Convex Boundary Research Articles

Peak sets in real-analytic convex boundaries

Throughtout this paper we will denote by D a smoothly bounded domain in C with defining function r. We denote by A∞(D) the set of holomorphic functions in D which have C∞-extension to D. First, we give the necessary definitions and notations needed in this paper. A closed subset K ⊂ ∂D is a peak set for A∞(D) if there exists a function f ∈ A∞(D) so that f = 1 on K and |f | 0 on D \K. Such a function g is called a strong support function for K. We denote by Tp(M) the real tangent space to a smooth manifold M at the point p ∈ M. For a point p ∈ M , the complex tangent space of M at p denoted by T C p (M) is the maximal complex subspace of Tp(M), of complex dimension n− 1 if M = ∂D. A C∞-submanifold M ⊆ ∂D is integral at p ∈M if Tp(M) ⊆ T C p (∂D). M is an integral manifold if it is integral at each point p ∈M. A C∞-submanifold M ⊂ ∂D is totally real if T C p (M) = {0} for every p ∈M. We denote by w(∂D) the set of weakly pseudoconvex boundary points. For p ∈ ∂D, we let Np denote the null space in T C p (∂D) of the Levi form at

On matrices satisfying a maximum principle with respect to a cone

A matrix equation A x = y is considered in the space R n that is ordered by a cone K. In case K = R n + it is known that a matrix A is said to satisfy the maximum principle if A is invertible and for each y ∈ R n + ⧹ {0} the solution x belongs to R n + and is such that x i = max 1 ⩽ k ⩽ n x k and y i > 0 for some i. This concept is generalized to finitely generated and to circular cones K ⊂ R n . This is achieved by “evaluating” x and y with the aid of elements of a given base for the polar cone K ∘ of K. The maximum principle is characterized geometrically by means of the behavior under A −1 of convex boundary parts of a base for K. A weighted maximum principle is investigated and an infinite dimensional example is indicated.

High-frequency surface wave excitation at a curved impedence boundary

We consider the situation when an arbitrary open, convex boundary, subject to an impedance condition, is insonified by a high–frequency time–harmonic plane wave. In particular, the inner diffraction structure close to the point where a propagating surface mode is excited is examined in detail, using the methods of multiple scales and asymptotic matching. This provides a complete description of the excitation process and of the outer ray form of the radiative surface mode for an abitrary, convex body. The calculation shows clearly the ideas behind the methodology and demonstrates how the ‘launching’ (or diffraction) coefficient for the amplitude of the surface mode can be calculated. It also helps to bring to the fore aspects of 0 (1) amplitude variations away from the point of excitiation which can be interpreted as an 0 (1/ k ) correction to the phase of the radiated field.

A stochastic boundary element solution applied to groundwater flow

A stochastic boundary element solution for two dimensional, transient groundwater flow in confined aquifers with random diffusivity is presented in this work. In particular, the fundamental solutions for the diffusion equation with a random coefficient are derived using the perturbation method. The resulting expressions for both piezometric head and flow solutions are validated through Monte-Carlo simulations. Subsequently, perturbations are again used for building a boundary integral equation-based solution for the covariance matrix of the response to arbitrarily posed boundary-value problems, with the mean solution being identical to the deterministic one due to the second order expansion used. The entire methodology is defined in the Laplace transform domain, and an efficient inversion scheme is employed for capturing the transient response in terms of a mean solution plus a variance. A series of examples serve to illustrate the methodology and the focus is on simple aquifers with parallel, concave and convex boundaries under piezometric heads with prescribed temporal variations which are arbitrary functions of time. It is shown that the medium stochasticity affects transient flow through the aquifer but has no effect under quasi-static conditions.

Analytic regularity for the Bergman kernel

Let Ω⊂ℝ 2 be a bounded, convex and open set with real analytic boundary. Let T Ω ⊂ℂ 2 be the tube with base Ω, and let ℬ be the Bergman kernel of T Ω . If Ω is strongly convex, then ℬ is analytic away from the boundary diagonal. In the weakly convex case this is no longer true. In this situation, we relate the off diagonal points where analyticity fails to the Trèves curves. These curves are symplectic invariants which are determined by the CR structure of the boundary of T Ω . Note that Trèves curves exist only when Ω has at least one weakly convex boundary point.

Semiclassical calculations of conductance in ballistic open quantum dots

Conductance of a ballistic quantum dot with convex boundary and two leads in arbitrary orientations is considered. The semiclassical conductance formula of Jalabert, Baranger, and Stone ( Phys. Rev. Lett., 1990, 65, 2442) [1] for collinear-lead dots is generalized to arbitrary lead orientation. The monodromy matrix for a convex billiard is derived, based on which a computation procedure is developed that allows for explicit numerical calculations of various quantities needed in the formula. This computation technique is applied to a circular billiard with three different lead orientations. Coherent backscattering as well as weak localization effects are obtained. Fluctuations in the reflection coefficients as a function of Fermi energy show high degree of regularities and strong autocorrelations reflecting the underlying regular classical dynamics.

A gradient estimate on a manifold with convex boundary

We present a simple probability approach for establishing a gradient estimate for a solution of an elliptic equation on a compact Riemannian manifold with convex boundary, or on a noncompact complete manifold. Our method can also be applied to derive a similar gradient estimate for a nonlinear parabolic equation, and an abstract gradient estimate for a Markov semigroup.

Interval Estimation of Eigenvalue Problem Based on Finite Element Sensitivity Analysis and Convex Model.

A formulation is presented for interval analysis of eigenvalue problem on the basis of the finite element sensitivity analysis and representation of uncertainty involved in a structural system by a convex model. The first-order approximation obtained by the sensitivity analysis is employed to express the response change due to the uncertainty that is assumed to be confined in a convex hull. The maximum and minimum of responses, by which the interval is bracketed, are searched on the convex boundary by the Lagrange multiplier method. The validity of the present formulation is demonstrated by a numerical example of the axial buckling load of an elastic straight column in the case of spatially uncertain Young's modulus.

Periodic orbits on Riemannian manifolds with convex boundary

We look for $T$-periodic solutions on a convex Riemannian manifold $\mathcal{M}$ of the differential equation <p align="center"> $D_s\dot x(s) + \nabla V_x(x(s),s) = 0$ <p align="left" class="times"> where $D_s\dot x(s)$ is the covariant derivative of $\dot x(s)$, $V$ is a $\mathcal{C}^2$ real function on $\mathcal{M}\times \mathbf{R}$, $T$-periodic in $s$. The manifold is allowed to be noncompact and to have boundary, so the action integral associated to the equation does not satisfy the Palais-Smale compactness condition. We overcome this problem under a assumption on the sectional curvature of $\mathcal{M}$ which allows to control the Morse index of the critical points of $f$ at "infinity". If $\mathcal{M}$ has a "rich" topology it is proved that there exist infinitely many periodic solutions.

Interval Estimation of Eigenvalue Problem Based on Finite Element Sensitivity Analysis and Convex Model.

A formulation is presented for interval analysis of eigenvalue problems on the basis of the finite element sensitivity analysis and the representation of uncertainties involved in a structural system by a convex model. The first-order approximation is employed to express the response change due to the uncertainties that are assumed to be confined in a convex hull by means of the sensitivity analysis. The maximum and minimum of responses, by which the interval is bracketed, are searched on the convex boundary by the Lagrange multiplier method. The validity of the present formulation is demonstrated by a numerical example of the axial buckling load of an elastic straight column in the case of uncertain Young's modulus.

Morse theory for trajectories of Lagrangian systems on Riemannian manifolds with convex boundary

In this paper we relate the set of the orbits of a second-order Lagrangian systems joining two points on an open set with convex boundary of a Riemannian manifold with the topological structure of the open set. Such relations are obtained by developing a Morse Theory for the action integral of the Lagrangian system. Because of the presence of the boundary, the action integral does not satisfy the Palais-Smale condition. We perturb the action integral with a family of smooth functionals, satisfying the Palais-Smale condition. The Morse Relations for the action integral are obtained as limit of the Morse Relations of the perturbing functionals. A relation between the Morse index and the energy of the orbits as critical points of the action integral is obtained.

Shape reconstruction of an impenetrable scattering body via the Rayleigh hypothesis

This work deals with the determination of the shape of a generally non-circular impenetrable cylinder from the way it scatters incident sound. A complete family (of generally non-orthogonal functions) representation of the scattered field is employed to match the total measured field. The data equation and state equation, derived from the Rayleigh hypothesis, are grouped into a single nonlinear cost functional which is minimized by means of the modified Levenberg - Marquardt algorithm to obtain the parametric equation of the boundary of the body in the cross section plane. Numerical examples of the results of the inversion scheme are given for cylinders with both convex and non-convex boundaries illuminated by a plane wave with frequency or angle-of-incidence diversity. Potential applications include robotics (artificial vision), non-destructive evaluation and medical imagery.

The influence of interacting strain rates on turbulence in convex boundary layers

A turbulent boundary layer subjected to multiple, additional strain rates, namely convex curvature coupled with zero, favorable, and adverse streamwise pressure gradients (ZPG, FPG, and APG) was investigated experimentally. The Reynolds stresses were suppressed, with respect to flat plate values, due primarily to the effects of strong convex curvature (δ0/R≊0.10). Combined with curvature, the FPG reduced the strength of the wake component, resulted in a greater suppression of the fluctuating velocity components, and a reduction of the primary Reynolds shear stress relative to the ZPG curved case, whereas the APG counteracted the stabilizing curvature effect. The additional influence of streamwise pressure gradients on the normal stresses is most apparent in the outer part of the boundary layer (y+≥100). However, the primary Reynolds shear stress is impacted by the additional strain rates throughout most of the boundary layer, especially for the strong pressure gradients. Using the uv2 quadrant burst detection method, including grouping of ejections, the normalized mean burst period in the convex boundary layer was found to increase in the presence of FPG and to decrease for APG. The time between bursts increased locally (by as much as 75%) in regions of locally strong acceleration. As the pressure gradient was relaxed to zero downstream, the bursting frequency in the case with stronger initial acceleration remained suppressed compared to that with milder initial acceleration, indicating a strain-rate history dependence.

Single-projection radiography for noncircular symmetries: Generalization of the Abel transform method

We present a new method which extends the use of the single projection radiographic Abel method, hitherto applicable only to objects of circular and elliptical cross sections, to objects having general, noncircular symmetries. This is done by developing a new integral equation that is similar in applications to Abel’s equation, and includes it as a special case. The use of the new equation is discussed for objects having a smooth and convex cross-section boundary (e.g., elliptic), a piecewise smooth convex boundary (e.g., bi-parabolic), and a boundary with regions of zero curvature (e.g., polygons). Specific examples are given for each of these three classes, and analytic inverses are calculated for these cases. Also, numerical inversion of the integral equation is given, showing satisfactory results. We show that in contrast to Abel’s equation in many cases the kernel of the integral equation is non-singular. Consequently, fairly simple inversion techniques are sufficient. Finally, the azimuthal variation of the transmitted intensity is employed to provide a convenient and fast nondestructive evaluation test of the deviation of the radiographed object from a prescribed symmetry.

A two-parameter study of the extent of chaos in a billiard system.

The billiard system of Benettin and Strelcyn [Phys. Rev. A 17, 773-785 (1978)] is generalized to a two-parameter family of different shapes. Its boundaries are composed of circular segments. The family includes the integrable limit of a circular boundary, convex boundaries of various shapes with mixed dynamics, stadiums, and a variety of nonconvex boundaries, partially with ergodic behavior. The extent of chaos has been measured in two ways: (i) in terms of phase space volume occupied by the main chaotic band; and (ii) in terms of the Lyapunov exponent of that same region. The results are represented as a kind of phase diagram of chaos. We observe complex regularities, related to the bifurcation scheme of the most prominent resonances. A detailed stability analysis of these resonances up to period six explains most of these features. The phenomenon of breathing chaos [Nonlinearity 3, 45-67 (1990)]-that is, the nonmonotonicity of the amount of chaos as a function of the parameters-observed earlier in a one-parameter study of the gravitational wedge billiard, is part of the picture, giving support to the conjecture that this is a fairly common global scenario. (c) 1996 American Institute of Physics.

A boundary value problem for Hermitian harmonic maps and applications

We study the existence and uniqueness problems for Hermitian harmonic maps from Hermitian manifolds with boundary to Riemannian manifolds of nonpositive sectional curvature and with convex boundary. The complex analyticity of such maps and the related rigidity problems are also investigated.

Shape reconstruction using diffracted waves and canonical solutions

This work deals with the determination of the shape of a generally-non-circular impenetrable cylinder from the way it scatters incident sound. A complete family (of generally non-orthogonal functions) representation of the scattered field is employed to match the total measured field. The resolution of the direct problem during the inversion is bypassed by assuming a priori that the coefficients in the field representation are locally those of an impenetrable circular cylinder. These coefficients are known explicitly to within a single parameter which is determined by resolution of a nonlinear equation. This parameter is none other than the length of the position vector joining the origin to the given point on the boundary of the cylinder, so that by varying the locations of the field measurement point and boundary point one generates a discrete form of the polar coordinate parametric equation of the boundary. Numerical examples of the results of the inversion scheme are given for cylinders with both convex (circular, elliptical) and non-convex boundaries.

Approximations for stochastic differential equations with reflecting convex boundaries

We consider convergence of a recursive projection scheme for a stochastic differential equation reflecting on the boundary of a convex domain G. If G satisfies Condition (B) in Tanaka (1979), we obtain mean square convergence, pointwise, with the rate O((δ log 1 δ ) 1 2 ) , and if G is a convex polyhedron we obtain mean square convergence, uniformly on compacts, with the rate O(δ log 1 δ ) . An application is given for stochastic differential equations with hysteretic components.

On a maximum principle for inverse monotone matrices

Let γ = ( γ1, …, γn) be a given vector with positive coordinates. A matrix A is said to satisfy the γ-maximum principle (γMP) if Ax = y, y ≥ 0 imply x ≥ 0 and max 1≤i≤n γ iχ i = max i ∈ N = (y) γ iχ i where N +) ( y) is the set of indices such that y is positive. For an invertible matrix A with positive inverse the γMP is characterized geometrically by means of the behavior under A −1 of convex boundary parts of the simplex generated in ( R n + by permissible multiples of the unit coordinate vectors. Some sufficient conditions and applications to M-matrices are given.

Convexity of the ideal boundary for complete open surfaces

For complete open surfaces admitting total curvature, we define several kinds of convexity for the ideal boundary, and provide examples of each of them. We also prove that a surface with most strongly convex ideal boundary is in fact a generalization of a Hadamard manifold in the sense that the ideal boundary consists entirely of Busemann functions.