Brownian motion in attenuated or renormalized inverse-square Poisson potential

We consider the parabolic Anderson problem with random potentials having inverse-square singularities around the points of a standard Poisson point process in $\mathbb{R}^d$, $d \geq 3$. The potentials we consider are obtained via superposition of translations over the points of the Poisson point process of a kernel $\mathfrak{K}$ behaving as $\mathfrak{K}(x) \approx \theta |x|^{-2}$ near the origin, where $\theta \in (0,(d-2)^2/16]$. In order to make sense of the corresponding path integrals, we require the potential to be either attenuated (meaning that $\mathfrak{K}$ is integrable at infinity) or, when $d=3$, renormalized, as introduced by Chen and Kulik in [8]. Our main results include existence and large-time asymptotics of non-negative solutions via Feynman-Kac representation. In particular, we settle for the renormalized potential in $d=3$ the problem with critical parameter $\theta = 1/16$, left open by Chen and Rosinski in [arXiv:1103.5717].Comment: 36 page

Fitting the Cusp Catastrophe in R: A cusp Package Primer

Of the seven elementary catastrophes in catastrophe theory, the âÂÂcuspâ model is the most widely applied. Most applications are however qualitative. Quantitative techniques for catastrophe modeling have been developed, but so far the limited availability of flexible software has hindered quantitative assessment. We present a package that implements and extends the method of Cobb (Cobb and Watson'80; Cobb, Koppstein, and Chen'83), and makes it easy to quantitatively fit and compare different cusp catastrophe models in a statistically principled way. After a short introduction to the cusp catastrophe, we demonstrate the package with two instructive examples.

R & D in education: the case of the laboratory school

This symposium presents ten years of an Israeli laboratory school, which was established on research and development (R & D) principles. First, David Chen presents four major requirements for R&D strategy: new theoretical foundation, educational research, laboratories schools and diffusion mechanisms. Second, Nir Chen presents the architectural perspective and describes ways by which school building promotes new pedagogy. Third, Yafa Ben-Amy explains how the school and, especially the "home" - the basic organizational unit, work. Finally, Dorit Tubin shows some of the evidence with regards to the schools succes

Homology and K-theory of the Bianchi groups

We reveal a correspondence between the homological torsion of the Bianchi groups and new geometric invariants, which are effectively computable thanks to their action on hyperbolic space. We use it to explicitly compute their integral group homology and equivariant $K$-homology. By the Baum/Connes conjecture, which holds for the Bianchi groups, we obtain the $K$-theory of their reduced $C^*$-algebras in terms of isomorphic images of the computed $K$-homology. We further find an application to Chen/Ruan orbifold cohomology. % {\it To cite this article: Alexander D. Rahm, C. R. Acad. Sci. Paris, Ser. I +++ (2011).