Synthesis,Spectroscopy, Crystal Structure Determination,and Quantum Chemical Calculations of BODIPY Dyes with Increasing Conformational Restriction and Concomitant Red‐Shifted Visible Absorption and Fluorescence Spectra

Starting from the conformationally unconstrained compound 3,5‐di‐(2‐bromophenoxy)‐4,4‐difluoro‐8‐(4‐methylphenyl)‐4‐bora‐3a,4a‐diaza‐s‐indacene ( 1 ), two BODIPY dyes ( 2 and 3 ) with increasingly rigid conformations were synthesized in outstanding total yields through palladium catalyzed intramolecular benzofuran formation. Restricted bond rotation of the phenoxy fragments leads to dyes 2 and 3 , which absorb and fluoresce more intensely at longer wavelengths relative to the unconstrained dye 1 . Reduction of the conformational flexibility in 2 and 3 leads to significantly higher fluorescence quantum yields compared to those of 1 . X‐ray diffraction analysis shows the progressively more extended planarity of the chromophore in line with the increasing conformational restriction in the series 1 → 2 → 3 , which explains the larger red shifts of the absorption and emission spectra. These conclusions are confirmed by quantum chemical calculations of the lowest electronic excitations in 1 , 1a , 2 , 2a , 3 and dyes of related chemical structures. The effect of the molecular structure on the visible absorption and fluorescence emission properties of 1 , 1a , 2 , 2a , 3 has been examined as a function of solvent by means of the new, generalized treatment of the solvent effect (J. Phys. Chem. B 2009 , 113, 5951–5960). Solvent polarizability is the primary factor responsible for the small solvent‐dependent shifts of the visible absorption and fluorescence emission bands of these dyes.     

Synthesis, characterization, and controlled aggregation of biotemplated polystyrene nanodisks

Cross-linked polystyrene nanodisks were prepared by controlled polymerization of styrene and divinylbenzene in the interior of bicelles, discoidal lipid aggregates. Aggregation behavior of polymer nanodisks was studied in water, organic solvents, and solid phase. Nanodisks form stable dispersions in aqueous solutions of surfactants, such as sodium dodecyl sulfate (SDS). Varying SDS/nanodisk ratio allowed us to control the size of nanodisk aggregates. Nanodisks are readily solubilized in nonpolar organic solvents, such as toluene and carbon tetrachloride, to yield stable monodisperse suspensions. These findings open opportunities for creating nanodisk-based nanocomposite materials. Stable nanodisk suspension in toluene enabled small angle neutron scattering (SANS) measurements. SANS data confirmed the nanodisk diameter and allowed accurate measurement of nanodisk thickness (19.5 ± 1.0 Å). In solid phase, nanodisks aggregate in sub-micron platelets.     

On the parameter choice in grad-div stabilization for the Stokes equations

Standard error analysis for grad-div stabilization of inf-sup stable conforming pairs of finite element spaces predicts that the stabilization parameter should be optimally chosen to be $\mathcal O(1)$ . This paper revisits this choice for the Stokes equations on the basis of minimizing the $H^{1}(\Omega )$ error of the velocity and the $L^{2}(\Omega )$ error of the pressure. It turns out, by applying a refined error analysis, that the optimal parameter choice is more subtle than known so far in the literature. It depends on the used norm, the solution, the family of finite element spaces, and the type of mesh. In particular, the approximation property of the pointwise divergence-free subspace plays a key role. With such an optimal approximation property and with an appropriate choice of the stabilization parameter, estimates for the $H^{1}(\Omega )$ error of the velocity are obtained that do not directly depend on the viscosity and the pressure. The minimization of the $L^{2}(\Omega )$ error of the pressure requires in many cases smaller stabilization parameters than the minimization of the $H^{1}(\Omega )$ velocity error. Altogether, depending on the situation, the optimal stabilization parameter could range from being very small to very large. The analytic results are supported by numerical examples. Applying the analysis to the MINI element leads to proposals for the stabilization parameter which seem to be new.     

Stability Analysis for maxwell’s Equation with a Thermal Effect in one Space Dimension

In this paper we study the asymptotic behavior of a system modeling heating of material by microwaves. Various assumptions have been made, concerning complexity (nonhomogeneous structure) and the two-phase state of the material. The mathematical model includes Maxwell’s and heat–transfer equations. Stability of solutions of the system is shown.     

Multiscale modeling of continuous fiber and fabric reinforced rubber components

Fabric or continuous fiber reinforced rubber components (e.g. tires, air springs, industrial hoses, conveyor belts or membranes) are underlying high deformations in application and show a complex, nonlinear material behavior. A particular challenge depicts the simulation of these composites. In this contribution we show the identification of the stress and strain distributions by using an uncoupled multiscale modeling method, see [1]. Within this method, two representation levels are described: One, the meso level, where all constituents of the composite are shown in a discrete manner by a representative volume element (RVE) and secondly, the macro level, where the structural behavior of the component is defined by a smeared anisotropic hyperelastic constitutive law. Uncoupled means that the RVE does not drive the macroscopic material behavior directly as in a coupled approach, where a RVE boundary value problem has to be solved at every integration point of the macro level. Thus an uncoupled approach leads to a tremendous reduction in numerical effort because the boundary value problem of a RVE just has to be solved at a point of interest, see [1]. However, the uncoupled scale transition has to fulfill the HILL–MANDEL condition of energetic equivalence of both scales. We show the calibration of material parameters for a given constitutive model for fiber reinforced rubber by fitting experimental data on the macro level. Additionally, we demonstrate the determination of effective properties of the yarns. Finally, we compare the energies of both scales in terms of compliance with the HILL–MANDEL condition by using the example of a biaxial loaded sample and discuss the consequences for the mesoscopic level. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal control of impulsive switched systems with minimum subsystem durations

This paper presents a new computational approach for solving optimal control problems governed by impulsive switched systems. Such systems consist of multiple subsystems operating in succession, with possible instantaneous state jumps occurring when the system switches from one subsystem to another. The control variables are the subsystem durations and a set of system parameters influencing the state jumps. In contrast with most other papers on the control of impulsive switched systems, we do not require every potential subsystem to be active during the time horizon (it may be optimal to delete certain subsystems, especially when the optimal number of switches is unknown). However, any active subsystem must be active for a minimum non-negligible duration of time. This restriction leads to a disjoint feasible region for the subsystem durations. The problem of choosing the subsystem durations and the system parameters to minimize a given cost function is a non-standard optimal control problem that cannot be solved using conventional techniques. By combining a time-scaling transformation and an exact penalty method, we develop a computational algorithm for solving this problem. We then demonstrate the effectiveness of this algorithm by considering a numerical example on the optimization of shrimp harvesting operations.     

Spatial Random Permutations and Infinite Cycles

We consider systems of spatial random permutations, where permutations are weighed according to the point locations. Infinite cycles are present at high densities. The critical density is given by an exact expression. We discuss the relation between the model of spatial permutations and the ideal and interacting quantum Bose gas.