Boundary Accessibility of a Domain Quasiconformally Equivalent to a Ball

A boundary point of a domain D in Rⁿ is said to be broadly accessibleif it ‘almost lies’ on the boundary of a round ballcontained in D. If f is a quasiconformal mapping of the unitball Bⁿ onto D, then it is shown that broadly accessible boundarypoints on D correspond under f to a set of full measure on Bⁿ.2000 Mathematics Subject Classification 30C65.     

Increased Accuracy Cubic Spline Solutions to Two-Point Boundary Value Problems

The relation between finite difference approximation and cubicspline solutions of a two-point boundary value problem for thedifferential equation y' +f(x)y^'+g(x)y = r(x) has been consideredin a previous paper. The present paper extends the analysisto the integral equation formulation of the problem. It is shownthat an improvement in accuracy (local truncation error O(h⁶)rather than O(h⁴)) now results from a cubic spline approximationand that for the particular case f(x) 0 the resulting recurrencerelations have a form and accuracy similar to the well-knownNumerov formula. For this case also a formula with local truncationerror O(h⁸) is derived.     

Two Interpolatory Cubic Splines

The third order and the uniform cubic spline are defined andshown to have O(h³) and O(h⁴) convergence respectively whenused for interpolation.     

Shape-preserving interpolation by G1 and G2 PH quintic splines

The interpolation of a planar sequence of points p₀, ..., p_Nby shape-preserving G¹ or G² PH quintic splines with specifiedend conditions is considered. The shape-preservation propertyis secured by adjusting ‘tension’ parameters thatarise upon relaxing parametric continuity to geometric continuity.In the G² case, the PH spline construction is based on applyingNewton–Raphson iterations to a global system of equations,commencing with a suitable initialization strategy—thisgeneralizes the construction described previously in NumericalAlgorithms 27, 35–60 (2001). As a simpler and cheaperalternative, a shape-preserving G¹ PH quintic spline schemeis also introduced. Although the order of continuity is lower,this has the advantage of allowing construction through purelylocal equations.     

On the Effect of Boundary Conditions in Collocation by Polynomial Splines for the Solution of Boundary Value Problems in Ordinary Differential Equations

Consider the numerical solution of a boundary-value problemfor a differential equation of order m using collocation ofa polynomial spline of degree n m on a uniform mesh of sizeh. We describe several collocation schemes which differ onlyin the boundary collocation conditions and which include a "natural"spline collocation scheme. Taking account of derived asymptoticerror bounds most of which are, roughly speaking, of O(hn^12m+1), we discuss the computational effectiveness of the variousschemes.     

End Conditions for Interpolatory Quintic Splines

Present address: Department of Mathematics, University of Tabriz, Tabriz, Iran. Accurate end conditions are derived for quintic spline interpolationat equally spaced knots. These conditions are in terms of availablefunction values at the knots and lead to O(h⁶) convergence uniformlyon the interval of interpolation.     

The Modulus of Smoothness and Discrete Data in a Square Domain

This paper relates to a function f on a two-dimensional squaredomain that is finite, infinite, or semi-infinite. It is shownthat, if the second difference of f with respect to a uniformsquare lattice of mesh-size 2^–n is uniformly of order2^–n with 0<<2, then the h-stepsize second differenceof f is uniformly of order h in any direction in the domain.This is deduced from a corresponding, more general weak-type(i.e. Marchaud-type) inequality.     

A Fourth-order Tridiagonal Finite Difference Method for General Non-linear Two-point Boundary Value Problems with Mixed Boundary Conditions

We present a new fourth-order finite difference method for thegeneral second-order non-linear differential equation y^N = f(x,y, y') subject to mixed two-point boundary conditions. An interestingfeature of our method is that each discretization of the differentialequation at an interior grid point is based on just three evaluationsof f. We establish, under appropriate conditions, O(h⁴)-convergenceof the finite difference scheme. In the case of linear differentialequations, our finite difference scheme leads to tridiagonallinear systems. Numerical examples are considered to demonstratecomputationally the fourth order of the method.     

A multigrid-type method for thin plate spline interpolation on a circle

We consider the problem of thin plate spline interpolation ton equally spaced points on a circle, where the number of datapoints is sufficiently large for work of O(n³ to be unacceptable.We develop an iterative multigrid-type method, each iterationcomprising ngrid stages, and n being an integer multiple of2^ngrid–1. We let the first grid, V₁ be the full set ofdata points, V say, and each subsequent (coarser) grid, V_k,k=2, 3,...,ngrid, contain exactly half of the data points ofthe preceding (finer) grid, these data points being equallyspaced. At each stage of the iteration, we correct our current approximationto the thin plate spline interpolant by an estimate of the interpolantto the current residuals on V_k, where the correction is constructedfrom Lagrange functions of interpolation on small local subsetsof p data points in V_k. When the coarsest grid is reached, however,then the interpolation problem is solved exactly on its q=n/2^ngrid–1points. The iterative process continues until the maximum residualdoes not exceed a specified tolerance. Each iteration has the effect of premultiplying the vector ofresiduals by an n x n matrix R, and thus convergence will dependupon the spectral radius, (R), of this matrix. We investigatethe dependence of the spectral radius on the values of n, p,and q. In all the cases we have considered, we find (R) <<1, and thus rapid convergence is assured.     

Closed-form Representations for Errors of Cubic Spline Approximations on Equally Spaced Knots

This paper considers estimating errors in the approximationof an arbitrary linear functional on C⁴[0, L] by use of cubicsplines on equally spaced knots. Using explicit formulas derivedfor the cubic spline approximations of f_k(x) = cos (kx/L), formulasare found for the cosine expansion of the Peano Kernel for theremainder functional.     

Sparse convolution quadrature for time domain boundary integral formulations of the wave equation

S. A. Sauter Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland ^{Many important physical applications are governed by the waveequation. The formulation as time domain boundary integral equationsinvolves retarded potentials. For the numerical solution ofthis problem, we employ the convolution quadrature method forthe discretization in time and the Galerkin boundary elementmethod for the space discretization. We introduce a simple apriori cut-off strategy where small entries of the system matricesare replaced by zero. The threshold for the cut-off is determinedby an a priori analysis which will be developed in this paper.This analysis will also allow to estimate the effect of additionalperturbations such as panel clustering and numerical integrationon the overall discretization error. This method reduces thestorage complexity for time domain integral equations from O(M2N)to O(M2N logM), where N denotes the number of time steps andM is the dimension of the boundary element space.}     

Repulsive Fixpoints of Analytic Functions with Applications to Complex Dynamics

Let G be a family of functions analytic in a domain D in thecomplex plane. It is proved that G is a normal family, providedthat for each fG, there exists k = k(f) > 1 such that thekth iterate f^k has no repulsive fixpoint in D. A new proof ofa result of Bergweiler and Terglane concerning the dynamicsof entire functions is also given.     

An Optimal Representation Formula for Carnot-Caratheodory Vector Fields

The simplest example of the sort of representation formula thatwe shall study is the following familiar inequality for a smooth,real-valued function f(x) defined on a ball B in N-dimensionalEuclidean space R^N: [formula] where f denotes the gradient of f, f_B is the average |B|^–1_Bf(y)dy, |B| is the Lebesgue measure of B, and C is a constantwhich is independent of f, x and B. This formula can be found,for example, in [4] and [12]; see also the closely related estimatesin [20, pp. 228{231]. Indeed, such a formula holds in any boundedconvex domain. 1991 Mathematics Subject Classification 31B10,46E35, 35A22.     

The <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-Approximation Order of Surface Spline Interpolation for 1 \leq <Emphasis Type="Italic">p</Emphasis> \leq 2

We show that the L_p-approximation order of surface spline interpolation equals m+1/p for p in the range 1 \leq p \leq 2, where m is an integer parameter which specifies the surface spline. Previously it was known that this order was bounded below by m + &frac; and above by m+1/p. With h denoting the fill-distance between the interpolation points and the domain , we show specifically that the L_p()-norm of the error between f and its surface spline interpolant is O(h^{m + 1/p}) provided that f belongs to an appropriate Sobolev or Besov space and that \subset R^d is open, bounded, and has the C^2m-regularity property. We also show that the boundary effects (which cause the rate of convergence to be significantly worse than O(h^2m)) are confined to a boundary layer whose width is no larger than a constant multiple of h |log h|. Finally, we state numerical evidence which supports the conjecture that the L_p-approximation order of surface spline interpolation is m + 1/p for 2 < p \leq \infty.     

Canonical curves in P3

Let Hilb^6t–3(P³) be the Hilbert scheme of closed 1-dimensionalsubschemes of degree 6 and arithmetic genus 4 in P³. Let H bethe component of Hilb^6t–3(P³) whose generic point correspondsto a canonical curve, that is, a complete intersection of aquadric and a cubic surface in P³. Let F be the vector spaceof linear forms in the variables z₁, z₂, z₃, z₄. Denote by F_dthe vector space of homogeneous forms of degree d. Set X = (f₂,f₃)where f₂ P(F₂) is a quadric surface, and f₃ P(F₃/f₂ ·F) is a cubic modulo f₂. Wehave a rational map, : X ... Hdefined by (f₂,f₃) f₂ f₃. It fails to be regular along thelocus where f₂ and f₃ acquire a common linear component. Ourmain result gives an explicit resolution of the indeterminaciesof as well as of the singularities of H. 2000 Mathematical Subject Classification: 14C05, 14N05, 14N10,14N15.     

OnG 2 continuous cubic spline interpolation

In this paper the interpolation byG ² continuous planar cubic Bézier spline curves is studied. The interpolation is based upon the underlying curve points and the end tangent directions only, and could be viewed as an extension of the cubic spline interpolation to the curve case. Two boundary, and two interior points are interpolated per each spline section. It is shown that under certain conditions the interpolation problem is asymptotically solvable, and for a smooth curvef the optimal approximation order is achieved. The practical experiments demonstrate the interpolation to be very satisfactory. Supported in prat by the Ministry of Science and Technology of Slovenjia, and in part by the NSF and SF of National Educational Committee of China.     

Pad?-approximant Bounds and Approximate Solutions for Kirkwood-Riseman Integral Equations

Two-point Pad? approximants are used to calculate tight upperand lower bounds on the quantity <?, f> associated withKirkwood-Riseman integral equations (1+yL)?=f, which arise inthe diffusion theory of flexible macromolecules. The self-adjointoperator L is an integral operator on –1 x 1, with weaklysingular kernel |x – x'|^–?, and the two specificcases (i) f = 1, (ii) f = x² are studied. In case (i) directbounds on <?, 1> are obtained; this quantity is inverselyproportional to the translational diffusion constant. In case(ii) bounds on <?, 1 > are found by a new technique involvingcombinations of bounds for the three cases f = 1, f = x² andf = bx²?b^–1. Various types of Pade and related approximantsare compared, using the information <f, Lⁿf>, n = –2,–1, 0, 1, 2, 3 and (an upper bound on L) for severalvalues of the positive parameter y. Pad?-approximant-generating trial vectors are investigated anda convergence theorem is established. The vector consistingof an optimum linear combination of L^–1f, f and Lf isfound to be an accurate approximation to a numerical solutionin case (ii), for all values of y and x. Specific analyticalexpressions are derived for the approximate solutions.     

On the Number of Singularities in Generic Deformations of Map Germs

Let f:Cⁿ, 0C^p, 0 be a K-finite map germ, and let i=(i₁, ...,i_k) be a Boardman symbol such that ⁱ has codimension n in thecorresponding jet space J^k(n, p). When its iterated successorshave codimension larger than n, the paper gives a list of situationsin which the number of ⁱ points that appear in a generic deformationof f can be computed algebraically by means of Jacobian idealsof f. This list can be summarised in the following way: f musthave rank n–i₁ and, in addition, in the case p=6, f mustbe a singularity of type ^i₁,i₂.     

Explicit Error Bounds for Periodic Splines of Odd Order on a Uniform Mesh

The dependence relationships connecting equal interval splinesand their derivatives are analysed to obtain the form of theerror term when the spline is replaced by a general function.The defining equations for periodic splines of odd order ona uniform mesh are then expressed in terms of a positive definitecirculant matrix A and attainable bounds determined for thecondition number of A and for the norm of A^-1. In conjunctionwith the error term associated with the dependence relationships,this enables explicit error bounds to be established for thederivatives at the knots of the spline function. Some subsidiary results in the paper also relate to B-splineson a uniform mesh.     

A Fourth-order Tridiagonal Finite Difference Method for General Two-point Boundary Value Problems with Non-linear Boundary Conditions

We present a fourth-order finite difference method for the generalsecond-order nonlinear differential equation y^" = f(x, y, y‘)subject to non-linear two-point boundary conditions g₁(y(a), — y^’()) = 0, g₂(y(b), y'(b)) = 0. When both the differential equation and the boundary conditionsare linear, the method leads to a tridiagonal linear system.We show that the finite difference method is O(h⁴)-convergent.Numerical examples are given to illustrate the method and itsfourth-order convergence. The present paper extends the methodgiven in Chawla (1978) to the case of non-linear boundary conditions.