arXiv subjects

Heinz Weisshaupt

Publications and source records attributed to Heinz Weisshaupt.

3 recordsLinked to original sources

arXiv2015

Dimension-raising Homomorphisms between Lattices of Convex Bodies

We settle the first unsolved case of a problem of P. M. Gruber, asked by him in 1991, namely, to investigate the homomorphisms from the lattice of convex bodies of ${\mathbb{E}}^c$ to the lattice of convex bodies of ${\mathbb{E}}^d$ for $c<d$. We completely describe these homomorphisms for the case $d=c+1$, for $c \ge 3$. The obtained result is then applied to characterize anti-homomorphisms and homomorphisms from lattices of convex bodies to lattices of convex functions.

math.MG↗

arXiv2011

Sensitivity analysis of one parameter semigroups exemplified by the Wright--Fisher diffusion

We consider the sensitivity, with respect to a parameter θ, of parametric families of operators A_θ, vectors π_θ corresponding to the adjoints A_θ^{*} of A_θ via A_θ^{*}π_θ=0 and one parameter semigroups t\mapsto e^{tA_θ}. We display formulas relating weak differentiability of θ\mapsto π_θ (at θ=0) to weak differentiability of θ\mapsto A_θ^{*}π_{0} and [e^{A_θt}]^{*}π_{0}. We give two applications: The first one concerns the sensitivity of the Ornstein--Uhlenbeck process with respect to its location parameter. The second one provides new insights regarding the Wright--Fisher diffusion for small mutation parameter.

math.FA↗

arXiv2010

The tree length of an evolving coalescent

A well-established model for the genealogy of a large population in equilibrium is Kingman's coalescent. For the population together with its genealogy evolving in time, this gives rise to a time-stationary tree-valued process. We study the sum of the branch lengths, briefly denoted as tree length, and prove that the (suitably compensated) sequence of tree length processes converges, as the population size tends to infinity, to a limit process with cadlag paths, infinite infinitesimal variance, and a Gumbel distribution as its equilibrium.

math.PR↗