arXiv subjects
Publications and source records attributed to Martin Schottenloher.
The unitary group $\mathrm U(\mathcal H)$ on an infinite dimensional complex Hilbert space $\mathcal H$ in its strong topology is a topological group and has some further nice properties, e.g. it is metrizable and contractible if $\mathcal H$ is separable. As an application Hilbert bundles are classified by homotopy.
Coexistence of competing species is, due to unavoidable fluctuations, always transient. In this Letter, we investigate the ultimate survival probabilities characterizing different species in cyclic competition. We show that they often obey a surprisingly simple, though non-trivial behavior. Within a model where coexistence is neutrally stable, we demonstrate a robust zero-one law: When the interactions between the three species are (generically) asymmetric, the `weakest' species survives at a probability that tends to one for large population sizes, while the other two are guaranteed to extinct. We rationalize our findings from stochastic simulations by an analytic approach.
In this article we propose a new and so-called holomorphic deformation scheme for locally convex algebras and Hopf algebras. Essentially we regard converging power series expansion of a deformed product on a locally convex algebra, thus giving the means to actually insert complex values for the deformation parameter. Moreover we establish a topological duality theory for locally convex Hopf algebras. Examples coming from the theory of quantum groups are reconsidered within our holomorphic deformation scheme and topological duality theory. It is shown that all the standard quantum groups comprise holomorphic deformations. Furthermore we show that quantizing the function algebra of a (Poisson) Lie group and quantizing its universal enveloping algebra are topologically dual procedures indeed. Thus holomorphic deformation theory seems to be the appropriate language in which to describe quantum groups as deformed Lie groups or Lie algebras.