On Powerful Normal Subgroups of a p-Group

 In this paper we investigate the action of a p-group G on its powerful normal subgroup N. We shall mainly be concerned to find conditions which guarantee that G acts uniserially on N. We shall also study what consequences uniserial action on N has on the structure of N. (Received 21 August 1998; in revised form 28 June 1999)     

A Universal Bound for the Low Eigenvalues of Neumann Laplacians on Compact Domains in a Hadamard Manifold

 Let M be an n-dimensional simply connected Hadamard manifold with Ricci curvature satisfying and be a bounded domain having smooth boundary. In this paper, we prove that the first n nonzero Neumann eigenvalues of the Laplacian on Ω satisfy , where is a computable constant depending only on and , Ω being the volume of Ω. This result generalizes the corresponding estimate for bounded domains in a Euclidean space obtained recently by M. S. Ashbaugh and R. D. Benguria. (Received 19 May 1998; in revised form 21 September 1998)     

The Space of Parabolic Affine Sphereswith Fixed Compact Boundary

 The aim of this paper is to study the moduli space of solutions of the Dirichlet problem associated to the equation of Monge-Ampeère type, det , on an exterior planar domain. We prove that this moduli space is either empty or a differentiable manifold of dimension five. (Received 23 April 1999)     

Invariant Cocycles Have Abelian Ranges

 Let R be a discrete nonsingular equivalence relation on a standard probability space , and let V be an ergodic strongly asymptotically central automorphism of R. We prove that every V-invariant cocycle with values in a Polish group G takes values in an abelian subgroup of G. The hypotheses of this result are satisfied, for example, if A is a finite set, a closed, shift-invariant subset, V is the shift, μ a shift-invariant and ergodic probability measure on X, the two-sided tail-equivalence relation on X, a shift-invariant subrelation which is μ-nonsingular, and a shift-invariant cocycle. (Received 15 September 2001)     

Diophantine Analysis of Conformal Iterated Function Systems

 A formula for the Hausdorff dimension of a subset of the limit set of a conformal infinite iterated function system is derived and expressed as a unique zero of the topological pressure of corresponding strongly H?lder family of functions. Received August 10, 2001; in revised form February 23, 2002     

Four positive solutions for the semilinear elliptic equation: -Delta u+u=a(x)u^p+f(x) in {mathbb R}^N

We consider the existence of positive solutions of the following semilinear elliptic problem in : where , , , and . Under the conditions: 1° for all , 2° as , 3° there exist and such that 4°, we show that (*) has at least four positive solutions for sufficiently small but . Received December 11, 1998 / Accepted July 16, 1999 / Published online April 6, 2000     

Minimizing the condition number of a positive definite matrix by completion

Summary. We consider the problem of minimizing the spectral condition number of a positive definite matrix by completion: \noindent where is an Hermitian positive definite matrix, a matrix and is a free Hermitian matrix. We reduce this problem to an optimization problem for a convex function in one variable. Using the minimal solution of this problem we characterize the complete set of matrices that give the minimum condition number. Received October 15, 1993     

Componentwise perturbation analyses for the QR factorization

Summary. This paper gives componentwise perturbation analyses for Q and R in the QR factorization A=QR, , R upper triangular, for a given real $m\times n$ matrix A of rank n. Such specific analyses are important for example when the columns of A are badly scaled. First order perturbation bounds are given for both Q and R. The analyses more accurately reflect the sensitivity of the problem than previous such results. The condition number for R is bounded for a fixed n when the standard column pivoting strategy is used. This strategy also tends to improve the condition of Q, so usually the computed Q and R will both have higher accuracy when we use the standard column pivoting strategy. Practical condition estimators are derived. The assumptions on the form of the perturbation are explained and extended. Weaker rigorous bounds are also given. Received April 11, 1999 / Published online October 16, 2000     

Maximal cofinitary groups

We discuss the cardinalities of maximal cofinitary groups under the assumption of . We also discuss various open questions in this area. Received: 24 July 1997     

Cyclic regulating balanced Butler groups

A necessary and sufficient condition is derived for an arbitrary cyclic regulating almost completely decomposable group to belong to a K(n)-class, n≥ 0. This allows one to formulate a Structure Theorem for clipped p-primary cyclic regulating groups in each K(n)-class. Using the Indecomposability Criterion we construct an example of an indecomposable balanced Butler group which does not have a unique regulating subgroup. Received: 3 November 1998     

On Hausdorff dimension of invariant measures arising from non-contractive iterated function systems

Under rather general assumptions we give an upper and lower bound estimate of the Hausdorff dimension of the invariant probability measure for an iterated function systems acting on a Polish space. Received: May 2, 2001; in final form: July 30, 2001?Published online: May 29, 2002     

A Procrustes problem on the Stiefel manifold

Summary. An orthogonal Procrustes problem on the Stiefel manifold is studied, where a matrix Q with orthonormal columns is to be found that minimizes for an matrix A and an matrix B with and . Based on the normal and secular equations and the properties of the Stiefel manifold, necessary conditions for a global minimum, as well as necessary and sufficient conditions for a local minimum, are derived. Received April 7, 1997 / Revised version received April 16, 1998     

Upper bound and stability of scaled pseudoinverses

Summary. For given matrices and where is positive definite diagonal, a weighed pseudoinverse of is defined by and an oblique projection of is defined by . When is of full column rank, Stewart [3] and O'Leary [2] found sharp upper bound of oblique projections which is independent of , and an upper bound of weighed pseudoinverse by using the bound of . In this paper we discuss the sharp upper bound of over a set of positive diagonal matrices which does not depend on the upper bound of , and the stability of over . Received September 29, 1993 / Revised version received October 31, 1994     

Stationary layered solutions in {\mathbb{R}}^2 for a class of non autonomous Allen-Cahn equations

We consider a class of non autonomous Allen-Cahn equations where is a multiple-well potential and is a periodic, positive, non-constant function. We look for solutions to (0.1) having uniform limits as corresponding to minima of W. We show, via variational methods, that under a nondegeneracy condition on the set of heteroclinic solutions of the associated ordinary differential equation the equation (0.1) has solutions which depends on both the variables x andy. In contrast, when a is constant such nondegeneracy condition is not satisfied and all such solutions are known to depend only on x. Received April 16, 1999 / Accepted October 1, 1999 / Published online June 28, 2000     

The condition number of real Vandermonde, Krylov and positive definite Hankel matrices

Summary. We show that the Euclidean condition number of any positive definite Hankel matrix of order may be bounded from below by with , and that this bound may be improved at most by a factor . Similar estimates are given for the class of real Vandermonde matrices, the class of row-scaled real Vandermonde matrices, and the class of Krylov matrices with Hermitian argument. Improved bounds are derived for the case where the abscissae or eigenvalues are included in a given real interval. Our findings confirm that all such matrices – including for instance the famous Hilbert matrix – are ill-conditioned already for “moderate” order. As application, we describe implications of our results for the numerical condition of various tasks in Numerical Analysis such as polynomial and rational i nterpolation at real nodes, determination of real roots of polynomials, computation of coefficients of orthogonal polynomials, or the iterative solution of linear systems of equations. Received December 1, 1997 / Revised version received February 25, 1999 / Published online 16 March 2000