L 1 andL ∞ Uniform convergence of a difference scheme for a semilinear singular perturbation problem

Summary A nonlinear difference scheme is given for solving a semilinear singularly perturbed two-point boundary value problem. Without any restriction on turning points, the solution of the scheme is shown to be first order accurate in the discreteL ¹ norm, uniformly in the perturbation parameter. When turning points are excluded, the scheme is first order accurate in the discreteL norm, uniformly in the perturbation parameter.Partly supported by the Arts Faculty Research Fund of University College, Cork     

Local solvability of a nonstationary leakage problem for an ideal incompressible fluid, 3

In this paper we prove the existence and uniqueness of solutions of the leakage problem for the Euler equations in bounded domain Ω C R³ with corners π/n, n = 2, 3… We consider the case where the tangent components of the vorticity vector are given on the part S₁ of the boundary where the fluid enters the domain. We prove the existence of an unique solution in the Sobolev space W_p^l(Ω), for arbitrary natural l and p > 1. The proof is divided on three parts: (1) the existence of solutions of the elliptic problem in the domain with corners where v – velocity vector, ω – vorticity vector and n is an unit outward vector normal to the boundary, (2) the existence of solutions of the following evolution problem for given velocity vector (3) the method of successive approximations, using solvability of problems (1) and (2).     

Primitive and mensurable hex-triangles

When the corners of a planar polygon P are restricted to lie in the set H of vertices of a tiling of the plane by hexagons of unit area, then very often the area of P is given by the Pick-type formula A(P)=b/4+i/2+c/12-1, where b and i count the number of points of H on the boundary P and in the interior of P respectively, and c is the boundary characteristic. We now characterize all primitive triangles for which this formula holds, and consider the magnitude of the error when it fails.     

Solvability of some overdetermined elliptic system in a domain with corners π/n

In the paper we prove the existence and uniqueness of solutions of the overdetermined elliptic system where b, ω are given functions, in a domain Ω C R³ with corners π/n, n = 2, 3, … The proof is divided on two steps, we construct a solution for the Laplace equation in a dihedral angle π/n, using the method of reflection and we get an estimate in the norms of the Sobolev spaces in some neighbourhood of the edge. In the dihedral angle system (A) reduces to the Dirichlet and Neumann problems for the Laplace equation. In the next step we prove the existence of solutions in the Sobolev spaces W_p^l(Ω) using the existence of generalized solutions of (A).     

Symplectic manifolds with contact type boundaries

Summary An example of a 4-dimensional symplectic manifold with disconnected boundary of contact type is constructed. A collection of other results about symplectic manifolds with contact-type boundaries are derived using the theory ofJ-holomorphic spheres. In particular, the following theorem of Eliashberg-Floer-McDuff is proved: if a neighbourhood of the boundary of (V, ) is symplectomorphic to a neighbourhood ofS ^2n–1 in standard Euclidean space, and if vanishes on all 2-spheres inV, thenV is diffeomorphic to the ballB ²ⁿ.Oblatum 19-III-1990Partially supported by NSF grant no: DMS 8803056     

Multiplicity and boundary behavior of quasiregular maps

We study the boundary behavior of a bounded quasiregular mapping f: Gⁿ. In the main results, Lindelöf-type problems are studied in connection with the local topological index i(x,f). The existence of certain types of limits at a given boundary point b G is shown. The assumptions involve local topological index of the mapping f on a given sequence of points approaching the boundary point b.The author was supported by the foundation Vilho, Yrjö ja Kalle Väisälän rahasto.     

On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles

Let ₁,..., _m bem simple Jordan curves in the plane, and letK ₁,...,K _m be their respective interior regions. It is shown that if each pair of curves _i , _j ,i j, intersect one another in at most two points, then the boundary ofK= _i ^=1m K _i contains at most max(2,6m – 12) intersection points of the curves ₁, and this bound cannot be improved. As a corollary, we obtain a similar upper bound for the number of points of local nonconvexity in the union ofm Minkowski sums of planar convex sets. Following a basic approach suggested by Lozano Perez and Wesley, this can be applied to planning a collision-free translational motion of a convex polygonB amidst several (convex) polygonal obstaclesA ₁,...,A _m . Assuming that the number of corners ofB is fixed, the algorithm presented here runs in timeO (n log² n), wheren is the total number of corners of theA _i 's.Work on this paper by the second author has been supported in part by a grant from the Bat-Sheva Fund at Israel. Work by the fourth author has been supported in part by a grant from the U.S.-Israeli Binational Science Foundation.     

Self-intersections of curves on surfaces

Let M be a compact orientable surface with nonempty boundary (x(M)<0) and fundamental group . Let be a geodesic on M (with a fixed hyperbolic structure), and let W be a (cyclically reduced) word in a fixed set of generators of which represents . In this paper, we give an algorithm to count the number of self-intersections of in terms of W, generalizing a result of Birman and Series, where an algorithm was given to decide if was simple. Some applications of the algorithm to surfaces with one boundary and the Markoff spectrum are also given.     

Bergman kernel and metric on non-smooth pseudoconvex domains

A stability theorem of the Bergman kernel and completeness of the Bergman metric have been proved on a type of non-smooth pseudoconvex domains defined in the following way:D = {z∈U|r(z)} <whereU is a neighbourhood of andr is a continuous plurisubharmonic function onU. A continuity principle of the Bergman Kernel for pseudoconvex domains with Lipschitz boundary is also given, which answers a problem of Boas.     

Martin Boundaries of Quasi-Sectorial Domains

Cranston and Salisbury have obtained an integral test for the existence of a maximal number of minimal Martin boundary points at 0 in certain domains (cf. [8]). This paper will extend the result as follows: let D R ^d be an open Greenian set (with respect to the Laplacian) consisting of n disjoint open connected cones with Lipschitz boundary and a subset of the boundary of these cones. Let be some local Kato measure supported by the boundary of the cones and consider the Schrödinger operator 1/2-µ. We will assume a boundary Harnack principle and give a sufficient integral criterion for the existence of exactly n minimal Martin boundary points at 0. In certain cases there is a necessary criterion, too. When the sufficient integral criterion holds we will give a necessary and a sufficient condition for the existence of a certain process related to the Schrödinger operator that connects two different admissible boundary points. In the paper of Cranston and Salisbury the case = 0, d = 2 is treated, but many of the arguments work as well in the general situation.     

On the mean width of random polytopes

Summary On the boundary of a d-dimensional convex body a probability distribution with a positive, continuous density function g is given. The convex hull of n points chosen independently according to g is a random polytope. The asymptotic behaviour (n) of the expected value of the mean width of the random polytope is determined.     

Total flux estimates for a finite element approximation of the Dirichlet problem using the boundary penalty method

Summary This paper considers a fully practical piecewise linear finite element approximation of the Dirichlet problem for a second order self-adjoint elliptic equation,Au=f, in a smooth region< ⁿ (n=2 or 3) by the boundary penalty method. Using an unfitted mesh; that is ^h , an approximation of with dist (, ^h )Ch ² is not in general a union of elements; and assuminguH ⁴ () we show that one can recover the total flux across a segment of the boundary of with an error ofO(h ²⁾. We use these results to study a fully practical piecewise linear finite element approximation of an elliptic equation by the boundary penalty method when the prescribed data on part of the boundary is the total flux.Supported by a SERC research studentship     

Legendre Gauss Spectral Collocation for the Helmholtz Equation on a Rectangle

A spectral collocation method with collocation at the Legendre Gauss points is discussed for solving the Helmholtz equation –u+(x,y)u=f(x,y) on a rectangle with the solution u subject to inhomogeneous Robin boundary conditions. The convergence analysis of the method is given in the case of u satisfying Dirichlet boundary conditions. A matrix decomposition algorithm is developed for the solution of the collocation problem in the case the coefficient (x,y) is a constant. This algorithm is then used in conjunction with the preconditioned conjugate gradient method for the solution of the spectral collocation problem with the variable coefficient (x,y).     

Compact perturbation of definite type spectra of self-adjoint quadratic operator pencils

Self-adjoint quadratic operator pencilsL()= ² A + B + C with a noninvertible leading operatorA are considered. In particular, a characterization of the spectral points of positive and of negative type ofL is given, and their behavior under a compact perturbation is studied. These results are applied to a pencil arising in magnetohydrodynamics.     

An exhaustion of locally symmetric spaces by compact submanifolds with corners

LetX be a Riemannian symmetric space of noncompact type and rank2 and let be a non-uniform, irreducible lattice. On the locally symmetric quotientV=/X we construct an exhaustion functionh:V[0,) whose sublevel sets {hs} are compact submanifolds ofV with corners. The top dimensional boundary faces of {hs} are parts of certain horospheres that join together at the corners. It can be shown that actually {hs} is a submanifold with corners isomorphic to the Borel-Serre compactification ofV.Oblatum 2-VIII-1993 & 19-XII-1994     

On the computation of multi-dimensional solution manifolds of parametrized equations

Summary A new algorithm is presented for computing vertices of a simplicial triangulation of thep-dimensional solution manifold of a parametrized equationF(x)=0, whereF is a nonlinear mapping fromR ⁿ toR ^m ,p=n–m>1. An essential part of the method is a constructive algorithm for computing moving frames on the manifold; that is, of orthonormal bases of the tangent spaces that vary smoothly with their points of contact. The triangulation algorithm uses these bases, together with a chord form of the Gauss-Newton process as corrector, to compute the desired vertices. The Jacobian matrix of the mapping is not required at all the vertices but only at the centers of certain local triangulation patches. Several numerical examples show that the method is very efficient in computing triangulations, even around singularities such as limit points and bifurcation points. This opens up new possibilities for determining the form and special features of such solution manifolds.Dedicated to Professor Ivo Babuka on the occasion of his sixtieth birthdayThis work was supported in part by the National Science Foundation under Grant DCR-8309926, the Office of Naval Research under contract N-00014-80-C-9455, and the Air Force Office of Scientific Research under Grant 84-0131     

Über die Anzahl der Lösungen der diskreten Theodorsen-Gleichung

Summary Discretization of the Theodorsen integral equation (T) yields the discrete Theodorsen-equation (T _d ), a system of 2N nonlinear equations. A so-called -condition may be fulfilled. It is known that (T) has exactly one continuous solution. This solution gives the boundary correspondence of the normalized conformal map of the unit disc onto a given domainG. It is also known that (T _d ) has one and only one solution if <1 and at least one solution if 1. We show here that for every 1 and N\ {1} there is a domainG satisfying an -condition such that (T _d ) has an infinite number of solutions. Moreover, givenK>0 and any domainG that fulfills an -condition, we will construct a domainG ₁ in the neighbourhood ofG that fulfills a max (1, +K)-condition such that (T _d ) forG ₁ has an infinite number of solutions. The underlying idea of the construction of those domains allows also to give important new facts about iterative methods for the solution of (T _d ), even in the case <1.

     

Open questions concerning Weiszfeld's algorithm for the Fermat-Weber location problem

The Fermat—Weber location problem is to find a point in ⁿ that minimizes the sum of the weighted Euclidean distances fromm given points in ⁿ . A popular iterative solution method for this problem was first introduced by Weiszfeld in 1937. In 1973 Kuhn claimed that if them given points are not collinear then for all but a denumerable number of starting points the sequence of iterates generated by Weiszfeld's scheme converges to the unique optimal solution. We demonstrate that Kuhn's convergence theorem is not always correct. We then conjecture that if this algorithm is initiated at the affine subspace spanned by them given points, the convergence is ensured for all but a denumerable number of starting points.