Convex Boundary Research Articles

Gradient estimates of Poisson equations on Riemannian manifolds and applications

For the reflected diffusion generated by L = Δ − ∇ V ⋅ ∇ on a connected and complete Riemannian manifold M with empty or convex boundary, we establish some sharp estimates of sup x ∈ M | ∇ G | ( x ) of the Poisson equation − L G = g in terms of the dimension, the diameter and the lower bound of curvature. Applications to transportation-information inequality, to Cheeger's isoperimetric inequality and to Gaussian concentration inequality are given. Several examples are provided.

An algorithm for constructing star-shaped drawings of plane graphs

A straight-line planar drawing of a plane graph is called a convex drawing if every facial cycle is drawn as a convex polygon. Convex drawings of graphs is a well-established aesthetic in graph drawing, however not all planar graphs admit a convex drawing. Tutte [W.T. Tutte, Convex representations of graphs, Proc. of London Math. Soc. 10 (3) (1960) 304–320] showed that every triconnected plane graph admits a convex drawing for any given boundary drawn as a convex polygon. Thomassen [C. Thomassen, Plane representations of graphs, in: Progress in Graph Theory, Academic Press, 1984, pp. 43–69] gave a necessary and sufficient condition for a biconnected plane graph with a prescribed convex boundary to have a convex drawing. In this paper, we initiate a new notion of star-shaped drawing of a plane graph as a straight-line planar drawing such that each inner facial cycle is drawn as a star-shaped polygon, and the outer facial cycle is drawn as a convex polygon. A star-shaped drawing is a natural extension of a convex drawing, and a new aesthetic criteria for drawing planar graphs in a convex way as much as possible. We give a sufficient condition for a given set A of corners of a plane graph to admit a star-shaped drawing whose concave corners are given by the corners in A, and present a linear time algorithm for constructing such a star-shaped drawing.

Local lens rigidity with incomplete data for a class of non-simple Riemannian manifolds

Let σ be the scattering relation on a compact Riemannian manifold M with nonnecessarily convex boundary, that maps initial points of geodesic rays on the boundary and initial directions to the outgoing point on the boundary and the outgoing direction. Let ` be the length of that geodesic ray. We study the question of whether the metric g is uniquely determined, up to an isometry, by knowledge of σ and ` restricted on some subset D. We allow possible conjugate points but we assume that the conormal bundle of the geodesics issued from D covers T ∗M ; and that those geodesics have no conjugate points. Under an additional topological assumption, we prove that σ and ` restricted to D uniquely recover an isometric copy of g locally near generic metrics, and in particular, near real analytic ones.

SPH particle boundary forces for arbitrary boundaries

This paper is concerned with approximating arbitrarily shaped boundaries in SPH simulations. We model the boundaries by means of boundary particles which exert forces on a fluid. We show that, when these forces are chosen correctly, and the boundary particle spacing is a factor of 2 (or more) less than the fluid particle spacing, the total boundary force on a fluid SPH particle is perpendicular to boundaries with negligible error. Furthermore, the variation in the force as a fluid particle moves, while keeping a fixed distance from the boundary, is also negligible. The method works equally well for convex or concave boundaries. The new boundary forces simplify SPH algorithms and are superior to other methods for simulating complicated boundaries. We apply the new method to (a) the rise of a cylinder contained in a curved basin, (b) the spin down of a fluid in a cylinder, and (c) the oscillation of a cylinder inside a larger fixed cylinder. The results of the simulations are in good agreement with those obtained using other methods, but with the advantage that they are very simple to implement.

Estimate of the Hölder constant for solutions to m-Hessian equations

An a priori estimate for the Holder constant with exponent 0 < α < 1 is obtained for smooth solutions to m-Hessian equations in a closed domain with (m − 1)-convex boundary of class C 2. The Holder constant depends on the L p -norm, p > [n(n + 1)]/2, of the right-hand side of the equation, and the estimate remains valid for approximate solutions. Bibliography: 4 titles.

The contact invariant in sutured Floer homology

We describe an invariant of a contact 3-manifold with convex boundary as an element of Juhasz’s sutured Floer homology. Our invariant generalizes the contact invariant in Heegaard Floer homology in the closed case, due to Ozsvath and Szabo.

The Singularities of the Distance Function Near Convex Boundary Points

In an open bounded set Ω, we consider the distance function from ∂Ω associated to a Riemannian metric with C1,1 coefficients. Assuming that Ω is convex near a boundary point x0, we show that the distance function is differentiable at x0 if and only if there exists the tangent space to ∂Ω at x0. Furthermore, if the distance function is not differentiable at x0 then there exists a Lipschitz continuous curve, with initial point at x0, such that the distance function is not differentiable along such a curve.

Partial open book decompositions and the contact class in sutured floer homology

We demonstrate how to combinatorially calculate the EH-class of a compatible contact structure in the sutured Floer homology group of a balanced sutured three manifold which is associated to an abstract partial open book decomposition. As an application we show that every contact three manifold (closed or with convex boundary) can be obtained by gluing tight contact handlebodies whose EH-classes are nontrivial.

Local A Posteriori Error Estimates for Convex Boundary Control Problems

In this paper, we derive some improved a posteriori error estimates for finite element approximation of Neumann boundary control problems. We first establish local upper a posteriori error estimates for both the state and the control approximation of the general convex boundary control problems. We then derive local upper and lower a posteriori error estimates for a class of control problems that frequently appear in applications.

Towards the analysis of materials with strain gradient effects using ‐Natural Element Method

AbstractThe notion of generalized continua unifies several classes of continuum theories that account for size dependence arising due to the underlying microstructure of the material. In gradient continua, besides the first gradient also the higher gradients of displacements are taken in to account. This results in higher order boundary conditions and C1 continuity requirements. α‐Natural Element Method has been used to this effect for numerical modeling of gradient elasticity. The α‐–Natural Element Method allows the construction of models entirely in terms of nodes and also ensures the linear precision of the interpolant over convex and non convex boundaries. C1 natural neighbor interpolants are achieved by a simple transformation of the Farins interpolant, which are basically obtained by embedding Sibsons natural neighbor coordinates in Bernstein–Bezier surface representation of a cubic simplex. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Physical interpretation of spiralling-columnar convection in a rapidly rotating annulus with radial propagation properties of Rossby waves

To aid the physical understanding of spiralling-columnar convection emerging in rapidly rotating spheres and spherical shells, two-dimensional thermal convection in a rapidly rotating annulus is investigated through the radial propagation properties of topographic Rossby waves. Two kinds of the boundaries containing the fluid in the axial direction are considered: a convex type modelling a spherical geometry and a concave type for comparison. The linear stability of a basic state with no motion and uniformly unstable stratification is examined and spirally elongated structures of critical convection are obtained for small Prandtl numbers. An analysis of the energy budget shows that a part of the kinetic energy generated in the region with slightly inclined boundaries is dynamically transferred and dissipates through viscosity in the region with strongly inclined boundaries. This indicates that the Rossby waves propagate from the region with slightly inclined boundaries to the region with strongly inclined boundaries. It is presented that the appearance of a spiral structure corresponds to an increase of the local radial wavenumber of the Rossby waves propagating in the radial direction. The flow patterns obtained using the dispersion relation of the Rossby waves coincide with those of the tailing part of the spiral structure obtained numerically. As the Prandtl number increases, the Rossby waves barely propagate because of strong viscous dissipation, and the flow pattern is localized in the region with slightly inclined boundaries. For convex boundaries with unstable stratification concentrating near the outer boundary and concave boundaries with unstable stratification confined near the inner boundary, the flow patterns tilt in the direction inverse to the case of uniform unstable stratification. The tilting direction of the flow pattern is not determined by the curvature of the boundaries considered but instead by the radial propagation direction of the Rossby waves excited by thermal convection.

Three-manifolds of positive curvature and convex weakly umbilic boundary

In this paper we study a boundary value problem in manifolds with weakly umbilic boundary (the Second Fundamental form of the boundary is a constant multiple of the metric). We show that if we start with a metric of positive curvature and convex boundary (positive Second Fundamental form), the Ricci flow uniformizes the curvature. In the case of a metric with rotational symmetry, we indicate how to weaken the hypothesis to positive Ricci curvature and positive Second Fundamental form.

Optimal routing algorithms for rectilinear pin clusters in high-density multichip modules

As the circuit densities and transistor counts are increasing, the package routing problem is becoming more and more challenging. In this article, we study an important routing problem encountered in typical high-end MCM designs: routing within dense pin clusters. Pin clusters are often formed by pins that belong to the same functional unit or the same data bus, and can become bottlenecks in terms of overall routability. Typically, these clusters have irregular shapes, which can be approximated with rectilinear convex boundaries. Since such boundaries have often irregular shapes, a traditional escape routing algorithm may give unroutable solutions. In this article, we study how the positions of escape terminals on a convex boundary affect the overall routability. For this purpose, we propose a set of necessary and sufficient conditions to model routability outside a rectilinear convex boundary. Given an escape routing solution, we propose an optimal algorithm to select the maximal subset of nets that are routable outside the boundary. After that, we focus on an integrated approach to consider routability constraints (outside the boundary) during the actual escape routing algorithm. Here, we propose an optimal algorithm to find the best escape routing solution that satisfies all routability constraints. Our experiments demonstrate that we can reduce the number of layers by 17% on the average, by using this integrated methodology.

Perpetual American options in incomplete markets: the infinitely divisible case

We consider the exercise of a number of American options in an incomplete market. In this paper we are interested in the case where the options are infinitely divisible. We make the simplifying assumptions that the options have infinite maturity, and the holder has exponential utility. Our contribution is to solve this problem explicitly and we show that, except at the initial time when it may be advantageous to exercise a positive fraction of his holdings, it is never optimal for the holder to exercise a tranche of options. Instead, the process of option exercises is continuous; however, it is singular with respect to calendar time. Exercise takes place when the stock price reaches a convex boundary which we identify.

Parameterization of non-convex surface and its application in game engine

Parameterization is the basic principle of 3D games modeling and the crucial technique in surface fitting.A new algorithm for mesh parameterization was proposed.Attach a convex boundary for non-convex mesh and then parameterize the interior vertices based on mean value coordinate method.This algorithm was applied to the modeling module of game engine,and the experimental results demonstrate the feasibility and efficiency of the method.

Integral Geometry of Tensor Fields on a Class of Non-Simple Riemannian Manifolds

We study the geodesic X-ray transform IΓ of tensor fields on a compact Riemannian manifold M with non-necessarily convex boundary and with possible conjugate points. We assume that IΓ is known for geodesics belonging to an open set Γ with endpoints on the boundary. We prove generic s-injectivity and a stability estimate under some topological assumptions and under the condition that for any ( x, ξ ) ∈ T*M , there is a geodesic in Γ through x normal to ξ without conjugate points.

Application of the natural element method to finite deformation inelastic problems in isotropic and fiber-reinforced biological soft tissues

In this paper, the application of the natural element method (NEM) to solve inelastic finite deformation problems in isotropic and fiber-reinforced materials is presented. As most meshless methods, the NEM does not require an explicit connectivity definition. Consequently, it is quite adequate to simulate large strain problems with important mesh distortions, reducing the need for remeshing and projection of results which becomes extremely important in three-dimensional problems. The NEM has also important advantages over other meshless methods, such as the interpolant character of the shape functions and the ability of exactly reproducing essential boundary conditions along convex boundaries. The α-NEM extension generalizes this behavior to non-convex boundaries. A fully three-dimensional finite-strain damage model for visco-hyperelastic fibrous soft tissue has been implemented to solve different problems. In order to show the performance of the constitutive model and its algorithmic counterpart some simple examples are included, comparing the obtained results with the ones obtained with a standard finite element approach. A more complex three-dimensional numerical application to ligament mechanics is also presented. This example clearly shows the important capabilities of the NEM in this kind of applications.

Hamilton–Jacobi Equations Arising from Boundary Control Problems with State Constraints

The analysis of a class of infinite-dimensional Hamilton-Jacobi-Bellman (HJB) equations is undertaken related to linear convex boundary control problems for PDEs with constraints on the state. A definition of generalized solution (namely, weak solution) of the HJB equation is provided (weak is the limit of strong, while strong is the limit of classical, as defined in [S. Faggian, Appl. Math. Optim., 51 (2005), pp. 123-162]). Consequently an existence and uniqueness result is provided for weak solutions of the HJB equation, as well as existence of an optimal control, being the limit of optimal controls of approximating problems. The study then describes an economic application to optimal investment with vintage capital, having positivity constraints on the capital.

Thermal instability in natural convection flow over a rotating convex surface subject to external magnetic field

This paper presents magnetic effect on the Goertler vortices in natural convection flow over a heated rotating convex surface. The onset position is characterized by the Grashof number, depends on the rotational Reynolds number, the Prandtl number, the Hartmann number, and the wave number. The buoyancy force, the Lorentz force, and the Coriolis force significantly affect the flow structure and heat transfer of the flow. However, centrifugal force has minor stabilizing effect on the flow. Negative rotation and external magnetic field stabilize the convex boundary layer flow. On the contrary, positive destabilizes the flow. The numerical data show agreement with the experimental results in the case of zero rotation.

Multiscale Active Contours

We propose a new multiscale image segmentation model, based on the active contour/snake model and the Polyakov action. The concept of scale, general issue in physics and signal processing, is introduced in the active contour model, which is a well-known image segmentation model that consists of evolving a contour in images toward the boundaries of objects. The Polyakov action, introduced in image processing by Sochen-Kimmel-Malladi in Sochen et al. (1998), provides an efficient mathematical framework to define a multiscale segmentation model because it generalizes the concept of harmonic maps embedded in higher-dimensional Riemannian manifolds such as multiscale images. Our multiscale segmentation model, unlike classical multiscale segmentations which work scale by scale to speed up the segmentation process, uses all scales simultaneously, i.e. the whole scale space, to introduce the geometry of multiscale images in the segmentation process. The extracted multiscale structures will be useful to efficiently improve the robustness and the performance of standard shape analysis techniques such as shape recognition and shape registration. Another advantage of our method is to use not only the Gaussian scale space but also many other multiscale spaces such as the Perona-Malik scale space, the curvature scale space or the Beltrami scale space. Finally, this multiscale segmentation technique is coupled with a multiscale edge detecting function based on the gradient vector flow model, which is able to extract convex and concave object boundaries independent of the initial condition. We apply our multiscale segmentation model on a synthetic image and a medical image.