Convex Boundary Research Articles

Invariant distributions and the geodesic ray transform

We establish an equivalence principle between the solenoidal injectivity of the geodesic ray transform acting on symmetric $m$-tensors and the existence of invariant distributions or smooth first integrals with prescribed projection over the set of solenoidal $m$-tensors. We work with compact simple manifolds, but several of our results apply to non-trapping manifolds with strictly convex boundary.

Monte Carlo study on abnormal growth of Goss grains in Fe-3%Si steel induced by second-phase particles

The selective abnormal growth of Goss grains in magnetic sheets of Fe-3%Si (grade Hi-B) induced by second-phase particles (AlN and MnS) was studied using a modified Monte Carlo Potts model. The starting microstructures for the simulations were generated from electron backscatter diffraction (EBSD) orientation imaging maps of recrystallized samples. In the simulation, second-phase particles were assumed to be randomly distributed in the initial microstructures and the Zener drag effect of particles on Goss grain boundaries was assumed to be selectively invalid because of the unique properties of Goss grain boundaries. The simulation results suggest that normal growth of the matrix grains stagnates because of the pinning effect of particles on their boundaries. During the onset of abnormal grain growth, some Goss grains with concave boundaries in the initial microstructure grow fast abnormally and other Goss grains with convex boundaries shrink and eventually disappear.

Marked boundary rigidity for surfaces

We show that, on an oriented compact surface, two sufficiently $C^{2}$-close Riemannian metrics with strictly convex boundary, no conjugate points, hyperbolic trapped set for their geodesic flows and the same marked boundary distance are isometric via a diffeomorphism that fixes the boundary. We also prove that the same conclusion holds on a compact surface for any two negatively curved Riemannian metrics with strictly convex boundary and the same marked boundary distance, extending a result of Croke and Otal.

Metrics with nonnegative Ricci curvature on convex three-manifolds

We prove that the space of smooth Riemannian metrics on the three-ball with non-negative Ricci curvature and strictly convex boundary is path connected; and, moreover, that the associated moduli space (i.e., modulo orientation-preserving diffeomorphisms of the three-ball) is contractible. As an application, using results of Maximo, Nunes, and Smith [MNS13], we show the existence of properly embedded free boundary minimal annulus on any three-ball with non-negative Ricci curvature and strictly convex boundary.

Lens rigidity for manifolds with hyperbolic trapped sets

For a Riemannian manifold ( M , g ) (M,g) with strictly convex boundary ∂ M \partial M , the lens data consist of the set of lengths of geodesics γ \gamma with end points on ∂ M \partial M , together with their end points ( x − , x + ) ∈ ∂ M × ∂ M (x_-,x_+)\in \partial M\times \partial M and tangent exit vectors ( v − , v + ) ∈ T x − M × T x + M (v_-,v_+)\in T_{x_-} M\times T_{x_+} M . We show deformation lens rigidity for such manifolds with a hyperbolic trapped set and no conjugate points. This class contains all manifolds with negative curvature and strictly convex boundary, including those with non-trivial topology and trapped geodesics. For the same class of manifolds in dimension 2 2 , we prove that the set of end points and exit vectors of geodesics (i.e., the scattering data) determines the Riemann surface up to conformal diffeomorphism.

On the Limiting Markov Process of Energy Exchanges in a Rarely Interacting Ball-Piston Gas

We analyse the process of energy exchanges generated by the elastic collisions between a point-particle, confined to a two-dimensional cell with convex boundaries, and a `piston', i.e. a line-segment, which moves back and forth along a one-dimensional interval partially intersecting the cell. This model can be considered as the elementary building block of a spatially extended high-dimensional billiard modeling heat transport in a class of hybrid materials exhibiting the kinetics of gases and spatial structure of solids. Using heuristic arguments and numerical analysis, we argue that, in a regime of rare interactions, the billiard process converges to a Markov jump process for the energy exchanges and obtain the expression of its generator.

Short-wave instability of an elastic plate in supersonic flow in the presence of the boundary layer

Panel flutter is a dangerous aeroelastic instability of the skin panels of supersonic flight vehicles. Though the linear stability of panels in uniform flow has been studied in detail, the influence of the boundary layer is still an open question. Most studies of panel flutter in the presence of the boundary layer are devoted to the ($1/7$)th-power velocity law and yield a stabilising effect of the boundary layer. Recently, Vedeneev (J. Fluid Mech., vol. 736, 2013, pp. 216–249) considered arbitrary velocity and temperature profiles and showed that, for a generalised convex boundary layer profile, a decrease of the growth rates of ‘supersonic’ perturbations (responsible for single-mode panel flutter) is accompanied by destabilisation of ‘subsonic’ perturbations that are neutral in uniform flow. However, this result is not self-consistent, as the long-wave expansion for solutions of the Rayleigh equation was used, whereas subsonic perturbations, generally speaking, cannot be considered as long waves. More surprising results are obtained for the boundary layer profile with a generalised inflection point, where the effect of the layer is destabilising even for ‘supersonic’ perturbations, and such waves can also have short lengths. In order to overcome this inconsistency, in this paper, we solve the Rayleigh equation numerically and investigate the stability of short-wave perturbation of the elastic plate in the presence of the boundary layer. As before, two problem formulations are investigated. First, we study running waves in an infinite plate. Second, we analyse eigenmodes of the plate of large finite length and use Kulikovskii’s global instability criterion. Based on the results of calculations, we confirm that the effect of the boundary layer with a generalised inflection point can be essentially destabilising. On the other hand, for generalised convex boundary layers, calculations show that, unlike the prediction of the long-wave approximation, the finite plate is fully stabilised for sufficiently thick boundary layers.

Stereo reconstruction using top-down cues

We present a framework which allows standard stereo reconstruction to be unified with a wide range of classic top-down cues from urban scene understanding. The resulting algorithm is analogous to the human visual system where conflicting interpretations of the scene due to ambiguous data can be resolved based on a higher level understanding of urban environments. The cues which are reformulated within the framework include: recognising common arrangements of surface normals and semantic edges (e.g. concave, convex and occlusion boundaries), recognising connected or coplanar structures such as walls, and recognising collinear edges (which are common on repetitive structures such as windows). Recognition of these common configurations has only recently become feasible, thanks to the emergence of large-scale reconstruction datasets. To demonstrate the importance and generality of scene understanding during stereo-reconstruction, the proposed approach is integrated with 3 different state-of-the-art techniques for bottom-up stereo reconstruction. The use of high-level cues is shown to improve performance by up to 15% on the Middlebury 2014 and KITTI datasets. We further evaluate the technique using the recently proposed HCI stereo metrics, finding significant improvements in the quality of depth discontinuities, planar surfaces and thin structures.

Gradient estimates for nonlinear elliptic equations with vanishing Neumann data in quasiconvex domains

We study the conormal derivative problem for an elliptic equation of p-Laplacian type with discontinuous coefficients in a non-smooth domain in order to look for the minimal assumptions necessary to have the nonlinear Calderón–Zygmund theory for such problem. Under the assumptions that the nonlinear operator is sufficiently close to the p-Laplacian operator in BMO semi-norm and the boundary of the domain can be locally approximated by the convex boundary, we prove that both the gradient and the associated nonhomogeneous term belong to the same Lq space for every q∈[p,∞). As far as the domain is concerned, our regularity assumption on the boundary is weaker than any other one reported in this direction.

Advancing Edge Speeds of Epithelial Monolayers Depend on Their Initial Confining Geometry

Collective cell migrations are essential in several physiological processes and are driven by both chemical and mechanical cues. The roles of substrate stiffness and confinement on collective migrations have been investigated in recent years, however few studies have addressed how geometric shapes influence collective cell migrations. Here, we address the hypothesis that the relative position of a cell within the confinement influences its motility. Monolayers of two types of epithelial cells—MCF7, a breast epithelial cancer cell line, and MDCK, a control epithelial cell line—were confined within circular, square, and cross-shaped stencils and their migration velocities were quantified upon release of the constraint using particle image velocimetry. The choice of stencil geometry allowed us to investigate individual cell motility within convex, straight and concave boundaries. Cells located in sharp, convex boundaries migrated at slower rates than those in concave or straight edges in both cell types. The overall cluster migration occurred in three phases: an initial linear increase with time, followed by a plateau region and a subsequent decrease in cluster speeds. An acto-myosin contractile ring, present in the MDCK but absent in MCF7 monolayer, was a prominent feature in the emergence of leader cells from the MDCK clusters which occurred every ~125 μm from the vertex of the cross. Further, coordinated cell movements displayed vorticity patterns in MDCK which were absent in MCF7 clusters. We also used cytoskeletal inhibitors to show the importance of acto-myosin bounding cables in collective migrations through translation of local movements to create long range coordinated movements and the creation of leader cells within ensembles. To our knowledge, this is the first demonstration of how bounding shapes influence long-term migratory behaviours of epithelial cell monolayers. These results are important for tissue engineering and may also enhance our understanding of cell movements during developmental patterning and cancer metastasis.

Relative symplectic caps, 4-genus and fibered knots

We prove relative versions of the symplectic capping theorem and sufficiency of Giroux’s criterion for Stein fillability and use these to study the 4-genus of knots. More precisely, suppose we have a symplectic 4-manifold X with convex boundary and a symplectic surface Σ in X such that ∂Σ is a transverse knot in ∂ X. In this paper, we prove that there is a closed symplectic 4-manifold Y with a closed symplectic surface S such that (X,Σ) embeds into (Y,S) symplectically. As a consequence we obtain a relative version of the symplectic Thom conjecture. We also prove a relative version of the sufficiency part of Giroux’s criterion for Stein fillability, namely, we show that a fibered knot whose mondoromy is a product of positive Dehn twists bounds a symplectic surface in a Stein filling. We use this to study 4-genus of fibered knots in $\mathbb {S}^{3} $ . Further, we give a criterion for quasipositive fibered knots to be strongly quasipositive.

Estimates of the first Steklov eigenvalue of properly embedded minimal hypersurfaces with free boundary

We consider a properly embedded minimal hypersurfacewith free boundary in a compact n-dimensional Riemannian manifold M be with nonnegative Ricci curvature and strictly convex boundary. Here, we obtain a new estimate from below for the first nonzero Steklov eigenvalue.

Local convex hull support and boundary estimation

In this paper we introduce a new estimator for the support of a multivariate density. It is defined as a union of convex hulls of observations contained in balls of fixed radius. We study the asymptotic behavior of this “local convex hull” for the estimation of the support and its boundary. When the support is smooth enough, the proposed estimator is proved to be, eventually almost surely, homeomorphic to the support. Numerical simulations on simulated data illustrate the performance of our estimator.

Hyperbolization of cusps with convex boundary

We prove that for every metric on the torus with curvature bounded from below by −1 in the sense of Alexandrov there exists a hyperbolic cusp with convex boundary such that the induced metric on the boundary is the given metric. The proof is by polyhedral approximation. This was the last open case of a general theorem: every metric with curvature bounded from below on a compact surface is isometric to a convex surface in a 3-dimensional space form.

Lens rigidity with trapped geodesics in two dimensions

We consider the scattering and lens rigidity of compact surfaces with boundary that have a trapped geodesic. In particular we show that the flat cylinder and the flat M\obius strip are determined by their lens data. We also see by example that the flat M\obius strip is not determined by it's scattering data. We then consider the case of negatively curved cylinders with convex boundary and show that they are lens rigid.

A CONTACT INVARIANT IN SUTURED MONOPOLE HOMOLOGY

We define an invariant of contact 3-manifolds with convex boundary using Kronheimer and Mrowka’s sutured monopole Floer homology theory ($SHM$). Our invariant can be viewed as a generalization of Kronheimer and Mrowka’s contact invariant for closed contact 3-manifolds and as the monopole Floer analogue of Honda, Kazez, and Matić’s contact invariant in sutured Heegaard Floer homology ($SFH$). In the process of defining our invariant, we construct maps on $SHM$ associated to contact handle attachments, analogous to those defined by Honda, Kazez, and Matić in $SFH$. We use these maps to establish a bypass exact triangle in $SHM$ analogous to Honda’s in $SFH$. This paper also provides the topological basis for the construction of similar gluing maps in sutured instanton Floer homology, which are used in Baldwin and Sivek [Selecta Math. (N.S.), 22(2) (2016), 939–978] to define a contact invariant in the instanton Floer setting.

Adsorption of water and methanol on highly graphitized thermal carbon black: The effects of functional group and temperature on the isosteric heat at low loadings

Isosteric heats of water and methanol adsorption on highly graphitized thermal carbon black have been studied using grand canonical Monte Carlo simulation to investigate the effects of the concentration of functional groups and temperature on the isosteric heats observed experimentally. The heat at zero loading is associated with the interaction between an adsorbate and the functional group. The variation in the heat with loading is due to a combination two effects: (1) Saturation of the functional groups by adsorbate molecules. (2) Adsorption at the convex boundary of the clusters, resulting in a lower number of neighbours for each new molecule adsorbed onto the clusters compared to the bulk liquid. The isosteric heat versus loading reaches a minimum and then increases again as the number of interactions with neighbouring molecules increases, and finally reaches the heat of liquefaction when the very large clusters approach liquid-like behaviour.

Reconstruction formulas for X-ray transforms in negative curvature

We give reconstruction formulas inverting the geodesic X-ray transform over functions (call it I_0) and solenoidal vector fields on surfaces with negative curvature and strictly convex boundary. These formulas generalize the Pestov-Uhlmann formulas in [PeUh] (established for simple surfaces) to cases allowing geodesics with infinite length on surfaces with trapping. Such formulas take the form of Fredholm equations, where the analysis of error operators requires deriving new estimates for the normal operator \Pi_0 = I_0^*I_0. Numerical examples are provided at the end.

Voronoi based discrete least squares meshless method for heat conduction simulation in highly irregular geometries

A new technique is used in Discrete Least Square Meshfree(DLSM) method to remove the common existing deficiencies of meshfree methods in handling of the problems containing cracks or concave boundaries. An enhanced Discrete Least Squares Meshless method named as VDLSM(Voronoi based Discrete Least Squares Meshless) is developed in order to solve the steady-state heat conduction problem in irregular solid domains including concave boundaries or cracks. Existing meshless methods cannot estimate precisely the required unknowns in the vicinity of the above mentioned boundaries. Conducted researches are limited to domains with regular convex boundaries. To this end, the advantages of the Voronoi tessellation algorithm are implemented. The support domains of the sampling points are determined using a Voronoi tessellation algorithm. For the weight functions, a cubic spline polynomial is used based on a normalized distance variable which can provide a high degree of smoothness near those mentioned above discontinuities. Finally, Moving Least Squares(MLS) shape functions are constructed using a varitional method. This straight-forward scheme can properly estimate the unknowns(in this particular study, the temperatures at the nodal points) near and on the crack faces, crack tip or concave boundaries without need to extra backward corrective procedures, i.e. the iterative calculations for modifying the shape functions of the nodes located near or on these types of the complex boundaries. The accuracy and efficiency of the presented method are investigated by analyzing four particular examples. Obtained results from VDLSM are compared with the available analytical results or with the results of the well-known Finite Elements Method(FEM) when an analytical solution is not available. By comparisons, it is revealed that the proposed technique gives high accuracy for the solution of the steady-state heat conduction problems within cracked domains or domains with concave boundaries and at the same time possesses a high convergence rate which its accuracy is not sensitive to the arrangement of the nodal points. The novelty of this paper is the use of Voronoi concept in determining the weight functions used in the formulation of the MLS type shape functions.

Instanton Floer homology and contact structures

We define an invariant of contact 3-manifolds with convex boundary using Kronheimer and Mrowka's sutured instanton Floer homology theory. To the best of our knowledge, this is the first invariant of contact manifolds -- with or without boundary -- defined in the instanton Floer setting. We prove that our invariant vanishes for overtwisted contact structures and is nonzero for contact manifolds with boundary which embed into Stein fillable contact manifolds. Moreover, we propose a strategy by which our contact invariant might be used to relate the fundamental group of a closed contact 3-manifold to properties of its Stein fillings. Our construction is inspired by a reformulation of a similar invariant in the monopole Floer setting defined by the authors in [1].