Lassi Päivärinta

Testing equality of relative survival patterns based on aggregated data.

The relative survival rate is defined as the ratio of the survival rate observed in a patient group under consideration to the survival rate expected in a group of people similar to the patient group at the beginning of the follow-up interval, with respect to all possible factors (e.g., age and sex) affecting survival, except the disease under study. Survival from cancer and other chronic diseases is often measured by this quantity, which is adjusted for the effect of mortality attributable to competing risks of death. In this paper, maximum likelihood ratio tests are constructed on the basis of aggregated data for testing the equality of relative survival rates between patient groups against proportional hazards and general alternative hypotheses. The tests are applied to the Finnish nationwide data on colon cancer patients with nonlocalized tumors as reported to the Finnish Cancer Registry. Simulation studies show that the maximum likelihood ratio tests compare favorably with alternative methods proposed earlier. Moreover, the maximum likelihood ratio tests are more extensive in coverage and are based on more applicable alternative hypotheses than the other test statistics. Finally, an extension to proportional hazards regression models of the relative survival rates is suggested.

A simple non‐linear model in incidence prediction

A simple model is proposed for incidence prediction. The model is non-linear in parameters but linear in time, following models in environmental cancer epidemiology. Assuming a Poisson distribution for the age and period specific numbers of incident cases approximate confidence and prediction intervals are calculated. The major advantage of this model over current models is that age-specific predictions can be made with greater accuracy. The model also preserves in the period of prediction the age pattern of incidence rates existing in the data. It may be fitted with any package which includes an iteratively reweighted least squares algorithm, for example GLIM. Cancer incidence predictions for the Stockholm-Gotland Oncological Region in Sweden are presented as an example.

RECOVERING SINGULARITIES FROM BACKSCATTERING IN TWO DIMENSIONS

We have shown that in two dimensions the leading singularities of the quantum mechanical scattering potential are determined by the backscattering data. We assume that the short range potential belongs to a suitable weighted Sobolev space, and by estimating the iterative terms in the Born-expansion we are able to show, that for example for Heaviside-type singularities across a smooth hypersurface, both the location and the size of the jump are recovered from backscattering. The main part of the proof consists in getting sharp enough estimates for the first non-linear Born-term. These estimates are proven using a recent characterization of W 1,p -functions due to P. Hajlasz, and a modification of the classical Triebels Maximal Inequality.

On recovering obstacles inside inhomogeneities

We consider acoustic scattering from an obstacle inside an inhomogeneous structure. We prove in the paper that if the outside inhomogeneity is known then the obstacle and possible inside inhomogeneity are uniquely determined by the fixed energy far field data. The proof is based on new mapping properties of layer potentials in spaces that specify one point.

An n-dimensional Borg-Levinson theorem for singular potentials

We prove in this paper that the boundary spectral data, i.e. the Dirichlet eigenvalues and normal derivatives of the eigenfunctions at the boundary uniquely determines a potential in L^p on bounded domains. This result generalizes the result of Nachman, Sylvester and Uhlmann to unbounded potentials. This result can be viewed as a generalization of the classical one-dimensional Borg-Levinson theorem.

Transmission eigenvalues and a problem of Hans Lewy

Abstract Subject to certain assumptions on the matrix N = N ( x ) we show that except for a discrete set of values of k (called transmission eigenvalues) the only solution of ∇·N∇u+k 2 u=0 Δ v+k 2 v=0 in D, u=v ∂u ∂ν = ∂v ∂ν on ∂D, where D⊂ R n is a domain with smooth boundary ∂D having unit outward normal ν is the trivial solution u = v =0. This problem has important applications to the inverse scattering problem for anisotropic media and is in a class of problems first considered by Lewy (Bull. Amer. Math. Soc. 65 (1959) 37–58).

New mapping properties for the resolvent of the Laplacian and recovery of singularities of a multi-dimensional scattering potential

We prove that in dimension three and higher potential scattering the leading order singularities (and in some special cases - all singularities) of unknown potential are obtained exactly by the Born approximation. The proof is based on new estimates for the continuous spectrum of the Laplacian in weighted L p-spaces. These estimates allow us to consider the potentials with stronger singularities than in previous publications.

The borderlines of invisibility and visibility in Calderón’s inverse problem

We consider the determination of a conductivity function in a two-dimensional domain from the Cauchy data of the solutions of the conductivity equation on the boundary. We prove uniqueness results for this inverse problem, posed by Calderon, for conductivities that are degenerate, that is, they may not be bounded from above or below. In particular, for scalar conductivities we solve the inverse problem in a class which is larger than

Analytic Methods for Inverse Scattering Theory

L^infty

On Determining Individual Behaviour from Population Data

. Also, we give new counterexamples for the uniqueness of the inverse conductivity problem. nWe say that a conductivity is visible if the inverse problem is solvable so that the inside of the domain can be uniquely determined, up to a change of coordinates, using the boundary measurements. The present counterexamples for the inverse problem have been related to the invisibility cloaking. This means that there are conductivities for which a part of the domain is shielded from detection via boundary measurements. Such conductivities are called invisibility cloaks. nIn the present paper we identify the borderline of the visible conductivities and the borderline of invisibility cloaking conductivities. Surprisingly, these borderlines are not the same. We show that between the visible and the cloaking conductivities there are the electric holograms, conductivities which create an illusion of a non-existing body. The electric holograms give counterexamples for the uniqueness of the inverse problem which are less degenerate than the previously known ones.