Stability of multistep-methods on variable grids

Summary This paper is concerned with the stability of multistep methods for ordinary initial-value problems on grids with variable mesh-sizes. A necessary and sufficient condition for stability is given from which generalizations of recent results by Gear et al. and by Zlatev can be obtained as special cases. As an application the stability of the variable BDF-formulas is treated.     

Point processes and multivariate extreme values

A new model for point processes is developed which assumes that the interarrival times are exponentially distributed and follow joint multivariate extreme value distributions. It is shown that such processes may arise via natural generating procedures, and that, under very weak assumptions, that they can be approximated as closely as desired by appropriate finite models.     

The uniqueness of the near hexagon on 729 points

We prove that any regular near hexagon with 729 vertices and lines of size 3 is derived from the ternary Golay code, thus settling the last case in doubt among the regular near hexagons with lines of size 3.     

Necessary conditions for normed convergence of critical multitype Bienaymé-Galton-Watson processes without variance

The condition on the offspring distribution in the critical multitype Bienaymé-Galton-Watson process without variance, which was previously shown to be sufficient for the existence of the analogue of the exponential limit law, is now shown also to be necessary. This completely extends previous one-type work of R. S. Slack.     

On the joint distribution of two discrete random variables

Summary LetX, Y be two discrete random variables with finite support andX≧Y. Suppose that the conditional distribution ofY givenX can be factorized in a certain way. This paper provides a method of deriving the unique form of the marginal distribution ofX (and hence the joint distribution of (X, Y)) when partial independence only is assumed forY andX−Y.     

Error bounds for computed eigenvalues and eigenvectors

Summary On the basis of an existence theorem for solutions of nonlinear systems, a method is given for finding rigorous error bounds for computed eigenvalues and eigenvectors of real matrices. It does not require the usual assumption that the true eigenvectors span the whole space. Further, a priori error estimates for eigenpairs corrected by an iterative method are given. Finally the results are illustrated with numerical examples.Dedicated to Professor Yoshikazu Nakai on his sixtieth birthday     

Effective computation of periodic orbits and bifurcation diagrams in delay equations

Summary By employing a numerical method which uses only rather classical tools of Numerical Analysis such as Newton's method and routines for ordinary differential equations, unstable periodic solutions of differential-difference equations can be computed. The method is applied to determine bifurcation diagrams with backward bifurcation.Dedicated to Professor Lothar Collatz on the occasion of his 70th birthdayThis paper has been read at the Conference of Numerical Mathematicians, Zeist, The Netherlands, October 12–15, 1979     

Invariant scale matrix hypothesis tests under elliptical symmetry

The usual assumption in multivariate hypothesis testing is that the sample consists of n independent, identically distributed Gaussian m-vectors. In this paper this assumption is weakened by considering a class of distributions for which the vector observations are not necessarily either Gaussian or independent. This class contains the elliptically symmetric laws with densities of the form f(X(n × m)) = ψ[tr(X ? M)′ (X ? M)Σ^?1]. For testing the equality of k scale matrices and for the sphericity hypothesis it is shown, by using the structure of the underlying distribution rather than any specific form of the density, that the usual invariant normal-theory tests are exactly robust, for both the null and non-null cases, under this wider class.     

Graph factorization,general triple systems,and cyclic triple systems

In this self-contained exposition, results are developed concerning one-factorizations of complete graphs, and incidence matrices are used to turn these factorization results into embedding theorems on Steiner triple systems. The result is a constructive graphical proof that a Steiner triple system exists for any order congruent to 1 or 3 modulo 6. A pairing construction is then introduced to show that one can also obtain triple systems which are cyclically generated.