Numerical methods for finding multiple eigenvalues of matrices depending on parameters

Summary. Let be a square matrix dependent on parameters and , of which we choose as the eigenvalue parameter. Many computational problems are equivalent to finding a point such that has a multiple eigenvalue at . An incomplete decomposition of a matrix dependent on several parameters is proposed. Based on the developed theory two new algorithms are presented for computing multiple eigenvalues of with geometric multiplicity . A third algorithm is designed for the computation of multiple eigenvalues with geometric multiplicity but which also appears to have local quadratic convergence to semi-simple eigenvalues. Convergence analyses of these methods are given. Several numerical examples are presented which illustrate the behaviour and applications of our methods. Received December 19, 1994 / Revised version received January 18, 1996     

Doppler cooling a single Ca+ ion with a violet extended-cavity diode laser

We present a scheme for employing a violet extended-cavity diode laser in experiments with single, trapped ions. For this the grating-stabilised laser is spatially and spectrally filtered and referenced to a Fabry–Pérot cavity. We measure an upper limit to the line width by observing a 305-kHz FWHM beat note with the second harmonic of a titanium sapphire laser. The laser is subsequently used to optically cool a single ⁴⁰Ca⁺ ion close to the Doppler limit. PACS 03.67.Lx; 32.80.Pj; 42.55.Px     

Hybrid catalysts: the synthesis, structure and ethene polymerisation activity of (salicylaldiminato)(pyrrolylaldiminato) titanium complexes

The mono(salicylaldiminato) complexes Ti{3-tBu-2-(O)C6H3CH=N(R)}Cl3(THF)(where R = C6H5, C6F5) react with the metallated pyrrolylaldiminato ligand, K[2-(C6H5NCH)C4H3N], to afford the first examples of hybrid salicylaldiminato-ligated octahedral titanium complexes; the pre-catalysts give from very high to extremely high ethene polymerisation productivities when activated with MAO.     

Procedures for the enhancement of selectivity in liquid phase chemiluminescence detection

An overview of liquid phase chemiluminescence (CL) processes is presented and the potential for CL detection in liquid chromatography (LC) is discussed, with particular reference to the luminol and peroxyoxalate reactions. Post column ion displacement from a solid phase reagent [a cation exchange resin in the copper(II) form] followed by catalysis of the luminol reaction is used for the quantification of mixtures of weak acids after separation by ion-exclusion chromatography. Polyaromatic hydrocarbons (PAHs) released during the combustion of biomass fuels are separated by reversed-phase chromatography and quantified by their sensitizing effect on the peroxyoxalate reaction. This procedure is also used for the determination of carboxylic acids in non-aqueous media after selective pre-column derivatization with a fluorescent label (9-anthracenemethanol).     

Ca(2.5)Sr(0.5)GaMn2O8: diamagnetic Ga in control of the structural and electronic properties of a bilayered manganate

The temperature dependence of the crystal structure and electronic properties of brownmillerite-like Ca(2.5)Sr(0.5)GaMn(2)O(8) has been studied by neutron powder diffraction and muSR spectroscopy. The results show that short-range 2D magnetic order begins to develop within the perovskite-like bilayers of MnO(6) octahedra approximately 50 K above the 3D Néel temperature of approximately 150 K. The bilayers show a structural response to the onset of magnetism throughout this temperature range whereas the GaO(4) layers that separate the bilayers only respond below the 3D ordering temperature. XANES spectroscopy shows that the sample contains Mn(3+) and Mn(4+) cations in a 1:1 ratio, and the behavior in the region of the Néel transition is interpreted as a local charge ordering. Electron diffraction and high-resolution electron microscopy have been used to show that the local microstructure is more complex than the average structure revealed by neutron diffraction, and that microdomains exist in which the GaO(4) tetrahedra show different orientations. It is argued that the bonding requirements of diamagnetic gallium control the electronic behavior within the perovskite-like bilayers.     

The synthesis, structure and ethene polymerisation catalysis of mono(salicylaldiminato) titanium and zirconium complexes

The silyl ethers 3-But-2-(OSiMe3)C6H3CH=NR (2a-e) have been prepared by deprotonation of the known iminophenols (1a-e) and treatment with SiClMe3 (a, R = C6H5; b, R = 2,6-Pri2C6H3; c, R = 2,4,6-Me3C6H2; d, R = 2-C6H5C6H4; e, R = C6F5). 2a-c react with TiCl4 in hydrocarbon solvents to give the binuclear complexes [Ti{3-But-2-(O)C6H3CH=N(R)}Cl(mu-Cl3)TiCl3] (3a-c). The pentafluorophenyl species 2e reacts with TiCl4 to give the known complex Ti{3-But-2-(O)C6H3CH=N(R)}2Cl2. The mononuclear five-coordinate complex, Ti{3-But-2-(O)C6H3CH=N(2,4,6-Me3C6H2)}Cl3 (4c), was isolated after repeated recrystallisation of 3c. Performing the dehalosilylation reaction in the presence of tetrahydrofuran yields the octahedral, mononuclear complexes Ti{3-But-2-(O)C6H3CH=N(R)}Cl3(THF) (5a-e). The reaction with ZrCl4(THF)2 proceeds similarly to give complexes Zr{3-But-2-(O)C6H3CH=N(R)}Cl3(THF) (6b-e). The crystal structures of 3b, 4c, 5a, 5c, 5e, 6b, 6d, 6e and the salicylaldehyde titanium complex Ti{3-But-2-(O)C6H3CH=O}Cl3(THF) (7) have been determined. Activation of complexes 5a-e and 6b-e with MAO in an ethene saturated toluene solution gives polyethylene with at best high activity depending on the imine substituent.     

Magnetism in geometrically frustrated YMnO3 under hydrostatic pressure studied with muon spin relaxation

The ferroelectromagnet YMnO3 consists of weakly coupled triangular layers of S=2 spins. Below T(N) approximately equal to 70 K muon-spin relaxation data show two oscillatory relaxing signals due to magnetic order, with no purely relaxing signals resolvable (which would require different coexisting spin distributions). The transition temperature T(N) increases with applied hydrostatic pressure, even though the ordered moment decreases. These results suggest that pressure increases both the exchange coupling between the layers and the frustration within the layers.     

Dramatic Influence of an Anionic Donor on the Oxygen‐Atom Transfer Reactivity of a MnV–Oxo Complex

Addition of an anionic donor to an Mn^V(O) porphyrinoid complex causes a dramatic increase in 2‐electron oxygen‐atom‐transfer (OAT) chemistry. The 6‐coordinate [Mn^V(O)(TBP₈Cz)(CN)]^? was generated from addition of Bu₄N⁺CN^? to the 5‐coordinate Mn^V(O) precursor. The cyanide‐ligated complex was characterized for the first time by Mn K‐edge X‐ray absorption spectroscopy (XAS) and gives Mn?O=1.53 Å, Mn?CN=2.21 Å. In combination with computational studies these distances were shown to correlate with a singlet ground state. Reaction of the CN^? complex with thioethers results in OAT to give the corresponding sulfoxide and a 2e^?‐reduced Mn^III(CN)^? complex. Kinetic measurements reveal a dramatic rate enhancement for OAT of approximately 24 000‐fold versus the same reaction for the parent 5‐coordinate complex. An Eyring analysis gives ΔH^≠=14 kcal mol^?1, ΔS^≠=?10 cal mol^?1 K^?1. Computational studies fully support the structures, spin states, and relative reactivity of the 5‐ and 6‐coordinate Mn^V(O) complexes.     

Novel diode laser-based sensors for gas sensing applications

The development of compact spectroscopic gas sensors and their applications to environmental sensing will be described. These sensors employ mid-infrared difference-frequency generation (DFG) in periodically poled lithium niobate (PPLN) crystals pumped by two single-frequency solid state lasers such as diode lasers, diode-pumped solid state, and fiber lasers. Ultrasensitive, highly selective, and real-time measurements of several important atmospheric trace gases, including carbon monoxide, nitrous oxide, carbon dioxide, formaldehyde [correction of formaldehye], and methane, have been demonstrated.