Gustaf Gripenberg

Computing the joint spectral radius

This paper presents algorithms for finding an arbitrarily small interval that contains the joint spectral radius of a finite set of matrices. It also presents a numerical criterion for verifying in certain cases that the joint spectral radius is the maximum of the spectral radii of the given matrices. Error bounds are derived for the case where calculations are done with finite precision and the matrices are not known exactly. The algorithms are implemented and applied to estimate Holder exponents of the orthonormal wavelets Nϕ constructed by Daubechies for 3 ⩽ N ⩽ 8.

Schauder estimates for equations with fractional derivatives

The equation D t (u− h1) +D x(u− h2) = f, 0 < α, β < 1, t, x ≥ 0, (∗) where Dα t and D β x are fractional derivatives of order α and β is studied. It is shown that if f = f(t, x), h1 = h1(x), and h2 = h2(t) are Hölder-continuous and f(0, 0) = 0, then there is a solution such that Dα t u and D β xu are Höldercontinuous as well. This is proved by first considering an abstract fractional evolution equation and then applying the results obtained to (∗). Finally the solution of (∗) with f = 1 is studied.

Approximation by neural networks with a bounded number of nodes at each level

It is shown that the general approximation property of feed-forward multilayer perceptron networks can be achieved in networks where the number of nodes in each layer is bounded, but the number of layers grows to infinity. This is the case provided the node function is twice continuously differentiable and not linear.

Expression of antibody activity measured by ELISA. Anti-ssDNA antibody activity characterized by the shape of the dose-response curve

To demonstrate the presence or absence of antibodies, results derived from a single serum dilution in an ELISA are sufficient. However, qualitative differences in antibodies are reflected by the shape of dose-response curves. A method based on approximating the absorbance value by a polynomial p(x) = a1x + a2x2, where 1/x is the dilution factor, was used to characterize the dose-response curves in an ELISA for anti=ssDNA antibodies. The parameters used are E = a1 and A = -a2(1)/a2. It can be argues that E gives an estimate of the effective amount of antibodies in the serum and that A is essentially a function of the reaction constants between antibody and antigen.

Periodic solutions of an epidemic model

SummaryThe existence of periodic solutions of the equation

On volterra equations of the first kind

x(t) = k\left( {P - \int_{ - \infty }^t A(t - s)x(s)ds } \right)\int_{ - \infty }^t a(t - s)x(s)ds

An Existence Result for a Nonlinear Volterra Integral Equation in a Hilbert Space

is established. This equation arises in the study of the spread of a disease which does not induce permanent immunity.

Nonexistence of Smooth Solutions for Shearing Flows in a Nonlinear Viscoelastic Fluid

The problem of the existence of a stationary distribution and the convergence towards it in a certain semistochastic model for the growth of a population is considered. It is assumed that the population grows according to a deterministic equation, but at certain times there are catastrophes, which lead to a decrease in the population level. The hazard function for the occurrence of catastrophes is a function of the population level only. The size of these jumps have a distribution that depends on the population size immediately before the catastrophe. A constructive method for finding the stationary distribution is given.