∈ [0, ∞], where f0 = lim|s|⟶0 f(s)/s. We use unilateral global bifurcation techniques and the approximation of connected components to prove our main results. [ABSTRACT FROM AUTHOR]

The renormalized quantum stress–energy tensor 〈 T μ ν 〉 ren for a massive spinor field around global monopoles is constructed within the framework of Schwinger–DeWitt approximation, valid whenever the Compton length of the quantum field is much less than the characteristic radius of the curvature of the background geometry. The results obtained show that the quantum massive spinor field in the global monopole spacetime violates all the pointwise energy conditions. [ABSTRACT FROM AUTHOR]

Whether borderline hip dysplasia is pathologic remains unclear. In order to evaluate the three-dimensional joint congruity, this study sought to answer the question: are borderline dysplastic hip curvature mismatch and eccentricity between the acetabulum and the femoral head different from dysplastic or control hips three-dimensionally? The 113 hips, categorized as: dysplastic (LCEA ≤ 20°), 47 hips; borderline (20° ≤ LCEA < 25°), 32 hips; and control (25° ≤ LCEA < 35°), 34 hips; were evaluated. Three-dimensional (3D) femoral and coxal bone models were reconstructed from CT images. Using a custom-written Visual C++ routine, the femoral head and acetabular radii of curvature, and the femoral head and the acetabular curvature center were calculated. Then the ratio of the acetabular radius to the femoral head radius (3D curvature mismatch ratio), and the distance between the acetabular curvature center and the femoral head center (3D center discrepancy distance) were calculated. These indices were compared statistically among the three groups using Tukey's post hoc test. The mean 3D curvature mismatch ratio in the borderline (1.13 ± 0.05) was smaller than in the dysplasia (1.23 ± 0.08, p < 0.001), and larger than in the control (1.07 ± 0.02, p < 0.001). The mean 3D center discrepancy distance in the borderline (3.2 ± 1.4 mm) was smaller than in the dysplasia (4.8 ± 2.3, p < 0.001) and larger than in the control (1.6 ± 0.7, p < 0.001). These results demonstrated that three-dimensional congruity of the borderline dysplastic hip is impaired, but its incongruity is not as severe as in dysplastic hips. The 3D curvature mismatch ratio and the 3D center discrepancy distance can be valuable signs of joint congruity in patients with borderline dysplasia. However, future studies are necessary to clarify any associations between curvature mismatch and pathogenesis of osteoarthritis in borderline dysplasia. [ABSTRACT FROM AUTHOR]

Based on assumptions that substorm field line dipolarization at geosynchronous altitudes is associated with the arrival of high-velocity magnetotail flow bursts referred to as bursty bulk flows, the following sequence of field line dipolarization is proposed: (1) slow magnetoacoustic wave excited through ballooning instability by enhanced inflows in pre-onset intervals towards the equatorial plane; (2) in the equatorial plane, slow magnetoacoustic wave stretching of the flux tube in dawn–dusk directions resulting in spreading plasmas in dawn–dusk directions and reduction in the radial pressure gradient in the flux tube. As a consequence of these processes, the flux tube assumes a new equilibrium geometry in which the curvature radius of new field lines increased in the meridian plane, suggesting an onset of field line dipolarization. The dipolarization processes associated with changing the curvature radius preceded classical dipolarization caused by a reduction of cross-tail currents and pileup of the magnetic fields. Increasing the curvature radius induced a convection surge in the equatorial plane as well as inductive westward electric fields of the order of millivolts per meter (mV m -1). Electric fields transmitted to the ionosphere produce an electromotive force in the E layer for generating a field-aligned current system of Bostrom type. This is also equivalent to the creation of an incomplete Cowling channel in the ionospheric E layer by the convection surge. [ABSTRACT FROM AUTHOR]

The torsional effect of the tracks on curved segment intensifies the ambient vibration response induced by Metro operation. This paper studies the ambient vibration responses induced by the operation of the Metro train on curved rail segments. By taking the curved segment of Hangzhou Metro Line 1 of China as an example, the wheel-rail model employing the multibody dynamics is demonstrated; the dynamic wheel-rail force of a B-type vehicle passing through a 350 m radius curved segment is also calculated. A finite element method model of the track-tunnel-soil-building is developed and verified by comparing the measured results with the simulated ones. Then, we analyzed the ambient vibration responses induced by the Metro operation. The results show that the horizontal vibration induced by the Metro train, on curved segments, cannot be ignored. When propagating on the ground, the variation trend of the horizontal vibration acceleration is greater than that of the vertical vibration. The horizontal vibration attenuates faster than the vertical vibration. The secondary vibration along the horizontal direction yields a significant amplifying effect upon the building. The vibration level increases with the height of the building, along the horizontal direction, but the vertical vibration level changes negligibly. The insight exhibits the influence level of the horizontal action of the Metro train wheels to the tracks, which can be referred by the practitioners for the planning and operation. [ABSTRACT FROM AUTHOR]

We present some estimate of the Laplacian Spectrum and of Topological Invariants for Riemannian manifold with pinched sectional curvature and with non-empty and non-convex boundary with finite injectivity radius. These estimates do not depend directly on the the lower bound of the boundary injectivity radius but on the bounds of the curvatures of the manifold and its boundary. [ABSTRACT FROM AUTHOR]

Copyright of Tunnel Construction / Suidao Jianshe (Zhong-Yingwen Ban) is the property of Tunnel Construction Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We investigated the golden rectangle and the related Fibonacci spiral and golden spiral. The coordinates of the shrinkage points of a golden rectangle were derived. Properties of shrinkage points were discussed. Based on these properties, we conduct a comparison study for the Fibonacci spiral and golden spiral. Their similarities and differences were looked into by examining their polar coordinate equations, polar radii, arm-radius angles, and curvatures. The golden spiral has constant arm-radius angle and continuous curvature, while the Fibonacci spiral has cyclic varying arm-radius angle and discontinuous curvature. [ABSTRACT FROM AUTHOR]

In this paper, we prove a localised version of the bounded L 2 -curvature theorem of (Klainerman et al. Invent Math 202(1):91–216, 2015). More precisely, we consider initial data for the Einstein vacuum equations posed on a compact spacelike hypersurface Σ with boundary, and show that the time of existence of a classical solution depends only on an L 2 -bound on the Ricci curvature, an L 4 -bound on the second fundamental form of ∂ Σ ⊂ Σ , an H 1 -bound on the second fundamental form, and a lower bound on the volume radius at scale 1 of Σ . Our localisation is achieved by first proving a localised bounded L 2 -curvature theorem for small data posed on B(0, 1), and then using the scaling of the Einstein equations and a low regularity covering argument on Σ to reduce from large data on Σ to small data on B(0, 1). The proof uses the author's previous works and the bounded L 2 -curvature theorem as black boxes. [ABSTRACT FROM AUTHOR]

In this paper, we consider the high-order geometric flows of a compact submanifolds M in a complete Riemannian manifold N with dim (N) = dim (M) + 1 = n + 1 , which were introduced by Mantegazza in the case the ambient space is an Euclidean space, and extend some results due to Mantegazza to the present situation under some assumptions on N. Precisely, we show that if m ∈ ℕ is strictly larger than the integer part of n / 2 and φ (t) is an immersion for all t ∈ [ 0 , T) and if 𝔉 m (φ 0) is bounded by a constant which relies on the injectivity radius R ̄ > 0 and sectional curvature K ̄ π ( K ̄ π ≤ 1) of N , then T must be ∞. [ABSTRACT FROM AUTHOR]

We prove that given a hyperbolic manifold endowed with an auxiliary Riemannian metric whose sectional curvature is negative and whose volume is sufficiently small in comparison to the hyperbolic one, we can always find for any radius at least 1 a ball in its universal cover whose volume is bigger than the hyperbolic one. This result is deduced from a nonsharp macroscopic version of a conjecture by R. Schoen about scalar curvature, whose proof is a variation of an argument due to M. Gromov and is based on a smoothing technique. We take the opportunity of this work to present a full account of this technique, which involves simplicial volume and deserves to be better known. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of H 2 -regular coordinates on Riemannian 3-manifolds with boundary, assuming only L 2 -bounds on the Ricci curvature, L 4 -bounds on the second fundamental form of the boundary, and a positive lower bound on the volume radius. The proof follows by extending the theory of Cheeger–Gromov convergence to include manifolds with boundary in the above low regularity setting. The main tools are boundary harmonic coordinates together with elliptic estimates and a geometric trace estimate, and a rigidity argument using manifold doubling. Assuming higher regularity of the Ricci curvature, we also prove corresponding higher regularity estimates for the coordinates. [ABSTRACT FROM AUTHOR]

The apex field enhancement factor (AFEF) γ a of a cylindrical emitter depends sensitively on its end-cap geometry. The hemispherical end-cap is well studied due to its simplicity, but, in general, a cylindrical emitter may terminate in a variety of end-cap shapes. It is well known that the AFEF depends on the ratio h / R a , where h is the total height of the emitter and R a is the apex radius of curvature. The authors show here that there can be a large variation in γ a as the end-cap geometry is altered while keeping h / R a fixed. They carry out a systematic numerical study and determine an approximate formula for γ a in terms of measurable end-cap geometry parameters such as its height H , the radius of the cylinder R , and the apex radius of curvature R a. They show that the formula is robust and can predict the net field emission current with errors generally less than 40 %. [ABSTRACT FROM AUTHOR]

A technique has been developed for measuring the radii of curvature of optical parts. A scheme of the device based on a wavefront sensor is proposed, which is characterized by a minimum measurement error. The theoretical substantiation of the measurement technique is given. A number of methods that make it possible to measure the radii of curvature of optical parts are considered. The reasons for an increase in the measurement error are analyzed. The results of measurements according to the given methodology with an original prototype device are considered. [ABSTRACT FROM AUTHOR]

We have calculated the electrical characteristics of a vacuum transistor withsharp, pointed emitter and collector with a radius of curvature R<20nm with the aim of revealing the advantages of such emitters. For such surfaces the traditional Fowler-Nordheim theory which pertains to a parallel plate configuration is not valid. We have used a 3-dimensional WKB theory capable of calculating the distribution of the current in space. We have shown that for values of emitter radius in the range of a few nanometers the current density is enclosed within a cone of angle of approximately 6°–16° thus minimizing current losses. The Id–Vd characteristics were calculated and found to exhibit a exponential behavior and a saturation region spanning currents from 10−14A to 10−6A by only 2 Voltsvolts change in gate voltage Vg, i.e. exhibiting an extraordinary transconductance. [ABSTRACT FROM AUTHOR]

We give, with its sample, a new type of canal surface constructed by means of the parallel transport frame of its spine curve in Euclidean 4-space IE4 . We investigate the curvature features of this surface with respect to the principal curvature functions according to parallel transport frame. Further, we give certain results about Weingarten type canal and tube surfaces. Finally, we give the visualizations of projections of this new type of canal surface in IE3 for various radius functions. [ABSTRACT FROM AUTHOR]

We establish a new lower bound for the null injectivity radius of a null cone. The idea is to use a function closely related to the null second fundamental form which codifies the directional expansion of the null cone along any null geodesic in it. This approach uses first derivatives of the metric instead of curvature bounds. The technique is applied to a family of null cones with the null conjugate radius less or equal to the null crossing radius, or folded due to the curvature in a precise way that we call geometric fold. This condition is necessary because there are trivial examples with bounded null injectivity radius due to global identifications. We show two examples where we compute the real null injectivity radius and the lower bound provided in this paper, in order to compare both quantities. We also give an analogous result for Riemannian manifolds. [ABSTRACT FROM AUTHOR]

We give, with its sample, a new type of canal surface constructed by means of the parallel transport frame of its spine curve in Euclidean 4-space IE4 . We investigate the curvature features of this surface with respect to the principal curvature functions according to parallel transport frame. Further, we give certain results about Weingarten type canal and tube surfaces. Finally, we give the visualizations of projections of this new type of canal surface in IE3 for various radius functions. [ABSTRACT FROM AUTHOR]

Copyright of China Mechanical Engineering is the property of Editorial Board of China Mechanical Engineering and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The objective of this non-randomized trial was comparison of two-year changes in dynamic corneal response (DCR) between 18 mW/cm2 (5- min) and 9 mW/cm2 (10-min) cross-linking (CXL) protocols, using novel stiffness parameters and correlating them to clinical indices. The two groups were evaluated before and 2 years after the procedure using Corvis ST (Oculus Optikgeräte GmbH, Germany) and DCR parameters such as deformation amplitude ratio at 1 mm and 2 mm (DA ratio-1 mm and DA ratio-2 mm) and integrated radius and stiffness parameters at A1 (SP-A1). Two-year follow-up was completed for 16 of the 30 eyes in the 5-min group and 21 of the 25 eyes in the 10-min group; data from those who were lost to follow-up was not included in the analyses. Mean age at baseline was 21.7 ± 4.9 and 21.5 ± 5.2 years in the 5- and 10-min groups, respectively (P = 0.895). At 2 years after CXL, in the 5-min group, the reduction in integrated radius (−1.12 ± 1.27 mm, P = 0.003) was significant, and the increase in SP-A1 (7.11 ± 14.86 mmHg/mm, P = 0.075) was borderline, while in the 10-min group, the decrease in DA ratio-2 mm (−0.43 ± 0.58, P = 0.003) and integrated radius (−1.89 ± 1.72 mm, P < 0.001), and increase in SP-A1 (7.67 ± 10.92 mmHg/mm, P = 0.004) were significant. In both groups, the strongest and significant correlation was observed between DCR parameters and changes in radius of curvature. In conclusion, results indicated corneal strengthening with both protocols especially with the 9 mW/cm2. Corvis ST indices can provide "in vivo" biomechanical evidence on the efficacy of CXL that may occur prior to clinical indices. [ABSTRACT FROM AUTHOR]

We study the Schwarz lemma for harmonic mappings in general setting and prove among the other results that: if is the class of harmonic mappings f of the unit disk into the geodesic disk of a Riemann surface , which is disjoint from its cut-locus with radius M satisfying some curvature restrictions, then there is a locally Lipschitz homeomorphism , convex in , with and such that for with we have for. [ABSTRACT FROM AUTHOR]

Copyright of Tunnel Construction / Suidao Jianshe (Zhong-Yingwen Ban) is the property of Tunnel Construction Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The current-voltage data of a gated metallic nanotipped pyramidal emitter are analyzed using recent advances in field emission theory such as curvature corrections to the tunneling potential and the generalized cosine law of local electrostatic field variation near the emitter apex. It is first shown numerically that the cosine law holds for gated emitters. The theory is then subjected to an experimental test [C. Lee, S. Tsujino, and R. J. Dwayne Miller, Appl. Phys. Lett. 113, 013505 (2018)] where minor uncertainties in the pyramid base length L b and the apex radius of curvature R a exist. It is found using comsol multiphysics that the best-fit theoretical prediction for the emission current corresponds to L b ≃ 1.275 μ m and R a ≃ 5.41 nm, both of which are within their respective uncertainties. The errors for the best-fit curve follow a known pattern with a change in the applied field, with higher field strengths (> --> 5 V/nm) reporting less than 10% error. [ABSTRACT FROM AUTHOR]

A famous theorem of Weyl states that if M is a compact submanifold of euclidean space, then the volumes of small tubes about M are given by a polynomial in the radius r, with coefficients that are expressible as integrals of certain scalar invariants of the curvature tensor of M with respect to the induced metric. It is natural to interpret this phenomenon in terms of curvature measures and smooth valuations, in the sense of Alesker, canonically associated to the Riemannian structure of M. This perspective yields a fundamental new structure in Riemannian geometry, in the form of a certain abstract module over the polynomial algebra R [ t ] that reflects the behavior of Alesker multiplication. This module encodes a key piece of the array of kinematic formulas of any Riemannian manifold on which a group of isometries acts transitively on the sphere bundle. We illustrate this principle in precise terms in the case where M is a complex space form. [ABSTRACT FROM AUTHOR]

We prove that for every r > 0 if a nonpositively curved (p, q)-map M contains no flat submaps of radius r, then the area ofM does not exceed Crn for some constant C. This strengthens a theorem of Ivanov and Schupp. We show that an infinite (p, q)-map which tessellates the plane is quasi-isometric to the Euclidean plane if and only if the map contains only finitely many nonflat vertices and faces. We also generalize Ivanov and Schupp's result to a much larger class of maps, namely to maps with angle functions. [ABSTRACT FROM AUTHOR]

The curvature of field emitter tips leads to an altered tunneling potential that assumes significance when the radius of curvature is small. The authors provide here an analytical curvature corrected formula for the field emission current from axially symmetric emitters aligned in the direction of the asymptotic electric field. Its domain of applicability is tested across a range of apex radius, R a , and local electric field, E a. It is found to give excellent results for R a > 5 nm and E a > 5 V / nm with errors generally less than 10 %. Surprisingly, for the uncorrected potential, the authors find the errors to be high even at R a = 100 nm (> 35% at E a = 3 V / nm) and conclude that curvature correction is essential for apex radius less than a micron. [ABSTRACT FROM AUTHOR]

The assessment of root canal curvature is essential for clinical and research purposes. This systematic review presents an overview of the published techniques for the measurement of root canal curvature features using imaging and to provide a critique of their clinical application. A database search in PubMed, PubMed Central, Embase, Scopus, EBSCO Dentistry & Oral Sciences Source and Virtual Health Library was conducted, using appropriate key words to identify measurement methods for root canal curvatures. The search strategy retrieved 10594 records in total, and 31 records fulfilled the inclusion criteria. From 2D image acquisitions, eleven studies measured exclusively the angle of curvature, an additional thirteen measured other curvature features (level, height, radius, length and shape). Seven reports described methods from 3D imaging (CBCT, μCT). Root canal curvatures should be measured, for clinical proposes, to facilitate endodontic treatment planning, and in research, to reduce the risk of selection bias. This review has revealed that there are many methods described in the literature; however, no consensus exists on which method should be used. Some of the methodologies have potential clinical translation, whereas others are suitable for research purpose only, as they require a specific software or radiographic exposure in the mesiodistal direction. [ABSTRACT FROM AUTHOR]