Numerical computation of stability and detection of Hopf bifurcations of steady state solutions of delay differential equations

The characteristic equation of a system of delay differential equations (DDEs) is a nonlinear equation with infinitely many zeros. The stability of a steady state solution of such a DDE system is determined by the number of zeros of this equation with positive real part. We present a numerical algorithm to compute the rightmost, i.e., stability determining, zeros of the characteristic equation. The algorithm is based on the application of subspace iteration on the time integration operator of the system or its variational equations. The computed zeros provide insight into the system’s behaviour, can be used for robust bifurcation detection and for efficient indirect calculation of bifurcation points. This revised version was published online in June 2006 with corrections to the Cover Date.     

Interpreting technical evidence from spectral imaging of paintings by Édouard Manet in the Courtauld Gallery

The paintings by Édouard Manet in The Courtauld Gallery Déjeuner sur l'herbe (1863–68), Marguerite de Conflans en Toilette de Bal (1870–1880), Banks of the Seine at Argenteuil (1874), and A Bar at the Folies–Bergère (1882) were investigated for the first time using a range of non-invasive in situ analyses. The aims of the study were to investigate the painting techniques and materials used for this group of works and to critically evaluate the technical evidence derived from the integrated use of imaging techniques and portable spectroscopic methods in this context. The paintings were investigated by means of macro X-ray fluorescence (MA-XRF), reflection spectral imaging, portable UV–Vis–NIR spectroscopy, portable Raman spectroscopy, and reflection FTIR. MA-XRF and reflection spectral imaging allowed visualising elements in the compositions that were not visible using traditional methods of technical study. For example, MA-XRF analysis of Déjeuner sur l'herbe revealed elements of the development of the composition that provided new evidence to consider its relationship to other versions of the composition. The study also highlighted questions about the interpretation of elemental distribution maps and spectral images that did not correspond to the reworking visible in X-radiographs. For example, in A Bar at the Folies–Bergère Manet made numerous changes during painting, which were not clearly visualised with any of the techniques used. The research has wider implications for the study of Impressionist paintings, as the results will support technical studies of works by other artists of the period who used similar materials and painting methods.     

Unusual frequency dispersion effects of the nonlinear optical response in highly conjugated (polypyridyl)metal-(porphinato)zinc(II) chromophores

The syntheses and electrooptic properties of a new family of nonlinear optical chromophores are reported. These species feature an ethyne-elaborated, highly polarizable porphyrinic component and metal polypyridyl complexes that serve as integral donor and acceptor elements. Examples of this structural motif include ruthenium(II) [5-(4'-ethynyl-(2,2';6',2' '-terpyridinyl))-10,20-bis(2',6'-bis(3,3-dimethyl-1-butyloxy)phenyl)porphinato]zinc(II)-(2,2';6',2' '-terpyridine)(2+) bis-hexafluorophosphate (Ru-PZn); osmium(II) [5-(4'-ethynyl-(2,2';6',2'-terpyridinyl))-10,20-bis(2',6'-bis(3,3-dimethyl-1-butyloxy)phenyl)porphinato]zinc(II)-(2,2';6',2'-terpyridine)(2+) bis-hexafluorophosphate (Os-PZn); ruthenium(II) [5-(4'-ethynyl-(2,2';6',2'-terpyridinyl))-15-(4'-nitrophenyl)ethynyl-10,20-bis(2',6'-bis(3,3-dimethyl-1-butyloxy)phen-yl)porphinato]zinc(II)-(2,2';6',2' '-terpyridine)(2+) bis-hexafluorophosphate (Ru-PZn-A); osmium(II) [5-(4'-ethynyl-(2,2';6',2' '-terpyridinyl))-15-(4'-nitrophenyl)ethynyl-10,20-bis(2',6'-bis(3,3-dimethyl-1-butyloxy)phenyl)porphinato]zinc(II)-(2,2';6',2' '-terpyridine)(2+) bis-hexafluorophosphate (Os-PZn-A); and ruthenium(II) [5-(4'-ethynyl-(2,2';6',2' '-terpyridinyl))osmium(II)-15-(4'-ethynyl-(2,2';6',2'-terpyridinyl))-10,20-bis (2',6'-bis(3,3-dimethyl-1-butyloxy)phenyl)porphinato]zinc(II)-bis(2,2';6',2'-terpyridine)(4+) tetrakis-hexafluorophosphate (Ru-PZn-Os). The frequency dependence of the dynamic hyperpolarizability of these compounds was determined from hyperRayleigh light scattering (HRS) measurements carried out at fundamental incident irradiation wavelengths (lambda(inc)) of 800, 1064, and 1300 nm. These data show that (i) coupled oscillator photophysics and metal-mediated cross-coupling can be exploited to elaborate high beta(0) supermolecules that exhibit significant excited-state electronic communication between their respective pigment building blocks; (ii) high-stability metal polypyridyl compounds constitute an attractive alternative to electron releasing dialkyl- and diarylamino groups, the most commonly used donor moieties in a wide range of established nonlinear optical dyes; (iii) this design strategy enables ready elaboration of chromophores having extraordinarily large dynamic hyperpolarizabilities (beta(lambda) values) at telecommunication relevant wavelengths; and (iv) porphyrin B- and Q-state-derived static hyperpolarizabilities (beta(0) values) can be designed to have the same or opposite sign in these species, thus providing a new means to regulate the magnitude of lambda(inc)-specific dynamic hyperpolarizabilities.     

Copper catalyzed asymmetric synthesis of chiral allylic esters

The complex derived from Taniaphos ligand 4 and CuBr*Me2S catalyzes the asymmetric addition of Grignard reagents to 3-bromopropenyl esters 1 to provide allylic esters 2 in high yields and high chemio-, regio-, and enantioselectivities. The work demonstrates that allylic asymmetric alkylation (AAA) can be done on substrates bearing a heteroatom at the gamma-position. The method is a practical route to chiral, nonracemic allylic alcohols. The use of functionalized substrates 1 or Grignard reagents leads to more complex products 2, which can be further manipulated as demonstrated in conversion to (S)-5-ethyl-2(5H)-furanone 6 and (S)-benzoic acid-cyclopent-2-enyl ester 7.     

Two-dimensional, pyrazine-based nonlinear optical chromophores with ruthenium(II) ammine electron donors

Six new nonlinear optical (NLO) chromophores with pyrazinyl-pyridinium electron acceptors have been synthesized by complexing a known pro-ligand with electron donating {Ru(II)(NH(3))(5)}(2+) or trans-{Ru(II)(NH(3))(4)(py)}(2+) (py = pyridine) centers. These cationic complexes have been characterized as their PF(6)(-) salts by using various techniques including electronic absorption spectroscopy and cyclic voltammetry. The visible d → π* metal-to-ligand charge-transfer (MLCT) absorptions gain intensity on increasing the number of Ru(II) centers from one to two, but remain at constant energy. One or two Ru(III/II) redox processes are observed which are reversible, quasi-reversible, or irreversible, while all of the ligand-based reductions are irreversible. Molecular first hyperpolarizabilities β have been determined by using hyper-Rayleigh scattering (HRS) at 1064 nm, and depolarization studies show that the NLO responses of the symmetric species are strongly two-dimensional (2D) in character, with dominant "off-diagonal" β(zyy) components. Stark (electroabsorption) spectroscopic measurements on the MLCT bands also allow the indirect determination of estimated static first hyperpolarizabilities β(0). Both the HRS and the Stark-derived β(0) values increase on moving from mono- to bimetallic complexes, and substantial enhancements in NLO response are achieved when compared with one-dimensional (1D) and 2D monometallic Ru(II) ammine complexes reported previously.     

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.