On the Order Reduction Approach for Singularly Perturbed Optimal Control Systems

The order reduction method for singularly perturbed optimal control systems consists of employing the system obtained while setting the small parameter to be zero. In many situations the differential-algebraic system thus obtained indeed provides an appropriate approximation to the singularly perturbed problem with a small parameter. In this paper we establish that if relaxed controls are allowed then the answer to the question whether or not this method is valid depends essentially on one simple parameter: the dimension of the fast variable, denoted n. More specifically, if n=1 then the order reduction method is indeed applicable, while if n>1 then the set of singularly perturbed optimal control systems for which it is not applicable is dense (in the L norm).     

Spectral and temporal holograms with nonlinear optics

In this Letter we show how encoding techniques for computer-generated holograms may be used to arbitrarily shape a nonlinearly generated spectrum and consequently the temporal shape by modulating the quadratic nonlinear coefficient. We give examples of a modulation pattern and a simple setup that can generate high-order Hermite-Gauss and Airy functions through difference-frequency generation from a transform-limited Gaussian pulse, under practical fabrication considerations.     

Wormlike aggregates from a supramolecular coordination polymer and a diblock copolymer

The formation of wormlike micelles in mixed systems of a supramolecular coordination polymer Zn-L2EO4 and a diblock copolymer P2MVP41-b-PEO205 is investigated by light scattering and Cryo-TEM. By direct mixing at a stoichiometric charge ratio, the above mixtures proved to be capable of formation of spherical micelles with a radius of about 25 nm (Yan et al. Angew. Chem., Int. Ed.; 2007, 46, 1807-1809). Lately, we find wormlike micelles with a hydrodynamic radius >150 nm in a mixture with excess positive charge, that is, a negative charge fraction f- < 0.5. The transformation between wormlike and spherical micelles can be realized by variation of the mixing ratio through different protocols. Upon addition of negatively charged Zn-L2EO4 to a mixture with excess positively charged P2MVP41-b-PEO205, most of the wormlike micelles are transformed into spherical ones; upon addition of positively charged P2MVP41-b-PEO205 to a mixture of pure spherical micelles, wormlike micelles can be produced again. The effect of sample preparation protocol, sample history, and concentration on this transformation process is systematically reported in this article. A possible mechanism for the formation of wormlike micelles is proposed.     

Measuring location, size, distribution, and loading of NiO crystallites in individual SBA-15 pores by electron tomography

By the combination of electron tomography with image segmentation, the properties of 299 NiO crystallites contained in 6 SBA-15 pores were studied. A statistical analysis of the particle size showed that crystallites between 2 and 6 nm were present with a distribution maximum at 3 and 4 nm, for the number-weighted and volume-weighted curves, respectively. Interparticle distances between nearest neighbors were 1-3 nm with very few isolated crystallites. In the examined pores, a local loading twice the applied average of 24 wt % NiO was found. This suggests that a very high local loading combined with a high dispersion is achievable.     

Molecular assemblies from imidazolyl-containing haloalkenes and haloalkynes: competition between halogen and hydrogen bonding

The structural characterization of molecular assemblies constructed from imidazolyl-containing haloalkenes and haloalkynes is reported. 1-(3-Iodopropargyl)imidazole (2) and 1-(2,3,3-triiodoallyl)imidazole (5) were synthesized from 1-propargylimidazole (1). In the solid state, these wholly organic modules self-assemble through N...I halogen-bonding interactions, thus giving rise to polymeric chains. The N...I interaction observed in 2 (d(N...I)=2.717 A, angle-spherical C(sp)-I...N=175.8 degrees) is quite strong relative to previously reported data. The N...I interaction in 5 (d(N...I)=2.901 A, angle-spherical C(sp2)-I...N=173.6 degrees) is weaker, in accordance with the order C(sp)-X<--base>C(sp2)-X<--base. Compound 5 was found to give a 1:1 cocrystal 4 with morpholinium iodide (6). In the X-ray crystal studies of 4, N...I halogen-bonding interactions similar to those observed in 5 were shown not to be present, as the arrangement of the molecules is governed by two interwoven hydrogen-bonding networks. The first network involves N-H...O interactions between nearby morpholinium cations, and the second network is based on N-H...N hydrogen bonding between morpholinium cations and imidazolyl groups. Both hydrogen-bonding schemes are charge-assisted. Halogen bonding is not completely wiped out, however, as the triiodoalkene fragment forms a halogen bond with an iodide anion in its vicinity (d(I...I)=3.470 A, angle-spherical C(sp2)-I...I=170.7 degrees). X-ray crystal studies of 6 show a completely different arrangement from that observed in 4, namely, N-H...O interactions are not present. In crystalline 6, morpholinium cations are interconnected through C-H...O bridges (d(H...O)=2.521 and 2.676 A), and the NH2+ groups interact with nearby iodide anions (d(H...I)=2.633 and 2.698 A).     

Molecular adjustment of the electronic properties of nanoporous electrodes in dye-sensitized solar cells

Molecular modification of dye-sensitized, mesoporous TiO2 electrodes changes their electronic properties. We show that the open-circuit voltage (V(oc)) of dye-sensitized solar cells varies linearly with the dipole moment of coadsorbed phosphonic, benzoic, and dicarboxylic acid derivatives. A similar dependence is observed for the short-circuit current density (I(sc)). Photovoltage spectroscopy measurements show a shift of the signal onset as a function of dipole moment. We explain the dipole dependence of the V(oc) in terms of a TiO2 conduction band shift with respect to the redox potential of the electrolyte, which is partially followed by the energy level of the dye. The I(sc) shift is explained by a dipole-dependent driving force for the electron current and a dipole-dependent recombination current.     

Linear and nonlinear optical properties of MgO:LiTaO3

We derive new temperature dependent Sellmeier equations for the extra-ordinary and ordinary refractive indices of 0.5% MgO-doped stoichiometric LiTaO₃ crystal. The equations are based on quasi phase-matched frequency conversion measurements, as well as on interferometric measurements of the thermal expansion and thermal dispersion. These equations fit experimental data over wide spectral ranges: 0.35–6 μm for the extra-ordinary wave and 0.375–3.75 μm for the ordinary wave, from room temperature up to 200°C. The nonlinear optical measurements set lower-limit values for the d ₃₃, d ₂₂, and d ₂₄ elements of the second-order susceptibility tensor χ ⁽²⁾ of 12.9, 1.54, and 0.46 pm/V, respectively. The interferometric measurements enable one to determine the linear and quadratic expansion coefficients of 1.45762e?5 1/°C, and 2.68608e?8 (1/°C)², respectively. The Sellmeier equations are in good agreement with previously published data.