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.     

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.     

The coexponent of a regular p -group

A sharp bound is derived for the nilpotency class of a regular p-group in terms of its coexponent, and is used to show that the number of groups of order pⁿ with a given fixed coexponent, is independent of n, for p and n sufficiently large. Explicit formulae are calculated in the case of coexponent 3.     

Smoothing noisy data with spline functions

Summary A procedure for calculating the trace of the influence matrix associated with a polynomial smoothing spline of degree2m–1 fitted ton distinct, not necessarily equally spaced or uniformly weighted, data points is presented. The procedure requires orderm ² n operations and therefore permits efficient orderm ² n calculation of statistics associated with a polynomial smoothing spline, including the generalized cross validation. The method is a significant improvement over an existing method which requires ordern ³ operations.     

Lamron ℓ-groups

The article introduces a new class of lattice-ordered groups. An ?-group G is lamron if Min(G)^?1 is a Hausdorff topological space, where Min(G)^?1 is the space of all minimal prime subgroups of G endowed with the inverse topology. It will be evident that lamron ?-groups are related to ?-groups with stranded primes. In particular, it is shown that for a W-object (G,u), if every value of u contains a unique minimal prime subgroup, then G is a lamron ?-group; such a W-object will be said to have W-stranded primes. A diverse set of examples will be provided in order to distinguish between the notions of lamron, stranded primes, W-stranded primes, complemented, and weakly complemented ?-groups.     

A class of direct methods for linear systems

Summary A class of methods of direct type for solving determined or underdetermined, full rank or deficient rank linear systems is presented and theoretically analyzed. The class can be considered as a generalization of the methods of Brent and Brown as restricted to linear systems and implicitly contains orthogonal,LU andLL ^T factorization methods.     

E-convex infix codes

Two partial orders that play an important role in the combinatorics of words, can be defined in a natural way on the free monoid X ^* generated by the finite alphabet X: the infix and the embedding orders. A set C of nonempty words is called an infix code (hypercode) over X if C is an antichain with respect to the infix (embedding) order. A set of words is said to be e-convex if it is convex with respect to the embedding order. Two characterizations of the e-convex infix codes are given as well as a sufficient condition for such codes to be finite. It is shown that the family EIC(X) of the e-convex infix codes with the empty word forms, under the operation of concatenation, a free submonoid of the free monoid B(X) of the biprefix codes and that the generating alphabet of EIC(X) is a sub-alphabet of the generating alphabet of B(X).This research was supported by Grant A7877 of the Natural Sciences and Engineering Council of Canada.     

Two families of mixed finite elements for second order elliptic problems

Summary Two families of mixed finite elements, one based on triangles and the other on rectangles, are introduced as alternatives to the usual Raviart-Thomas-Nedelec spaces. Error estimates inL ² () andH ^–5 () are derived for these elements. A hybrid version of the mixed method is also considered, and some superconvergence phenomena are discussed.     

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.     

Approximation by cubicC 1-splines on arbitrary triangulations

Summary In this paper, we present an efficient representation for bivariate piecewise cubicC ¹-splines on arbitrary triangulations. A numerical method is discussed for computing the dimension of the spaceS ₃ ¹ () of these splines. We consider subspaces ofS ₃ ¹ () satisfying certain boundary conditions. Some applications are given where piecewise cubicC ¹-functions are used to solve interpolation problems and least squares approximation problems.     

A consturction of monotonically convergent sequences from successive approximations in certain Banach spaces

Summary LetA: XX (whereX=C ^q [a, b] orL ^p [a, b]) be a contraction having the fixed pointf. In this note, using ideas from [1–8], we obtain a modified successive approximation sequence which approximatesf and which has certain properties regarding monotonicity too.     

Regularity theorems and optimal error estimates for linear parabolic Cauchy problems

Summary We prove some regularity results for the solution of a linear abstract Cauchy problem of parabolic type. As an application, we study the approximation of the solution by means of an implicit-Euler discretization in time, which is stable with respect to a wide class of Galerkin approximation methods in space. The error is evaluated in norms of typeL ²(0, ,L ²) andL ²(0, ,V)(H ₀₀ ^1/2 (0, ,H)+H ¹(0, ,V)), whereVHV are Hilbert spaces (the embeddings are supposed to be dense and continuous). We prove error estimates which are optimal with respect to the regularity assumptions on the right-hand side of the equation.The author was supported by G.N.A.F.A. and I.A.N. of C.N.R. and by M.P.I.     

Parametrization via secant length and application to path following

Summary A possible way for parametrizing the solution path of the nonlinear systemH(u)=0, H: ^{n+1 n} consists of using the secant length as parameter. This idea leads to a quadratic constraint by which the parameter is introduced. A Newton-like method for computing the solution for a given parameter is proposed where the nonlinear system is linearized at each iterate, but the quadratic parametrizing equation is exactly satisfied. The localQ-quadratic convergence of the method is proved and some hints for implementing the algorithm are givenDedicated to Professor Lothar Collatz on the occasion of his 75th birthday     

Error bounds for computed eigenvalues and eigenvectors. II

Summary In this paper, motivated by Symm-Wilkinson's paper [5], we describe a method which finds the rigorous error bounds for a computed eigenvalue ⁽⁰⁾ and a computed eigenvectorx ⁽⁰⁾ of any matrix A. The assumption in a previous paper [6] that ⁽⁰⁾,x ⁽⁰⁾ andA are real is not necessary in this paper. In connection with this method, Symm-Wilkinson's procedure is discussed, too.     

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.     

Extended iterative methods for the solution of operator equations

Summary Given an iterative methodM ₀, characterized byx ^(k+1=G ₀(x( ^k )) (k0) (x(⁰) prescribed) for the solution of the operator equationF(x)=0, whereF:XX is a given operator andX is a Banach space, it is shown how to obtain a family of methodsM _p characterized byx ^(k+1=G _p (x( ^k )) (k0) (x(⁰) prescribed) with order of convergence higher than that ofM _o. The infinite dimensional multipoint methods of Bosarge and Falb [2] are a special case, in whichM ₀ is Newton's method.Analogues of Theorems 2.3 and 2.36 of [2] are proved for the methodsM _p, which are referred to as extensions ofM ₀. A number of methods with order of convergence greater than two are discussed and existence-convergence theorems for some of them are proved.Finally some computational results are presented which illustrate the behaviour of the methods and their extensions when used to solve systems of nonlinear algebraic equations, and some applications currently being investigated are mentioned.     

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.     

On the eigenvalue distribution of a class of preconditioning methods

Summary A class of preconditioning methods depending on a relaxation parameter is presented for the solution of large linear systems of equationAx=b, whereA is a symmetric positive definite matrix. The methods are based on an incomplete factorization of the matrixA and include both pointwise and blockwise factorization. We study the dependence of the rate of convergence of the preconditioned conjugate gradient method on the distribution of eigenvalues ofC ^–1 A, whereC is the preconditioning matrix. We also show graphic representations of the eigenvalues and present numerical tests of the methods.     

On a difference scheme of exponential type for a nonlinear singular perturbation problem

Summary A difference scheme of exponential type for solving a nonlinear singular perturbation problem is analysed. Although this scheme is not of monotone type, aL ¹ convergence result is obtained. Relations between this scheme and Engquist-Osher scheme are also discussed.