A new premium calculation principle based on Orlicz norms

A multiplicative equivalent of the zero utility premium calculation principle is introduced. If the utility function happens to be a normalized Young function the new premium calculation principle is related to Orlicz norms.     

Estimators of covariances in time series models

The theory of Minimum Norm Quadratic Estimators for estimating variances and covariances is applied to show that some commonly used estimators of covariances in time series models are easily derived using the above principle. In applying the theory MINQE, it is observed that no unbiased estimator exists in the class of invariant quadratics.     

Centered Poisson Approximation via Stein's Method

The centered Poisson approximation can be considered as a link between the Poisson and normal laws. It has the structure similar to the Poisson distribution but depends on two parameters. The results obtained via the Stein–Chen method for sums of independent and dependent indicators show that the centred Poisson approximation is a strong alternative to the standard Poisson and normal approximations and, potentially, is more widely applicable.     

An inequality for the norm of a polynomial factor

Let be a monic polynomial of degree , with complex coefficients, and let be its monic factor. We prove an asymptotically sharp inequality of the form , where denotes the sup norm on a compact set in the plane. The best constant in this inequality is found by potential theoretic methods. We also consider applications of the general result to the cases of a disk and a segment.

     

On Weighted Norms of Riesz Transforms Equal to One

We consider the vector Riesz transform ^t-(t+s)/2 div^s of even order s + t in the weighted space L ₂(ⁿ;|x|^a). We establish that for t s, n >3 its norm is equal to one on some interval of values of a, while inside the interval a stronger estimate for a subordinate norm is valid.     

A Faster and Simpler Recursive Algorithm for the LAPACK Routine DGELS

We present new algorithms for computing the linear least squares solution to overdetermined linear systems and the minimum norm solution to underdetermined linear systems. For both problems, we consider the standard formulation min AX – B _F and the transposed formulation min A ^T X – B _F , i.e, four different problems in all. The functionality of our implementation corresponds to that of the LAPACK routine DGELS. The new implementation is significantly faster and simpler. It outperforms the LAPACK DGELS for all matrix sizes tested. The improvement is usually 50–100% and it is as high as 400%. The four different problems of DGELS are essentially reduced to two, by use of explicit transposition of A. By explicit transposition we avoid computing Householder transformations on vectors with large stride. The QR factorization of block columns of A is performed using a recursive level-3 algorithm. By interleaving updates of B with the factorization of A, we reduce the number of floating point operations performed for the linear least squares problem. By avoiding redundant computations in the update of B we reduce the work needed to compute the minimum norm solution. Finally, we outline fully recursive algorithms for the four problems of DGELS as well as for QR factorization.This revised version was published online in October 2005 with corrections to the Cover Date.     

The inverse problem of bisymmetric matrices with a submatrix constraint

An n×n real matrix A is called a bisymmetric matrix if A=A^T and A=S_nAS_n, where S_n is an n×n reverse unit matrix. This paper is mainly concerned with solving the following two problems: Problem I Given n×m real matrices X and B, and an r×r real symmetric matrix A₀, find an n×n bisymmetric matrix A such that where A([1: r]) is a r×r leading principal submatrix of the matrix A. Problem II Given an n×n real matrix A^*, find an n×n matrix  in S_E such that where ∥·∥ is Frobenius norm, and S_E is the solution set of Problem I. The necessary and sufficient conditions for the existence of and the expressions for the general solutions of Problem I are given. The explicit solution, a numerical algorithm and a numerical example to Problem II are provided. Copyright © 2003 John Wiley & Sons, Ltd.     

Polynomials,Symmetric Multilinear Forms and Weak Compactness

In this paper we obtain some versions of weak compactness James theorem, replacing bounded linear functionals by polynomials and symmetric multilinear forms.     

Characterizing Cohen-Macaulay local rings by Frobenius maps

Let be a commutative noetherian local ring of prime characteristic. Denote by the ring regarded as an -algebra through -times composition of the Frobenius map. Suppose that is F-finite, i.e., is a finitely generated -module. We prove that is Cohen-Macaulay if and only if the -modules have finite Cohen-Macaulay dimensions for infinitely many integers .

     

Rectifiability of Measures with Locally Uniform Cube Density

The conjecture that Radon measures in Euclidean space with positivefinite density are rectifiable was a central problem in GeometricMeasure Theory for fifty years. This conjecture was positivelyresolved by Preiss in 1986, using methods entirely dependenton the symmetry of the Euclidean unit ball. Since then, dueto reasons of isometric immersion of metric spaces into l andthe uncommon nature of the sup norm even in finite dimensions,a popular model problem for generalising this result to non-Euclideanspaces has been the study of 2-uniform measures in . The rectifiability or otherwise of these measureshas been a well-known question. In this paper the stronger result that locally 2-uniform measuresin are rectifiable is proved. This is the first result that proves rectifiability, from aninitial condition about densities, for general Radon measuresof dimension greater than 1 outside Euclidean space. 2000 MathematicalSubject Classification: 28A75.