A geometrical approach to bifurcation for nonlinear boundary value problems

Summary In this note we use a new averaging method, which was introduced in [2], to explain the geometrical behaviour of systems governed by nonlinear boundary value problems of the formy+g(y)=K sin(t),y(0)=y(/)=0. We show by numerical computations that global features of the solutions (such as the number of solutions, their magnitude, bifurcation behaviour, etc.) agree in both the original and averaged model. As an example, the pendulum equation is discussed in detail.

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Zusammenfassung In dieser Arbeit benutzen wir eine neue, in [2] eingeführte Mittelwertmethode, um das geometrische Verhalten nichtlinearer Randwertprobleme der Formy+g(y)=K sin(t),y(0)=y(/)=0. zu erklären. Wir belegen durch numerische Untersuchungen, daß globale Eigenschaften der Lösungen (wie z. B. die Anzahl der Lösungen, ihre Größenordnung, das Verzweigungsverhalten usw.) in der originalen und genäherten Gleichung übereinstimmen. Als Beispiel wird die Pendelgleichung ausführlich diskutiert.

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Supported by the Deutsche Forschungsgemeinschaft under grant No. BA 735/3-1     

Two-photon four-level hologram recording in poly-(alkyl-α-cyanoacrylates)

We have investigated the suitability of poly-(alkyl--cyanoacrylates) (p-(alkyl-CAc)) for two-photon four-level holographic recording. It is shown that these materials follow indeed a two-photon four-level mechanism with the second absorption step in the near infrared. Light induces the polymerization of residual monomer in the matrix. The reaction is associated with positive density changes. The sensitivity ofp-(alkyl-CAc) depends strongly on composition and age of the sample. Maximum sensitivities achieved are comparable to the so far best-known two-photon four-level ir-sensitive systems.     

Two-photon absorption in quadrupolar pi-conjugated molecules: influence of the nature of the conjugated bridge and the donor-acceptor separation

Quadrupolar-type substitution of pi-conjugated chromophores with donor and acceptor groups has been shown to increase their two-photon absorption (TPA) response by up to two orders of magnitude. Here, we apply highly correlated quantum-chemical calculations to evaluate the impact of the nature of conjugated bridge and the charge-transfer distance on that enhancement. We compare chromophores with phenylenevinylene-, thienylenevinylene-, polyene-, and indenofluorene-type backbones substituted by dimethylamino and cyano groups. In all compounds, we find a strongly TPA-active A(g) state (either 2A(g) or 3A(g)) in the low-energy region, as well as a higher lying TPA-active state (mA(g)) at close to twice the energy of the lowest lying one-photon allowed state; the smaller energy detuning in the mA(g) states results in very large TPA cross sections delta. We also investigate the influence of the degree of ground-state polarization on TPA. Independent of the nature of the backbone and the donor-acceptor separation, delta displays the same qualitative evolution with a maximum before the cyanine-like limit; the highest TPA cross sections are calculated for distirylbenzene- and polyene-based systems.     

Origin of high second- and third-order nonlinear optical response in ammonio/borato diphenylpolyene zwitterions: the remarkable role of polarized aromatic groups

We present a quantum-chemical analysis of the molecular structure and second- and third-order polarizabilities in a series of promising nonlinear optical (NLO) chromophores, the zwitterionic ammonio/borato diphenylpolyenes, R3N+Ph(C=C)nPhB-R3, whose synthesis has been reported recently. The molecular geometries are obtained via MP2/6-31G optimization, while the NLO properties are calculated with the INDO Hamiltonian using the sum-over-states and finite-field real-space methods. The real-space approach allows the direct evaluation of the NLO-active segments of the molecules, while the sum-over-states results illustrate the virtual excitations and charge-transfer pathways that are essential in the NLO response. Both methods highlight the remarkable and unexpected result that it is the strongly polarized phenylene groups that play the key role in generating a high NLO response.     

X-ray structure and variable temperature NMR spectra of [meso-triarylcorrolato]copper(III)

The diamagnetic square planar d(8) complexes [meso-arylcorrolato]copper(III) become paramagnetic upon warming, indicative of the equilibrium between the [corrolato]copper(III) and the [corrolato](+)* copper(II) forms of the complex. [meso-Triphenylcorrolato]copper(III) was structurally characterized and found to be saddled.     

Separation and detection of oxidation products of fluorodeoxyglucose and glucose by high-performance liquid chromatography-electrospray ionisation mass spectrometry

2-Deoxy-2-[18F]fluoro-D-glucose ([18F] FDG), the most popular positron emitting radiopharmaceutical, may oxidise by autoradiolysis in aqueous solution. The aim of this work was to use LC-MS for determination of the oxidation products of fluorodeoxyglucose and glucose (Glc) obtained by oxidation with Fenton's reagent. Asahipak NH2P-50 polyamide silica column and acetonitrile-0.025% aqueous ammonium formate (80:20 (v/v)) eluent were utilised with an Agilent 1100 HPLC-MS instrument. Ten major oxidation products of FDG and Glc were separated and identified by mass spectrometry: 2-fluorogluconic acid, 2-fluoroglucuronic acid, 2-oxoerythronic acid, arabinose, arabonic acid, araburonic acid, erythrose, erythrulose, gluconic acid, and glucuronic acid. The most intensive electrospray ionisation signals were found in the negative ion spectra and were due to HCOO- adducts, the other acids being in their lactone forms.     

Hg2(OH)[BF4], the First Basic Mercurous Tetrafluoroborate

The first basic mercurous tetrafluoroborate, Hg₂(OH)[BF₄], was obtained through a synproportionation reaction of red mercuric oxide, HgO, and elemental mercury in a 35% solution of HBF₄ in water as colourless single crystals. The crystal structure (orthorhombic, Pbca, Z = 8, a = 985.0(1), b = 991.0(1), c = 1141.0(2) pm) contains [(HO)_1/2—Hg—Hg—(OH)_1/2]⁺ zig‐zag chains that are further connected via weak Hg—OH interchain interactions to layers between which the [BF₄]^— are located in an up and down fashion.     

GnRH binding RNA and DNA Spiegelmers: a novel approach toward GnRH antagonism

Mirror-image oligonucleotide ligands (Spiegelmers) that bind to the pharmacologically relevant target gonadotropin-releasing hormone I (GnRH) with high affinity and high specificity have been identified using the Spiegelmer technology. GnRH is a decapeptide that plays an important role in mammalian reproduction and sexual maturation and is associated with several benign and malignant diseases. First, aptamers that bind to D-GnRH with dissociation constants of 50-100 nM were isolated out of RNA and DNA libraries. The respective enantiomers of the DNA and RNA aptamers were synthesized, and their binding to L-GnRH was shown. These Spiegelmers bind to L-GnRH with similar affinity to that of the corresponding aptamers that bind to D-GnRH. We further demonstrated dose-dependent inhibition of GnRH-induced Ca(2+) release in Chinese hamster ovary cells that were stably transfected with the human GnRH receptor.     

The synthesis and the molecular structure of hexakis(carbonyl)hexafluoroantimonato(V)tungsten(II) undecafluorodiantimonate(V), [W(CO)6(FSbF5)][Sb2F11

The reaction of tungsten hexacarbonyl, W(CO)6, with antimony(V) fluoride, SbF5, in the conjugate Br?nsted-Lewis superacid HF-SbF5 at 40 degrees C produces quantitatively the salt [W(CO)6(FSbF5)][Sb2F11] as the main product. The observed 2e- oxidation without any loss of CO is unprecedented. The cation [W(CO)6(FSbF5)]+ is seven coordinated with a distorted C2v capped trigonal prismatic structure. [W(CO)6(FSbF5)][Sb2F11] crystallizes in the monoclinic space group P21 (No. 4). a = 8.2051(12) A, b = 16.511(3) A, c = 8.1432(2) A, beta = 111.5967(6) degrees, V = 1025.8(2) A3, Z = 2. Number of reflections measured = 9112, unique 4410. Residuals on F, I > 3 sigma (I): R (Rw) = 0.023 (0.023). In the [W(CO)6(FSbF5)]+ cation the FSbF5 group is very tightly coordinated to tungsten with the bridging fluorine nearly equidistant from W and Sb. The details of the molecular structure are compared to those to polymeric [[Mo(CO)4]2(cis-mu-F2SbF4)3]x[Sb2F11]x reported by us very recently.     

Filter Diagonalization with Finite Fourier Transform Eigenfunctions

Below, we briefly report on the progress in the development of the Filter Diagonalization technique when filtering is carried out with the aid of Finite Fourier Transform (FFT) eigenfunctions. During recent years interest in these functions, also known as ‘prolates’, or ‘slepians’, has increased among scientists doing research in the field of signal processing. The main explanation to this follows from the set of very special extremal and orthogonality properties exibited by the FFT eigenfunctions. Recent results of Walter and Shen on sampling with prolate spheroidal functions will necessary produce a new wave of interest. In the presented, Filter diagonalization machinery, we show that the sampling formula of Walter and Shen simplifies essentially the computation of matrix elements as certain 2D–integrals involving FFT eigenfunctions.