Functionalized hyperbranched polymers via olefin metathesis

Hyperbranched polymers are highly branched, three-dimensional macromolecules which are closely related to dendrimers and are typically prepared via a one-pot polycondensation of AB_(n≥2) monomers.^1 Although hyperbranched macromolecules lack the uniformity of monodisperse dendrimers, they still possess many attractive dendritic features such as good solubility, low solution viscosity, globular structure, and multiple end groups.^1-3 Furthermore, the usually inexpensive, one-pot synthesis of these polymers makes them particularly desirable candidates for bulk-material and specialty applications. Toward this end, hyperbranched polymers have been investigated as both rheology-modifying additives to conventional polymers and as substrate-carrying supports or multifunctional macroinitiators, where a large number of functional sites within a compact space becomes beneficial

Towards the quantum Brownian motion

We consider random Schr\"odinger equations on \bR^d or \bZ^d for $d\ge 3$ with uncorrelated, identically distributed random potential. Denote by $\lambda$ the coupling constant and $\psi_t$ the solution with initial data $\psi_0$. Suppose that the space and time variables scale as $x\sim \lambda^{-2 -\kappa/2}, t \sim \lambda^{-2 -\kappa}$ with $0< \kappa \leq \kappa_0$, where $\kappa_0$ is a sufficiently small universal constant. We prove that the expectation value of the Wigner distribution of $\psi_t$, \bE W_{\psi_{t}} (x, v), converges weakly to a solution of a heat equation in the space variable $x$ for arbitrary $L^2$ initial data in the weak coupling limit $\lambda \to 0$. The diffusion coefficient is uniquely determined by the kinetic energy associated to the momentum $v$.Comment: Self-contained overview (Conference proceedings). The complete proof is archived in math-ph/0502025. Some typos corrected and new references added in the updated versio

Induced Gauge Theory on a Noncommutative Space

We discuss the calculation of the 1-loop effective action on four dimensional, canonically deformed Euclidean space. The theory under consideration is a scalar $\phi^4$ model with an additional oscillator potential. This model is known to be re normalisable. Furthermore, we couple an exterior gauge field to the scalar field and extract the dynamics for the gauge field from the divergent terms of the 1-loop effective action using a matrix basis. This results in proposing an action for noncommutative gauge theory, which is a candidate for a renormalisable model.Comment: 8 page

Probabilistic simulation for the certification of railway vehicles

The present dynamic certification process that is based on experiments has been essentially built on the basis of experience. The introduction of simulation techniques into this process would be of great interest. However, an accurate simulation of complex, nonlinear systems is a difficult task, in particular when rare events (for example, unstable behaviour) are considered. After analysing the system and the currently utilized procedure, this paper proposes a method to achieve, in some particular cases, a simulation-based certification. It focuses on the need for precise and representative excitations (running conditions) and on their variable nature. A probabilistic approach is therefore proposed and illustrated using an example. First, this paper presents a short description of the vehicle / track system and of the experimental procedure. The proposed simulation process is then described. The requirement to analyse a set of running conditions that is at least as large as the one tested experimentally is explained. In the third section, a sensitivity analysis to determine the most influential parameters of the system is reported. Finally, the proposed method is summarized and an application is presented

Randomization and the Gross-Pitaevskii hierarchy

We study the Gross-Pitaevskii hierarchy on the spatial domain $\mathbb{T}^3$. By using an appropriate randomization of the Fourier coefficients in the collision operator, we prove an averaged form of the main estimate which is used in order to contract the Duhamel terms that occur in the study of the hierarchy. In the averaged estimate, we do not need to integrate in the time variable. An averaged spacetime estimate for this range of regularity exponents then follows as a direct corollary. The range of regularity exponents that we obtain is $\alpha>\frac{3}{4}$. It was shown in our previous joint work with Gressman that the range $\alpha>1$ is sharp in the corresponding deterministic spacetime estimate. This is in contrast to the non-periodic setting, which was studied by Klainerman and Machedon, in which the spacetime estimate is known to hold whenever $\alpha \geq 1$. The goal of our paper is to extend the range of $\alpha$ in this class of estimates in a \emph{probabilistic sense}. We use the new estimate and the ideas from its proof in order to study randomized forms of the Gross-Pitaevskii hierarchy. More precisely, we consider hierarchies similar to the Gross-Pitaevskii hierarchy, but in which the collision operator has been randomized. For these hierarchies, we show convergence to zero in low regularity Sobolev spaces of Duhamel expansions of fixed deterministic density matrices. We believe that the study of the randomized collision operators could be the first step in the understanding of a nonlinear form of randomization.Comment: 51 pages. Revised versio

Underdetermined-order recursive least-squares adaptive filtering: The concept and algorithms

Published versio