Operator algebras and a theorem of Dieudonne

LetA be aC*-algebra with second dualA″. Let (φ _n)(n=1,...) be a sequence in the dual ofA such that limφ _n(a) exists for eacha εA. In general, this does not imply that limφ _n(x) exists for eachx εA″. But if limφ _n(p) exists whenever p is the range projection of a positive self-adjoint element of the unit ball ofA, then it is shown that limφ _n(x) does exist for eachx inA″. This is a non-commutative generalisation of a celebrated theorem of Dieudonné. A new proof of Dieudonné’s theorem, for positive measures, is given here. The proof of the main result makes use of Dieudonné’s original theorem.     

Fabrication and Characterization of Monolithically Fused Wavelength-Independent 1 × 4 Couplers

A new method for the fabrication of a monolithic 1 × 4 single-mode fused coupler is described together with details of its performance in terms of coupling ratio, spectral response and polarization sensitivity. The fabricated device exhibits ultra-broadband performance with low polarization dependent loss. The coupling pattern exhibits identical coupling to all interacting fibers, enabling an easy control on fabrication parameters of the device.     

Observation and evaluation of the synchronous ligand atom pyramidal inversions in binuclear rhenium carbonyl halide complexes of disulphides and diselenides: an X-ray crystal structure of [Re2Br2(CO)6(C6H5CH2SeSeCH2C6H5)]

The dinuclear complexes [Re₂X₂(CO)₆(RCH₂EECH₂R)] (X = Cl or Br, R = Ph or Me₃Si, E = S or Se) have been prepared and characterized. A variable temperature ¹H NMR study on these complexes demonstrated the pyramidal atomic inversion process at the coordinated sulphur and selenium atoms. Total band-shape fittings were used to yield activation parameters for the rate process, in which the two sulphur or selenium atoms undergo synchronous or correlated inversion.     

Synthetic control of excited-state properties in cyclometalated Ir(III) complexes using ancillary ligands

The synthesis and photophysical characterization of a series of (N,C(2')-(2-para-tolylpyridyl))2 Ir(LL') [(tpy)2 Ir(LL')] (LL' = 2,4-pentanedionato (acac), bis(pyrazolyl)borate ligands and their analogues, diphosphine chelates and tert-butylisocyanide (CN-t-Bu)) are reported. A smaller series of [(dfppy)2 Ir(LL')] (dfppy = N,C(2')-2-(4',6'-difluorophenyl)pyridyl) complexes were also examined along with two previously reported compounds, (ppy)2 Ir(CN)2- and (ppy)2 Ir(NCS)2- (ppy = N,C(2')-2-phenylpyridyl). The (tpy)2 Ir(PPh2CH2)2 BPh2 and [(tpy)2 Ir(CN-t-Bu)2](CF3SO3) complexes have been structurally characterized by X-ray crystallography. The Ir-C(aryl) bond lengths in (tpy)2 Ir(CN-t-Bu)2+ (2.047(5) and 2.072(5) A) and (tpy)2 Ir(PPh2CH2)2 BPh2 (2.047(9) and 2.057(9) A) are longer than their counterparts in (tpy)2 Ir(acac) (1.982(6) and 1.985(7) A). Density functional theory calculations carried out on (ppy)2 Ir(CN-Me)2+ show that the highest occupied molecular orbital (HOMO) consists of a mixture of phenyl-pi and Ir-d orbitals, while the lowest unoccupied molecular orbital is localized primarily on the pyridyl-pi orbitals. Electrochemical analysis of the (tpy)2 Ir(LL') complexes shows that the reduction potentials are largely unaffected by variation in the ancillary ligand, whereas the oxidation potentials vary over a much wider range (as much as 400 mV between two different LL' ligands). Spectroscopic analysis of the cyclometalated Ir complexes reveals that the lowest energy excited state (T1) is a triplet ligand-centered state (3LC) on the cyclometalating ligand admixed with 1MLCT (MLCT = metal-to-ligand charge-transfer) character. The different ancillary ligands alter the 1MLCT state energy mainly by changing the HOMO energy. Destabilization of the 1MLCT state results in less 1MLCT character mixed into the T1 state, which in turn leads to an increase in the emission energy. The increase in emission energy leads to a linear decrease in ln(k(nr)) (k(nr) = nonradiative decay rate). Decreased 1MLCT character in the T1 state also increases the Huang-Rhys factors in the emission spectra, decreases the extinction coefficient of the T1 transition, and consequently decreases the radiative decay rates (k(r)). Overall, the luminescence quantum yields decline with increasing emission energies. A linear dependence of the radiative decay rate (k(r)) or extinction coefficient (epsilon) on (1/deltaE)2 has been demonstrated, where deltaE is the energy difference between the 1MLCT and 3LC transitions. A value of 200 cm(-1) for the spin-orbital coupling matrix element 3LC absolute value(H(SO)) 1MLCT of the (tpy)2 Ir(LL') complexes can be deduced from this linear relationship. The (fppy)2 Ir(LL') complexes with corresponding ancillary ligands display similar trends in excited-state properties.     

Conductimetric and potentiometric titrations of phenolic acids with triethylamine and tetrabutylammonium hydroxide in acetonitrile

Nineteen phenolic acids, namely phenol, 2-nitrophenol, 3-nitrophenol, 4-nitrophenol, 2,4-dinitrophenol, 2,5-dinitrophenol, 2,6-dinitrophenol, 2,4,6-trinitrophenol, 4-chlorophenol, 2,4,6-trichlorophenol, 1,2-dihydroxybenzene, 1,3-dihydroxybenzene, 1,4-dihydroxybenzene, 1,2,3-trihydroxybenzene, 2-aminophenol, 3-aminophenol, 4-aminophenol, 3-cresol and 4-cresol, were titrated conductimetrically and potentiometrically with triethylamine and tetrabutylammonium hydroxide in acetonitrile under a nitrogen atmosphere at 25°C. Most exhibited normal behaviour on titration with triethylamine, but only 2,4-dinitrophenol, 2,6-dinitrophenol and 2,4,6-trinitrophenol gave useful characteristic and stoichiometric conductimetric titration curves. None of the compounds gave a meaningful conductimetric titration curve on titration with tetrabutylammonium hydroxide, which is a much stronger base than triethylamine. With three exceptions, all compounds exhibited stoichiometric well shaped potentiometric titration curves. Potentiometric titration curves of the phenolic acids with tetrabutylammonium hydroxide in acetonitrile are very similar to those of strong acids obtained in aqueous media with strong bases. Acetonitrile is a suitable solvent for titrating some of the weak phenolic acids conductimetrically with triethylamine, and is an excellent medium for titrating them potentiometrically with tetrabutylammonium hydroxide under a nitrogen atmosphere at 25°C.     

Transport characterisation of a PIM system used for the extraction of Pb(II) using d2ehpa as carrier

The facilitated transport of lead(II) through polymeric inclusion membranes consisting of cellulose triacetate as polymeric support, bis-(2-ethylhexyl)-phosphoric acid (d2ehpa) as carrier, and tris-(2-butoxyethyl)phosphate as plasticiser (tbep), is investigated. The influence of some of the aqueous and membrane components on the permeability of Pb(II) was studied. The maximum flux obtained with these membranes is 3.5×10⁻⁶ mol m⁻² s⁻¹, which is of the same order of magnitude of those reported for supported liquid membranes and is in the upper range of those reported for polymeric inclusion membranes. Aqueous and membranes resistances were determined from a model that describes the transport mechanism across the membranes using the stoichiometric relationship and the extraction equilibrium constant value of 6.2×10⁻⁴ determined independently by solid–liquid extraction. An activation energy of 11 kJ mol⁻¹ was also determined for Pb(II) migration, which suggest that the transport of Pb(II) is controlled by a membrane diffusion mechanism. Membrane characterisation was performed using several techniques including atomic force microscopy, scanning electron microscopy coupled with energy-dispersive spectroscopy, and thermal analysis.     

A Robust,Precious‐Metal‐Free Dye‐Sensitized Photoanode for Water Oxidation: A Nanosecond‐Long Excited‐State Lifetime through a Prussian Blue Analogue

Herein, we establish a simple synthetic strategy affording a heterogeneous, precious metal‐free, dye‐sensitized photoelectrode for water oxidation, which incorporates a Prussian blue (PB) structure for the sensitization of TiO₂ and water oxidation catalysis. Our approach involves the use of a Fe(CN)₅ bridging group not only as a cyanide precursor for the formation of a PB‐type structure but also as an electron shuttle between an organic chromophore and the catalytic center. The resulting hetero‐functional PB‐modified TiO₂ electrode demonstrates a low‐cost and easy‐to‐construct photoanode, which exhibits favorable electron transfers with a remarkable excited state lifetime on the order of nanoseconds and an extended light absorption capacity of up to 500 nm. Our approach paves the way for a new family of precious metal‐free robust dye‐sensitized photoelectrodes for water oxidation, in which a variety of common organic chromophores can be employed in conjunction with CoFe PB structures.     

NUMERICAL METHOD FOR SOLVING LINEAR BOUNDARY VALUE PROBLEMS BY THE CHEBYSHEV τ-METHOD

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