A Pregenerator for Burgers Equation¶Forced by Conservative Noise

We consider the stationarity of a Burgers equation with an external random force of gradient type in one space dimension. The expected stationary measure is the white noise measure on the space of tempered distributions. As a consequence, the nonlinearity of the formal equation u _t +λu u _x =νu _xx +η _x is ill-defined. Introducing a pregenerator we can formulate a generalized martingale problem leading to a meaningful version of the formal equation which was an open problem. Received: 9 March 2001 / Accepted: 10 October 2001     

Heavy element research at GSI — Recent results and perspectives

The nuclear shell model predicts that the next doubly magic shellclosure beyond ²⁰⁸Pb is at a proton number between Z=114 and 126 and at a neutron number N=184. The outstanding aim of experimental investigations is the exploration of this region of spherical ‘Superheavy Elements’. This article describes the experiments that were performed recently at the GSI SHIP. They resulted in an unambiguous identification of elements 110 to 112. They were negative so far in searching for elements 113, 116, and 118. The measured decay data are compared with theoretical predictions. Some aspects concerning the reaction mechanism are also presented.     

Synthesis and solid-state organization of coil-ring-coil block copolymers

Shape-persistent macrocycles based on the phenyl-ethynyl-butadienyl backbone containing two extraannular hydroxyl groups were prepared by the oxidative coupling of the appropriate phenylethynyl oligomers. Carbodiimide-directed coupling with independently synthesized polystyrene carboxylic acid oligomers led to ABA coil-ring-coil block copolymers in which the central macrocycle serves as rigid and the polystyrene oligomers as flexible elements. Depending on the size of the coil blocks, these structures aggregate in cyclohexane into supramolecular hollow cylindrical brushes in which the rigid core is surrounded by the flexible matrix. However, in the solid state it is not possible to identify a morphology in which isolated channels based on aggregated macrocycles are embedded in a matrix of polystyrene. Detailed X-ray and electron diffraction studies on samples prepared from a solution in cyclohexane under equilibrium conditions show that the material adopts a lamellar morphology in the solid state in which columns of macrocycles are aggregated into layers which are separated by polystyrene.     

Solvent triggering between conformational states in amphiphilic shape-persistent macrocycles

The amphiphilic shape-persistent macrocycle 1 containing four phenol-OH groups as polar side groups and four hexyloxy groups as nonpolar side groups in an adaptable arrangement was recrystallized from solvents of different polarity. X-ray crystallography reveals that the conformation of the macrocycle is solvent dependent such that in the pyridine solvate only two of the nonpolar side groups point outward while in the THF solvate all four of them point outward. Moreover, in the latter case the three-dimensional packing leads to the formation of a supramolecular channel structure with a large pore size.     

Simultaneous Raman and fluorescence spectroscopy of single conjugated polymer chains

Simultaneous surface enhanced Raman scattering (SERS) and fluorescence is demonstrated from single conjugated polymer chains. As resonance enhancement of SERS depends on the spectral overlap of the polymer's absorption and the incident laser, resonance Raman and fluorescence effectively probe the absorbing and emitting part of the polymer, respectively. The optical phonon energies change along the polymer chain, providing a window to spatially track excited state relaxation. Whereas a mean spatial redistribution of the excitation is witnessed by a change in vibronic fingerprint following interchromophoric energy transfer, intrachromophoric exciton self-trapping leaves the vibrations unchanged.     

Transfer Operators and State Spaces for Discrete Multidimensional Linear Systems

This paper treats multidimensional discrete input-output systems from the constructive point of view. We adapt and improve recursive algorithms, derived earlier by E. Zerz and the second author from standard Gröbner basis algorithms, for the solution of the canonical Cauchy problem for linear systems of partial difference equations with constant coefficients on the lattices N = ^r1 × ^r2. These recursive algorithms, in turn, furnish four other solution methods for the initial value problem, namely by transfer operators, by canonical Kalman global state equations, by parametrizations of controllable systems and, for systems with proper transfer matrix and left bounded input signals, by convolution with the transfer matrix. In the 2D-case N = ² the last method was studied by S. Zampieri. Minimally embedded systems are studied and give rise to especially simple Kalman equations. The latter also imply a useful characterization of the characteristic or polar variety of the system by eigenvalue spectra. For N = ^r we define reachability of a system and prove that controllability implies reachability, but not conversely. Moreover we solve, in full generality, the modelling problem which was introduced and partially solved by F. Pauer and S. Zampieri. Various algorithms have been implemented by the first author in axiom, and examples are demonstrated by means of computer generated pictures. Related work on state space representations has been done by the Padovian and Groningian system theory schools.     

Self-diffusion in core-shell composite 12CO2/13CO2 nanoparticles

A new technique for the generation of multilayered molecular nanoparticles is presented. Core-shell composite nanoparticles of CO(2) with varied composition have been investigated by Fourier-transform infrared spectroscopy over 600 s at 78 K. The isotopically different zones of the particles turned out to have completely different spectra in the nu(3) region: a tub structure (mantle) and a head-and-shoulders structure (core). From the aggregation behavior of both components the particle formation time was found to be 0.1 s. Low-temperature self-diffusion of airborne molecular nanoparticles has been monitored for the first time. The self-diffusion coefficient for (12)CO(2)/(13)CO2 nanocomposites at 78 K was determined from the time evolution of the nu(1) + nu(3) combination band to about 7 x 10(-20) m(2)/s. The work represents a major advance toward nanoengineering of molecular nanoparticles at low temperatures.