Integration of polyharmonic functions

The results in this paper are motivated by two analogies. First, m-harmonic functions in R(n) are extensions of the univariate algebraic polynomials of odd degree 2m-1. Second, Gauss' and Pizzetti's mean value formulae are natural multivariate analogues of the rectangular and Taylor's quadrature formulae, respectively. This point of view suggests that some theorems concerning quadrature rules could be generalized to results about integration of polyharmonic functions. This is done for the Tchakaloff-Obrechkoff quadrature formula and for the Gaussian quadrature with two nodes.

Gaussian extended cubature formulae for polyharmonic functions

The purpose of this paper is to show certain links between univariate interpolation by algebraic polynomials and the representation of polyharmonic functions. This allows us to construct cubature formulae for multivariate functions having highest order of precision with respect to the class of polyharmonic functions. We obtain a Gauss type cubature formula that uses ℳ values of linear functional (integrals over hyperspheres) and is exact for all 2ℳ-harmonic functions, and consequently, for all algebraic polynomials of n variables of degree 4ℳ - 1.

Isolated singularities of polyharmonic operator in even dimension

We consider the equation Delta(2)u = g(x, u) >= 0 in the sense of distribution in Omega' = Omega\textbackslash {0} where u and -Delta u >= 0. Then it is known that u solves Delta(2)u = g(x, u) + alpha delta(0) - beta Delta delta(0), for some nonnegative constants alpha and beta. In this paper, we study the existence of singular solutions to Delta(2)u = a(x) f (u) + alpha delta(0) - beta Delta delta(0) in a domain Omega subset of R-4, a is a nonnegative measurable function in some Lebesgue space. If Delta(2)u = a(x) f (u) in Omega', then we find the growth of the nonlinearity f that determines alpha and beta to be 0. In case when alpha = beta = 0, we will establish regularity results when f (t) <= Ce-gamma t, for some C, gamma > 0. This paper extends the work of Soranzo (1997) where the author finds the barrier function in higher dimensions (N >= 5) with a specific weight function a(x) = |x|(sigma). Later, we discuss its analogous generalization for the polyharmonic operator.

Higher order Sobolev Spaces and polyharmonic boundary value problems in Carnot Groups

The main task of this work is to present a concise survey on the theory of certain function spaces in the contexts of Hörmander vector fields and Carnot Groups, and to discuss briefly an application to some polyharmonic boundary value problems on Carnot Groups of step 2.

An approximate method for the solution of an influence function foil rolling model

Fleck and Johnson (Int. J. Mech. Sci. 29 (1987) 507) and Fleck et al. (Proc. Inst. Mech. Eng. 206 (1992) 119) have developed foil rolling models which allow for large deformations in the roll profile, including the possibility that the rolls flatten completely. However, these models require computationally expensive iterative solution techniques. A new approach to the approximate solution of the Fleck et al. (1992) Influence Function Model has been developed using both analytic and approximation techniques. The numerical difficulties arising from solving an integral equation in the flattened region have been reduced by applying an Inverse Hilbert Transform to get an analytic expression for the pressure. The method described in this paper is applicable to cases where there is or there is not a flat region.

Smoothing a Discrete Hazard Function for the Number of Patients Colonized with Methicillin-Resistance Staphylococcus Aureus in an Intensive Care Unit

A new method for estimating the time to colonization of Methicillin-resistant Staphylococcus Aureus (MRSA) patients is developed in this paper. The time to colonization of MRSA is modelled using a Bayesian smoothing approach for the hazard function. There are two prior models discussed in this paper: the first difference prior and the second difference prior. The second difference prior model gives smoother estimates of the hazard functions and, when applied to data from an intensive care unit (ICU), clearly shows increasing hazard up to day 13, then a decreasing hazard. The results clearly demonstrate that the hazard is not constant and provide a useful quantification of the effect of length of stay on the risk of MRSA colonization which provides useful insight.