Correction of finite element estimates for Sturm-Liouville eigenvalues

Summary It is shown that a simple asymptotic correction technique of Paine, de Hoog and Anderssen reduces the error in the estimate of thekth eigenvalue of a regular Sturm-Liouville problem obtained by the finite element method, with linear hat functions and mesh lengthh, fromO(k ⁴ h ²) toO(k h ²). The result still holds when the matrix elements are evaluated by Simpson's rule, but if the trapezoidal rule is used the error isO(k ² h ²). Numerical results demonstrate the usefulness of the correction even for low values ofk.     

A note on skew differential operators on commutative rings

Let A be a commutative integral domain that is a finitely generated algebra over a field k of characteristic 0 and let ø be a k-algebra automorphism of A of finite order m. In this note we study the ring D(A;ø of differential operators introduced by A.D. Bell. We prove that if A is a free module over the fixed sub-ring A ^ø, with a basis containing 1, then D(A;ø) is isomorphic to the matrix ring M_m(D(A ^ø). It follows from Grothendieck's Generic Flatness Theorem that for an arbitrary A there is an element c?Asuch that D(A[c^-1];ø)?M _m(D(A[c^-1]^ø)). As an application, we consider the structure of D(A;ø)when A is a polynomial or Laurent polynomial ring over k and ø is a diagonalizable linear automorphism.     

Correction of Numerov's eigenvalue estimates

Summary The error in the estimate of thekth eigenvalue of a regular Sturm-Liouville problem obtained by Numerov's method with mesh lengthh isO(k ⁶ h ⁴). We show that a simple correction technique of Paine, de Hoog and Anderssen reduces the error to one ofO(k ³ h ⁴). Numerical examples demonstrate the usefulness of this correction even for low values ofk.     

AC 0-collocation-like method for two-point boundary value problems

Summary A method of a collocation type based onC ⁰-piecewise polynomial spaces is presented for a two-point boundary value problem of the second order. The method has an optimal order of convergence under smoothness requirements on the exact solution which are weaker than forC ¹-collocation methods. If the differential operator is symmetric, a modification of this method leads to a symmetric system of linear equations. It is shown that if the collocation solution is a piecewise polynomial of degree not greater thanr, the method is stable and convergent with orderh ^r inH ¹-norm. A similar symmetric modification forC ⁰-colloction-finite element method [7] is also obtained. Superconvergence at the nodes is established.     

Starlike domains of convergence for Newton's method at singularities

Summary Given a solutionx ^* of a system of nonlinear equationsf with singular Jacobian f(x ^*) we construct an open starlike domainR of initial points, from which Newton's method converges linearly tox ^*. Under certain conditions the union of those straight lines throughx ^*, that do not intersect withR is shown to form a closed set of measure zero, which is necessarily disjoint from any starlike domain of convergence. The results apply to first and higher order singularities.     

Compact,convex sets in R n and a new banach lattice. I.- Theory.

The present paper shows that compact, non-empty convex sets in R ⁿ form a wedge in a well-defined Banach lattice, which turns out to be isometrically Riesz-isomorphic to the continuous functions in S ^n–1, the unit sphere of R ⁿ . Among other results, we obtain Dini-like convergence results for sets, linking order- and norm-convergence.     

On character amenability of semigroup algebras

We study the character amenability of semigroup algebras. We work on general semigroups and certain semigroups such as inverse semigroups with a finite number of idempotents, inverse semigroups with uniformly locally finite idempotent set, Brandt and Rees semigroup and study the character amenability of the semigroup algebra l¹(S) in relation to the structures of the semigroup S. In particular, we show that for any semigroup S, if ?¹(S) is character amenable, then S is amenable and regular. We also show that the left character amenability of the semigroup algebra ?¹(S) on a Brandt semigroup S over a group G with index set J is equivalent to the amenability of G and J being finite. Finally, we show that for a Rees semigroup S with a zero over the group G, the left character amenability of ?¹(S) is equivalent to its amenability, this is in turn equivalent to G being amenable.     

Estimations de l'erreur d'approximation sur un domaine borné deR n parD m -splines d'interpolation et d'ajustement discrètes

Résumé Considérant un espace discretV _h associé àH ^m (), une fonctionfH ^m+1 () et laD ^m -spline d'interpolation discrète _h ^d def dansV _h (cf. [1]), on établit des estimations de l'erreurf– _h ^d en fonction de la distance de Hausdorffd de et de l'ensembleA ^d des points de données, du type |– _h ^d | _l, =o(d ^m–l ), en utilisant des résultats de Duchon [5].De la même façon, on établit des estimations de l'erreurf– _h ^d , oùfH ^m (),m entier >m, et _h ^d désigne laD ^m -spline d'ajustement discrète def dansV _h de paramètre >0 (cf. [1]), du type |– _h ^d | _l, =o(d ^m–l )+O(d ^n/2–1/2). La méthode suivie est applicable auxD ^m -splines d'ajustement surR ⁿ de Duchon [4].

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Approximation error estimates on a bounded domain inR ⁿ for interpolating and smoothing discreteD ^m -splines

Summary Considering a discrete spaceV _h associated withH ^m (), a functionfH ^m+1 () and the interpolation discreteD ^m -spline _h ^d off inV _h (cf. [1]), and using Duchon's results [5], we establish estimates of the errorf– _h ^d . These estimates are of the type |– _h ^d | _l, =o(d ^m–l ), whered is the Hausdorffs distance between and the setA ^d of data points.In the same way, we establish estimates of the errorf– _h ^d , wherefH ^m (), andm>m, and _h ^d is the smoothing discreteD ^m -spline off inV _h associated to the parameter >0 (cf. [1]). These estimates are of the type |– _h ^d | _l, =o(d ^m–l )+O(d ^n/2–1/2). The proposed method can be applied to the smoothingD ^m -splines in #x211D; ⁿ of Duchon [4].

     

Monotonie bei den Quadraturverfahren von Gauß und Newton-Cotes

Summary LetQ _n be the quadrature rule of Gauss or Newton-Cotes withn abscissas. It is proven here, thatf ⁽²ⁿ⁾0 impliesQ _n ^G [f]Q _m ^G [f] (for allm>n) andQ _2n–1 ^NC [f]Q _2n ^NC [f]Q _2n+1 ^NC [f]. It follows that the sequenceQ _n[f] (n=1, 2, ...) is monotone, if all derivatives off are positive.

     

Characterizations of Some Product Manifolds by Their Spectrum

Let Sⁿ(c) denote the n-dimensional Euclidean sphere of constant sectional curvature c and denote by CPⁿ(c) the complex projective space of complex dimension n and of holomorphic sectional curvature c. In this paper, we obtain some characterizations of the manifolds S²(c) × S²(c′), S⁴(c) × S⁴(c′), CP²(c) × CP²(c′) by their spectrum.     

Essential Norms of Composition Operators

We obtain simple estimates for the essential norm of a composition operator acting from the Hardy space H ^p to H ^q , p > q, in one or several variables. When p = and q = 2 our results give an exact formula for the essential norm.     

Two composition methods for solving certain systems of linear equations

Summary This note is concerned with the following problem: Given a systemA·x=b of linear equations and knowing that certains of its subsystemsA ¹·x ¹=b ¹, ...,A ^m ·x ^m =b ^m can be solved uniquely what can be said about the regularity ofA and how to find the solutionx fromx ¹, ...,x ^m ? This question is of particular interest for establishing methods computing certain linear or quasilinear sequence transformations recursively [7, 13, 15].Work performed under NATO Research Grant 027-81     

Ganea comonads

We construct for all topological space X and all n∈N a natural section e ⁿ _X :G ⁿ X→G ⁿ G ⁿ X of the Ganea projection :G ⁿ G ⁿ X→G ⁿ X and show that the triple (G ⁿ ,g ⁿ ,e ⁿ ) is a comonad on Top ^*. Received: 6 March 2000     

The convergence of matrices generated by rank-2 methods from the restricted β-class of Broyden

Summary It is shown that the matricesB _k generated by any method from the restricted -class of Broyden converge, if the method is applied to the unconstrained minimization of a functionfC ²(R ⁿ ) with Lipschitz continuous ² f(x) and if the method is such that it generates vectorsx _k converging sufficiently fast to a local minimumx ^* off with positive definite ² f(x ^*). This result not only holds for constant step sizes _k 1 in each iterationx _k x _k+1=x _k – _k B _k ^–1 f(x _k ) of these methods but also for step sizes determined by asymptotically exact line searches. The paper generalizes corresponding results of Ge Ren-Pu and Powell [6] for the DFP and BFGS methods used in conjunction with step sizes _k 1.Dedicated to Professor F.L. Bauer on the occasion of his 60th birthday     

On the <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-conjecture for locally compact groups

Let G be a locally compact group. For 1 < p < ∞, it is well-known that f * g exists and belongs to L^p(G) for all f, g L^p (G) if and only if G is compact. Here, for 2 < p < ∞, we show that f * g exists for all f, g L^p(G) if and only if G is compact. We also show that this result does not remain true for 1 < p ≤ 2. Received: 23 April 2006     

Optimal solution of nonlinear equations satisfying a Lipschitz condition

Summary For a given nonnegative we seek a pointx ^* such that |f(x ^*)| wheref is a nonlinear transformation of the cubeB=[0,1] ^m into (or ^p ,p>1) satisfying a Lipschitz condition with the constantK and having a zero inB.The information operator onf consists ofn values of arbitrary linear functionals which are computed adaptively. The pointx ^* is constructed by means of an algorithm which is a mapping depending on the information operator. We find an optimal algorithm, i.e., algorithm with the smallest error, which usesn function evaluations computed adaptively. We also exhibit nearly optimal information operators, i.e., the linear functionals for which the error of an optimal algorithm that uses them is almost minimal. Nearly optimal information operators consists ofn nonadaptive function evaluations at equispaced pointsx _j in the cubeB. This result exhibits the superiority of the T. Aird and J. Rice procedure ZSRCH (IMSL library [1]) over Sobol's approach [7] for solving nonlinear equations in our class of functions. We also prove that the simple search algorithm which yields a pointx ^*=x _k such that is nearly optimal. The complexity, i.e., the minimal cost of solving our problem is roughly equal to (K/) ^m .     

Complete submanifolds in Euclidean spaces¶with parallel mean curvature vector

In this paper, we prove that n-dimensional complete and connected submanifolds with parallel mean curvature vector H in the (n+p)-dimensional Euclidean space E ^{n + p} are the totally geodesic Euclidean space E ⁿ , the totally umbilical sphere S ⁿ (c) or the generalized cylinder S ^{n − 1} (c) ×E ¹ if the second fundamental form h satisfies <h>²≤n ²|H|²/ (n− 1). Received: 28 November 2000 / Revised version: 7 May 2001     

On the convergence of difference schemes for the approximation of solutionsu ∈ W 2 m (m>0.5) of elliptic equations with mixed derivatives

Summary The paper deals with such estimates of the rate of convergence of difference methods, which are compatible with the smoothness of the exact solutionu W ₂ ^m (),m>0.5, of elliptic equations with mixed derivatives: The error in the norm of the discrete Sobolev spaceW ₂ ^s (), denoting the set of grid points, is shown to be of the orderO(|h| ^m–s), 0s<m.     

Smoothness of solutions of a nonlinear ode

Smoothness of aC -functionf is measured by (Carleman) sequence {M _k} ₀ ; we sayfC _M [0, 1] if|f ^(k) (t)|CR ^k M _k,k=0, 1, ... withC, R>0. A typical statement proven in this paper isTHEOREM: Let u, b be two C -functions on [0, 1]such that (a) u=u ²+b, (b) |b ^(k) (t)|CR ^k (k!) , >1,k_–.Then |u^(k)(t)|C₁R^k((k–1)!),k_–.The first author acknowledges the hospitality of Mathematical Research Institute of the Ohio State University during his one month visit there in the spring of 1999